# Limits of Hodge Structures via holonomic D-modules

###### Abstract.

We construct the limiting mixed Hodge structure of a degeneration of compact
Kähler manifolds over the unit disk with a possibly non-reduced normal
crossing singular central fiber via holonomic ${\mathscr{D}}$-modules, which
generalizes some results of Steenbrink. Our limiting mixed Hodge structure
does not carry a ${\mathbb{Q}}$-structure; instead we use sesquilinear
pairings on ${\mathscr{D}}$-modules as a replacement. The associated graded
quotient of the weight filtration of the limiting mixed Hodge structure can be
computed by the cohomology of the cyclic coverings of certain intersections of
components of the central fiber.

Qianyu Chen

##  1\. Introduction

###  1.1. Limits of Hodge structures

Consider a degeneration of compact Kähler manifolds $f:X\rightarrow\Delta$
over the unit disk $\Delta$. The cohomology of each smooth fiber carries a
polarizable Hodge structure. It is natural to ask how the family of Hodge
structures on the cohomologies of smooth fibers degenerate and how the
cohomology of the central fiber relates to that of nearby fibers. These are
two classical and central questions in Hodge theory. Before Saito’s theory of
mixed Hodge modules [Sai88, Sai90], Schmid showed the existence of a limiting
mixed Hodge structure for an abstract polarized variation of Hodge structures
over the unit disk [Sch73] using Lie theoretic methods, and later Cattani,
Kapplan and Schmid extend this to polydisks [CKS86]. For the variation of
Hodge structures coming from a semistable family of Kähler manifolds over a
$1$-dimensional base, the limiting mixed Hodge structure was first established
by Steenbrink [Ste76] whose construction is equivalent to Schmid’s in [Sch73]
but purely geometric:

######  Theorem (Steenbrink).

Let $f:X\rightarrow\Delta$ be a proper holomorphic morphism which is smooth
away from the origin, whose central fiber $Y$ is reduced simple normal
crossing. Suppose $X$ is Kähler of dimension $n+1$. Then the hypercohomology
$H^{k}(X,\Omega^{{}\bullet+n}_{X/\Delta}(\log Y)|_{Y})$ of the relative log de
Rham complex restricted on $Y$ admits a limiting mixed Hodge structure with a
${\mathbb{Q}}$-structure whose graded quotient of the weight filtration can be
expressed in terms of the cohomology of certain intersections of components of
$Y$ via spectral sequences.

Let us briefly explain Steenbrink’s result. Suppose we are in the setting of
the theorem but $Y$ is possibly non-reduced. Denote by $X^{*}=X\setminus Y$
and $\Delta^{*}=\Delta\setminus\\{0\\}$. Then the higher direct image of the
relative de Rham complex $R^{k}f_{*}\Omega^{\bullet+n}_{X^{*}/\Delta^{*}}$ is
a vector bundle, where the shifting is needed to adopt the convention of the
theory of perverse sheaves and ${\mathscr{D}}$-modules; it underlies a
polarizable variation of Hodge structure of weight $n$ over the punctured disk
$\Delta^{*}$. Recall that a polarized variation of Hodge structure of weight
$n$ over a complex manifold $Z$ is an integrable connection
$({\mathcal{V}},\nabla)$ together with a so-called Hodge filtration by
subundles $F^{\bullet}{\mathcal{V}}$ and a flat Hermitian pairing
$S:{\mathcal{V}}\otimes_{\mathbb{C}}\overline{\mathcal{V}}\rightarrow\mathscr{C}^{\infty}_{Z}$
satisfying $(1)$ Griffith transversality $\nabla
F^{\bullet}{\mathcal{V}}=\Omega_{Z}\otimes F^{\bullet-1}{\mathcal{V}}$, and
$(2)$ each fiber of $({\mathcal{V}},F^{\bullet}{\mathcal{V}},S)$ is a
polarized Hodge structure of weight $n$. However, the higher direct image of
the relative de Rham complex $\Omega^{\bullet+n}_{X/\Delta}$ does not give
anything interesting when $Y$ is singular. Steenbrink discovered a natural
extension of the vector bundle
$R^{k}f_{*}\Omega^{\bullet+n}_{X^{*}/\Delta^{*}}$ over the origin via the
relative log de Rham complex. Let

$\Omega_{X/\Delta}(\log Y)=\Omega_{X}(\log Y)/f^{*}\Omega_{\Delta}(\log
0)\text{ \,and\, }\Omega^{p}_{X/\Delta}(\log
Y)=\bigwedge^{p}\Omega_{X/\Delta}(\log Y),$

where $\Omega_{X}(\log Y)$ is the sheaf of meromorphic one-forms with log
poles along $Y$. Then the relative log de Rham complex is defined to be

$\Omega^{\bullet+n}_{X/\Delta}(\log
Y)=\\{{\mathscr{O}}_{X}\rightarrow\Omega_{X/\Delta}(\log
Y)\rightarrow\cdots\rightarrow\Omega^{n}_{X/\Delta}(\log Y)\\}[n].$

Steenbrink showed in [Ste76] that $R^{k}f_{*}\Omega^{\bullet}_{X/\Delta}(\log
Y)$ is a locally free integrable logarithmic connection with a pole along the
origin whose residue $R$ has eigenvalues in $[0,1)\cap{\mathbb{Q}}$ for each
$k\in{\mathbb{Z}}$. It follows from Grauert’s theorem that there exists a
canonical isomorphism

$R^{k}f_{*}\Omega^{\bullet+n}_{X/\Delta}(\log Y)\otimes{\mathbb{C}}(p)\simeq
H^{k}(Y,\Omega^{\bullet+n}_{X/\Delta}(\log Y)|_{X_{p}})$

for every fiber $X_{p}$ over any point $p\in\Delta$, where ${\mathbb{C}}(p)$
denotes the residue field of $p$. The vector bundle
$R^{k}f_{*}\Omega^{\bullet}_{X/\Delta}(\log Y)$ is Deligne’s canonical
extension [Del70] of $R^{k}f_{*}\Omega^{\bullet+n}_{X^{*}/\Delta^{*}}$ with
eigenvalues of the residues of the log connection in the interval in $[0,1)$.
Now we can think of the space $H^{k}(Y,\Omega^{\bullet+n}_{X/\Delta}(\log
Y)|_{Y})$ as a canonical specialization of
$H^{k}(X_{p},\Omega^{\bullet+n}_{X_{p}})$ for general fibers $X_{p}$. In fact,
the limiting Hodge filtration is induced by the stupid filtration defined by,

$F^{-\ell}\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)=\\{\Omega^{{}-\ell}_{X/\Delta}(\log
Y)\rightarrow\cdots\rightarrow\Omega^{{}n}_{X/\Delta}(\log Y)\\}[n+\ell],$

for each $\ell\in{\mathbb{Z}}$. This extends the Hodge filtration
$F^{\bullet}R^{k}f_{*}\Omega^{\bullet+n}_{X^{*}/\Delta^{*}}$ for the variation
of Hodge structure $R^{k}f_{*}\Omega^{\bullet+n}_{X^{*}/\Delta^{*}}$ which is
also induced by the stupid filtration on the complex
$\Omega^{\bullet+n}_{X^{*}/\Delta^{*}}$. When $Y$ is reduced, the residue $R$
is nilpotent on the hypercohomology of $\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)|_{Y}$ for every $k$ so it gives a monodromy filtration
$W_{\bullet}=W_{\bullet}(R)$ on $H^{k}(X,\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)|_{Y})$ uniquely characterized by two properies: (1) $RW_{\bullet}\subset
W_{\bullet-2}$ and (2)
$R^{r}:{\mathrm{g}}{\mathrm{r}}^{W}_{r}\rightarrow{\mathrm{g}}{\mathrm{r}}^{W}_{-r}$
is an isomorphism for every $r\geq 0$. The filtration $W_{\bullet}(R)$ is
called the monodromy filtration because $\exp\left(-2\pi\sqrt{-1}R\right)$ is
the monodromy induced by the generator of the fundamental group of
$\Delta^{*}$. Steenbrink showed that the monodromy filtration is the weight
filtration of the limiting mixed Hodge structure when $f$ is projective, and
this was later generalized to the Kähler case by Guillén and Navarro Aznar in
[GNA90].

Steenbrink later pointed out the limiting mixed Hodge structure he constructed
only depends on the log structure associated to the semistable family
$f:X\rightarrow\Delta$ [Ste95]. Inspired by the idea in [Ste95], Fujisawa
extended Steenbrink’s results in [Ste76, Ste95] to semistable Kähler families
over the polydisk and furthermore to the log geometry setting [Fuj99, Fuj08,
Fuj14]. Recently, Nakkajima announced a simpler proof of Fujisawa’s results
[Nak21].

###  1.2. Main results

We revisit Steenbrink’s theorem and construct the limiting mixed Hodge
structure of the degeneration over the unit disk $\Delta$ with a simple normal
crossing central fiber $Y$ which is possibly non-reduced via the theory of
holonomic ${\mathscr{D}}$-modules. Although we can run Mumford’s semistable
reduction [KKMSD73], which is a sequence of base changes, normalizations and
blow-ups, on every degeneration of Kähler manifolds over the unit disk to
obtain a semistable degeneration, it is still interesting to remove the
assumption that $Y$ is reduced in Steenbrink’s theorem since the semistable
reduction may not be canonical. When $Y$ is non-reduced, the residue is no
longer nilpotent; instead, we need to consider the Jordan-Chevallay
decomposition of $R$. Here is our main theorem:

######  Theorem A.

Let $f:X\rightarrow\Delta$ be a proper holomorphic morphism which smooth away
from the origin, whose central fiber $Y$ is possibly non-reduced simple normal
crossing. Assume that $X$ is Kähler of dimension $n+1$. Let $R_{n}$ (resp.
$R_{s}$) denote the nilpotent (resp. semisimple) part of the Jordan-Chevalley
decomposition of the residue operator $R$ on
$\bigoplus_{k}H^{k}(X,\Omega^{{}\bullet+n}_{X/\Delta}(\log Y)|_{Y})$. Then
each eigenspace of $R_{s}$ on

$\bigoplus_{k,\ell}{\mathrm{g}}{\mathrm{r}}^{W}_{\ell}H^{k}(X,\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)|_{Y})$

underlies a limiting polarized bigraded Hodge-Lefschetz structure over
${\mathbb{C}}$ of central weight $n$, where $W_{\bullet}=W_{\bullet}(R_{n})$
is the monodromy filtration associated to $R_{n}$.

A polarized bigraded Hodge-Lefschetz structure is essentially a direct sum of
polarized Hodge structures of different weights preserving by an
${\mathfrak{s}}{\mathfrak{l}}_{2}({\mathbb{C}})\times{\mathfrak{s}}{\mathfrak{l}}_{2}({\mathbb{C}})$-action.
In the setting of Theorem A, the
${\mathfrak{s}}{\mathfrak{l}}_{2}({\mathbb{C}})\times{\mathfrak{s}}{\mathfrak{l}}_{2}({\mathbb{C}})$-action
will be induced by the operator $R_{n}$ and $2\pi\sqrt{-1}L$ where
$L=\omega\wedge$ is the Lefschetz operator for a Kähler form $\omega$. In
particular, each component
${\mathrm{g}}{\mathrm{r}}^{W}_{\ell}H^{k}(X,\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)|_{Y})$ is a Hodge structure of weight $n+k+\ell$ and there are two Hard
Lefschetz type isomorphisms of Hodge structures:

  * •

$\left(2\pi\sqrt{-1}L\right)^{k}:{\mathrm{g}}{\mathrm{r}}^{W}_{\ell}H^{-k}(X,\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)|_{Y})\rightarrow{\mathrm{g}}{\mathrm{r}}^{W}_{\ell}H^{k}(X,\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)|_{Y})(k)$ for $k\geq 0,\ell\in{\mathbb{Z}}$;

  * •

$R^{\ell}_{n}:{\mathrm{g}}{\mathrm{r}}^{W}_{\ell}H^{k}(X,\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)|_{Y})\rightarrow{\mathrm{g}}{\mathrm{r}}^{W}_{-\ell}H^{k}(X,\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)|_{Y})(-\ell)$ for $\ell\geq 0,k\in{\mathbb{Z}}$.

Theorem A implies that each $H^{k}(X,\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)|_{Y})$ still underlies a limiting mixed Hodge structure of weight $n+k$
whose weight filtration is given by $W_{\bullet}=W_{\bullet}(R_{n})$ when the
central fiber is non-reduced. We refer to $\mathsection$2.4 for the definition
of polarized bigraded Hodge-Lefschetz structures. Our argument also says that
the limiting mixed Hodge structure can be computed in terms of the cohomology
of certain cyclic coverings of intersections of components of $Y$ via spectral
sequences.

Steenbrink proved, loosely speaking, that
$\Omega^{{}\bullet+n}_{X/\Delta}(\log Y)|_{Y}$ is isomorphic to
$\psi_{f}\left({\mathbb{C}}_{X}[n+1]\right)$ in the derived category of
complex vector spaces ${\mathbf{D}}^{b}(X,{\mathbb{C}})$ where $\psi_{f}$
denotes the nearby cycles functor, so that the function $p\mapsto\dim
H^{k}(X,\Omega^{{}\bullet+n}_{X/\Delta}(\log Y)|_{X_{p}})$ is constant on
$\Delta$. Thanks to Grauert’s theorem, the sheaf
$R^{k}f_{*}\Omega^{\bullet+n}_{X/\Delta}(\log Y)$ is locally free. The log
connection on $R^{k}f_{*}\Omega^{{}\bullet+n}_{X/\Delta}(\log Y)$ is the
higher direct image of an operator
$\nabla\in{\mathrm{E}}{\mathrm{n}}{\mathrm{d}\,}_{{\mathbf{D}}^{b}(X,{\mathbb{C}})}\left(\Omega^{\bullet+n}_{X/\Delta}(\log
Y)\right)$, which fits in a distinguished triangle in
${\mathbf{D}}^{b}(X,{\mathbb{C}})$

${f^{*}\Omega_{\Delta}\otimes\Omega^{{}\bullet+n-1}_{X/\Delta}(\log
Y)}$${\Omega^{\bullet+n}_{X}(\log Y)}$${\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)}$${f^{*}\Omega_{\Delta}\otimes\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)}$$\scriptstyle{\nabla}$

if we trivialize $f^{*}\Omega_{\Delta}$. The induced operator
$[\nabla]:\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)|_{Y}\rightarrow\Omega^{{}\bullet+n}_{X/\Delta}(\log Y)|_{Y}$ has a
characteristic polynomial whose roots are in $[0,1)\cap{\mathbb{Q}}$. The
action of $[\nabla]$ on the hypercohomology of
$\Omega^{{}\bullet+n}_{X/\Delta}(\log Y)|_{Y}$ is identical to the residue
operator $R$ of the log connection. So the methods of studying the monodromy
filtration of $R$ on the cohomology is to make the monodromy filtration of
$[\nabla]$ on the complex $\Omega^{{}\bullet+n}_{X/\Delta}(\log Y)|_{Y}$
explicit. One of the main difficulties that we encounter is the construction
of the rational monodromy filtration on the complex
$\Omega^{{}\bullet+n}_{X/\Delta}(\log Y)|_{Y}$ because the operator $[\nabla]$
only lives in the derived category. Steenbrink resolves the relative log de
Rham complex using a certain double complex and then he works out the
monodromy filtration directly in the case that $Y$ is reduced. He also needs
to show that the monodromy filtrations are defined over ${\mathbb{Q}}$, using
some complicated topological argument, so that all the data gives a rational
cohomological mixed Hodge complex.

Through the Riemann-Hilbert correspondence [Kas84, Meb84], there should be a
regular holonomic ${\mathscr{D}}$-module whose de Rham complex is isomorphic
to $\Omega^{{}\bullet+n}_{X/\Delta}(\log Y)|_{Y}$ in
${\mathbf{D}}^{b}(X,{\mathbb{C}})$ since the nearby cycle functor preserves
perversity [Beĭ87]. On the ${\mathscr{D}}$-module side, we can derive the
monodromy filtration easily by local calculations on a single
${\mathscr{D}}$-module which bypasses the derived categories. More
importantly, we give a concrete description of the primitive parts of the
associated quotient of the monodromy filtrations. Instead of using
${\mathbb{Q}}$-structures, we consider sesuqilinear pairings on
${\mathscr{D}}$-modules, which play the role of a polarization on a Hodge
structure. In fact, the polarization on the bigraded Hodge-Lefschetz structure
in Theorem A will be induced by a sesquilinear pairing. Although part of the
topological data is lost, the sesquilinear pairings that we shall use can be
constructed pure algebraically and only involve symbolic calculations. The
local calculation and the sesquilinear pairing justify the fact that the
monodromy filtration of $[\nabla]$ is the correct choice for the weight
filtration. Our method also allows us to construct naturally the limit when
$Y$ is non-reduced.

As an application, we establish the local invariant theorem, which is a piece
in the Clemens-Schmid sequence [Cle77], when $Y$ is non-reduced. The local
invariant cycle theorem first was proved by Deligne in an algebraic setting
when the base is a scheme [Del71, Theorem 4.1.1] and later treated in[Ste76],
[Cle77] and [GNA90] for a semistable Kähler degeneration. It also generalized
to mixed Hodge module theory by Saito [Sai88, Sai90].

######  Theorem B (local invariant cycle theorem).

Suppose we are in the same setting as in Theorem A. Then the following
sequence of mixed Hodge structures is exact:

${H^{\ell}(Y,{\mathbb{C}})}$${H^{\ell}(X,\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)|_{Y})}$${H^{\ell}(X,\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)|_{Y})(-1).}$$\scriptstyle{R}$

In other words, all cohomology classes invariant under the monodromy action
comes from the cohomologies of $Y$.

###  1.3. Strategy of the construction

Let $f:X\rightarrow\Delta$ be a proper holomorphic morphism smooth away from
the origin such that the central fiber $Y$ is simple normal crossing but not
necessarily reduced. Assume that $X$ is Kähler of dimension $n+1$ and
$Y=\sum_{i\in I}e_{i}Y_{i}$ where the $Y_{i}$’s are smooth components and $I$
a finite index set. We adopt the convention that if $F^{\bullet}$ denotes a
decreasing filtration then $F_{-\bullet}=F^{\bullet}$ denotes the
corresponding increasing filtration and vice versa.

We first give a different proof of the local freeness of
$R^{k}f_{*}\Omega^{\bullet+n}_{X/\Delta}(\log Y)$ which only uses the fact
that $[\nabla]$ has eigenvalues in $[0,1)$ (Theorem 3.2). Then we translate
the data of the relative log de Rham complex to the ${\mathscr{D}}$-module
side (see $\S$4):

######  Theorem C.

There exists a filtered holonomic ${\mathscr{D}}_{X}$-module
$({\mathcal{M}},F_{\bullet}{\mathcal{M}})$ whose de Rham complex
${\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}}$ with the induced filtration
$F_{\bullet}{\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}}$ is isomorphic to
$\Omega^{{}\bullet+n}_{X/\Delta}(\log Y)|_{Y}$ with the stupid filtration in
the derived category of filtered complex of ${\mathbb{C}}$-vector spaces.
Moreover, there exists an operator
$R:({\mathcal{M}},F_{\bullet}{\mathcal{M}})\rightarrow({\mathcal{M}},F_{\bullet+1}{\mathcal{M}})$
whose eigenvalues are in $[0,1)\cap{\mathbb{Q}}$ such that
${\mathrm{D}}{\mathrm{R}}_{X}R$ can be identified with $[\nabla]$ via the
above isomorphism.

Then we investigate the Jordan block of the operator $R$. Let
${\mathcal{M}}_{\geq\alpha}$ (resp. ${\mathcal{M}}_{>\alpha}$) be the
submodule of ${\mathcal{M}}$ spaned by the generalized eigen-modules
$\ker(R-\lambda)^{\infty}$ for $\lambda\geq\alpha$ (resp. $\lambda>\alpha$).
Let
${\mathcal{M}}_{\alpha}={\mathcal{M}}_{\geq\alpha}/{\mathcal{M}}_{>\alpha}$.
Note that ${\mathcal{M}}_{\alpha}$ is canonically isomorphic to
$\ker(R-\alpha)^{\infty}$ and therefore $R_{\alpha}=R-\alpha$ acts nilpotently
on ${\mathcal{M}}_{\alpha}$. Using an idea of Saito [Sai90], we filter
${\mathcal{M}}_{\alpha}$ by

$F_{\ell}{\mathcal{M}}_{\alpha}=\frac{F_{\ell}{\mathcal{M}}\cap{\mathcal{M}}_{\geq\alpha}+{\mathcal{M}}_{>\alpha}}{{\mathcal{M}}_{>\alpha}},\quad\text{
for }\ell\in{\mathbb{Z}}.$

The filtration $F_{\bullet}{\mathcal{M}}_{\alpha}$ is different from the naive
one $F_{\bullet}{\mathcal{M}}\cap\ker(R-\alpha)^{\infty}$. The reason why we
do not use the naive filtration is that $F_{\bullet}{\mathcal{M}}_{\alpha}$
not only gives the correct weight but is also easy to work out. We prove that
any power of the operator $R_{\alpha}$ is strict with respect to
$F_{\bullet}{\mathcal{M}}_{\alpha}$. Namely, for every $\ell\geq 0$, we have
the relation
$R_{\alpha}^{\ell}F_{\bullet}{\mathcal{M}}_{\alpha}=F_{\bullet+\ell}{\mathcal{M}}\cap
R_{\alpha}^{\ell}{\mathcal{M}}_{\alpha}$ (Theorem 5.1 for the case $Y$ is
reduced and Theorem 7.5 for the general case). This implies that the monodromy
filtration $W_{\bullet}{\mathcal{M}}_{\alpha}$ and
$F_{\bullet}{\mathcal{M}}_{\alpha}$ interacts very well. Note that the
monodromy filtration associated to $R_{\alpha}$ is the same as the one of
$R_{n}$ on ${\mathcal{M}}_{\alpha}$, the nilpotent part of $R$ in Jordan-
Chevalley decomposition. We have the induced good filtrations

$F_{\bullet}W_{r}{\mathcal{M}}_{\alpha}=F_{\bullet}{\mathcal{M}}\cap
W_{r}{\mathcal{M}}_{\alpha}\quad\text{ and }\quad
F_{\bullet}{\mathrm{g}}{\mathrm{r}}^{W}_{r}{\mathcal{M}}_{\alpha}=F_{\bullet}W_{r}{\mathcal{M}}_{\alpha}/F_{\bullet}W_{r-1}{\mathcal{M}}_{\alpha}.$

Denote by ${\mathcal{P}}_{\alpha,\ell}=\ker
R_{\alpha}^{\ell+1}\cap{\mathrm{g}}{\mathrm{r}}^{W}_{\ell}{\mathcal{M}}_{\alpha}$
the $\ell$-th primitive for $\ell\geq 0$, which is isomorphic to

$\frac{\ker R_{\alpha}^{\ell+1}}{\ker
R_{\alpha}^{\ell}+\mathrm{im\,}R_{\alpha}\cap\ker R_{\alpha}^{\ell+1}}.$

We endow it with the induced good filtration
$F_{\bullet}{\mathcal{P}}_{\alpha,\ell}=\mathrm{im}\,(F_{\bullet}{\mathcal{M}}\cap\ker
R^{\ell+1}_{\alpha}\rightarrow{\mathcal{P}}_{\alpha,\ell}).$ As a corollary of
the strictness of every power of $R_{\alpha}$, the Lefschetz decomposition of
${\mathrm{g}}{\mathrm{r}}^{W}{\mathcal{M}}_{\alpha}$ respects the good
filtrations, i.e.

$F_{\bullet}{\mathrm{g}}{\mathrm{r}}^{W}_{r}{\mathcal{M}}_{\alpha}=\bigoplus_{\begin{subarray}{c}\ell\geq
0,-\frac{r}{2}\end{subarray}}R_{\alpha}^{\ell}F_{\bullet-\ell}{\mathcal{P}}_{\alpha,r+2\ell}\quad\text{
for }r\geq 0.$

See Theorem 5.6 for the case $Y$ is reduced and Theorem 7.8 for the general
case. This corollary suggests that it suffices to study the hypercohomology of
each primitive part. The primitive parts will be the source for the pure
polarized Hodge structures.

We will construct a sesquilinear pairing
$S_{\alpha}:{\mathcal{M}}_{\alpha}\otimes_{\mathbb{C}}\overline{{\mathcal{M}}_{\alpha}}\rightarrow\mathfrak{C}_{X}$
using the Mellin transformation [Sab02], where
$\overline{{\mathcal{M}}_{\alpha}}$ is the naive conjugation of
${\mathcal{M}}_{\alpha}$ and $\mathfrak{C}_{X}$ is the sheaf of currents. Both
${\mathcal{M}}_{\alpha}\otimes_{\mathbb{C}}\overline{{\mathcal{M}}_{\alpha}}$
and $\mathfrak{C}_{X}$ canonically carry
${\mathscr{D}}_{X}\otimes_{\mathbb{C}}{\mathscr{D}}_{\overline{X}}$-module
structures where ${\mathscr{D}}_{\overline{X}}$ denotes the sheaf of anti-
holomorphic differential operators and the sesquilinear pairing is just a
morphism of
${\mathscr{D}}_{X}\otimes_{\mathbb{C}}\overline{{\mathscr{D}}_{X}}$-modules. A
good reference is the MHM project [SS] by Sabbah and Schnell. The sesquilinear
pairings on ${\mathcal{M}}_{\alpha}$ is an analogy of a polarization on a
Hodge structure: a complex polarized Hodge structure of weight $n$ can be
described as a filtered vector space $(V,F^{\bullet})$ with a Hermitian
pairing $S$ such that $(-1)^{n-p}S$ is a Hermitian inner product on $F^{p}\cap
G^{n-p}$ where $G^{n-p}$ is the $S$-orthogonal complement of $F^{p+1}$. The
sesquilinear pairing $S_{\alpha}$ induces the second filtration on the
hypercohomology of ${\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}}_{\alpha}$. The
Mellin transformation is used here to extract the principal part of the
asymptotic expansion of fiberwise integration of log forms since the log forms
may not be integrable. We refer to the $\S$2.1 for the definition of
sesquilinear pairings on ${\mathscr{D}}$-module

The operator $R_{\alpha}$ is self-adjoint with respect to the pairing
$S_{\alpha}:{\mathcal{M}}_{\alpha}\otimes_{\mathbb{C}}\overline{\mathcal{M}}_{\alpha}\rightarrow\mathfrak{C}_{X}$,
i,e, $S_{\alpha}(-,R_{\alpha}-)=S_{\alpha}(R_{\alpha}-,-)$. See $\S$6 for the
case that $Y$ is reduced $\S$8 for the general case. This implies we have an
induced pairing on the associated graded modules:

$S_{\alpha,r}:{\mathrm{g}}{\mathrm{r}}^{W}_{r}{\mathcal{M}}_{\alpha}\otimes_{\mathbb{C}}{\mathrm{g}}{\mathrm{r}}^{W}_{-r}{\mathcal{M}}_{\alpha}\rightarrow\mathfrak{C}_{X}.$

Then
$P_{R_{\alpha}}S_{\alpha,r}=S_{\alpha,r}\circ\left({\mathrm{i}}{\mathrm{d}\,}\otimes_{\mathbb{C}}R_{\alpha}^{r}\right)$
defines a sesquilinear pairing on the primitive part
${\mathcal{P}}_{\alpha,r}$.

######  Theorem D.

The cohomologies of the de Rham complex of ${\mathcal{P}}_{\alpha,r}$

$\bigoplus_{\ell\in{\mathbb{Z}}}H^{\ell}(X,{\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{P}}_{\alpha,r})$

together with the filtration induced by $F_{\bullet}{\mathcal{P}}_{\alpha,r}$
and the sesquilinear pairing induced by $P_{R_{\alpha}}S_{\alpha,r}$ determine
a polarized Hodge-Lefschetz structure of central weight $n+r$ with
${\mathfrak{s}}{\mathfrak{l}}_{2}({\mathbb{C}})$-action induced by
$2\pi\sqrt{-1}L$.

A polarized Hodge-Lefschetz structure basically is a direct sum of Hodge
structures of different weights preserving by an
${\mathfrak{s}}{\mathfrak{l}}_{2}({\mathbb{C}})$-action modeled by the direct
sum of all the cohomologies of a compact Kähler manifold. We refer to $\S$2.3
for the definition of polarized Hodge-Lefschetz structures. To illustrate the
idea of Theorem D, assume for a moment that $Y$ is reduced. Then the
endomorphism $R$ will be nilpotent and this implies that
${\mathcal{M}}={\mathcal{M}}_{0}$. Denote by $Y^{J}=\bigcap_{i\in J}Y_{i}$ for
any non-empty subset $J$ of $I$. Let $\tau^{J}:Y^{J}\rightarrow X$ be the
closed embedding and
$\tau^{(r+1)}:\tilde{Y}^{(r+1)}=\coprod_{\\#J=r+1}Y^{J}\rightarrow X$ be the
natural morphism for every $r\geq 0$. For simplicity, suppose
${\mathcal{P}}_{r}={\mathcal{P}}_{0,r}$. We will show that there exists a
filtered isomorphism (Theorem 5.7)

$\phi_{r}:({\mathcal{P}}_{r},F_{\bullet}{\mathcal{P}}_{r})\rightarrow\tau^{(r+1)}_{+}\omega_{\tilde{Y}^{(r+1)}}(-r).$

Here, the Tate twist of a filtered ${\mathscr{D}}$-module is
$({\mathcal{N}},F_{\bullet}{\mathcal{N}})(-r)=({\mathcal{N}},F_{\bullet+r}{\mathcal{N}})$.
Moreover, the isomorphism respects the pairing $P_{R}S_{r}$ on
${\mathcal{P}}_{r}$ (Theorem 6.5):

$P_{R}S_{r}(-,-)=\frac{(-1)^{r}}{(r+1)!}\tau^{(r+1)}_{+}S_{{\tilde{Y}^{(r+1)}}}(\phi_{r}-,{\phi_{r}-}),$

where $S_{{\tilde{Y}^{(r+1)}}}$ is the standard pairing on
$\omega_{\tilde{Y}^{(r+1)}}$. Therefore, the $k$-th hypercohomology of the de
Rham complex ${\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{P}}_{r}$ is isomorphic to
$H^{n-r+k}(\tilde{Y}^{(r+1)},{\mathbb{C}})(-r)$ as polarized Hodge structures
of weight $n+r+k$. Summing all the hypercohomologies of
${\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{P}}_{r}$, we get a polarized Hodge-
Lefschetz structure of central weight $n+r$ with
${\mathfrak{s}}{\mathfrak{l}}_{2}({\mathbb{C}})$-action induced by
$2\pi\sqrt{-1}L$. For the case when $Y$ is non-reduced, we will identify the
primitive part ${\mathcal{P}}_{\alpha,r}$ with certain filtered holonomic
${\mathscr{D}}$-modules coming from the cyclic coverings (Theorem 7.13), and
the identification also respects the sesquilinear pairing (Theorem 8.10). As a
direct consequence, we obtain

######  Theorem E.

Let
$V^{\alpha}_{\ell,k}=H^{\ell}(X,{\mathrm{g}}{\mathrm{r}}^{W}_{k}{\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}}_{\alpha})$
be the relabelling of the first page of the weight spectral sequence. Then
$V^{\alpha}=\bigoplus_{k,\ell\in{\mathbb{Z}}}V^{\alpha}_{\ell,k}$ is a
polarized bigraded Hodge-Lefschetz structure of central weight $n$ with the
polarization induced by $S_{\alpha}$ and
${\mathfrak{s}}{\mathfrak{l}}_{2}({\mathbb{C}})\times{\mathfrak{s}}{\mathfrak{l}}_{2}({\mathbb{C}})$-action
induced by $2\pi\sqrt{-1}L$ and $R_{\alpha}$. Moreover, the differential
$d_{1}$ of the first page of weight spectral is a differential of polarized
bigraded Hodge-Lefschetz structure.

By a formal argument of Guillén and Navarro Aznar [GNA90], which follows some
ideas of Deligne and Saito, we have

######  Corollary F.

We have the following statements:

  1. (1)

the Hodge spectral sequence degenerates at $\prescript{F}{}{E_{1}}$;

  2. (2)

the weight spectral sequence degenerates at $\prescript{W}{}{E_{2}}$;

  3. (3)

the $\alpha$-generalized eigenspace of the bigraded vector space
$\prescript{W}{}{E_{2}}=\bigoplus_{\ell,k\in{\mathbb{Z}}}{\mathrm{g}}{\mathrm{r}}^{W}_{\ell}H^{k}(Y,\Omega^{\bullet}_{X/\Delta}(\log
Y)|_{Y})$ with respect to $R$ is a polarized bigradged Hodge-Lefschetz
structure of central weight $n$ with polarization induced by $S_{\alpha}$ and
${\mathfrak{s}}{\mathfrak{l}}_{2}({\mathbb{C}})\times{\mathfrak{s}}{\mathfrak{l}}_{2}({\mathbb{C}})$-action
induced by $2\pi\sqrt{-1}L$ and $R_{\alpha}$.

Note that the third statement in the above Corollary is equivalent to the
Theorem A; therefore, we finish the proof of Theorem A. See Theorem 6.6 and
Corollary 6.7, when $Y$ is reduced. See Theorem 8.11 and Corollary 8.12, when
$Y$ is allowed to be non-reduced,.

###  1.4. Outline

We first review basic notions on holonomic filtered ${\mathscr{D}}$-modules,
integrable logarithmic connections and polarized bigraded Hodge-Lefschetz
structures in $\S$2. Then we set up the relative log de Rham complex and
construct a log connection on its higher direct images in $\S$3. We transfer
all of the data on the relative log de Rham complex into a filtered holonomic
${\mathscr{D}}$-module in $\S$4. To avoid the messy calculations, we first
prove everything in the reduced case in $\S$5 and $\S$6. The idea for the non-
reduced case is almost the same but requires some Hodge theory of cyclic
coverings. We construct some ${\mathscr{D}}$-modules in $\S$7.4 as the summand
of the primitive part and prove their hypercohomogies underlies canonical
polarized Hodge structures in $\S$8.1. Lastly, we prove the local invariant
cycle theorem in $\S$9.

###  1.5. Acknowledgement

The author thanks his advisor Christian Schnell for introducing this topic to
the author, and also for sharing ideas and discussing details during our
weekly meetings. Many ideas of this paper should be attributed to him. The
author also would like to thank Guodu Chen and Nathan Chen for reading a draft
of this paper and useful discussions.

##  2\. Preliminaries

###  2.1. Filtred ${\mathscr{D}}$-modules with sesquilinear pairings

We will work with right ${\mathscr{D}}$-modules unless further specified. Let
$Z$ be a complex manifold of dimension $n$ and denote by $\Omega^{p}_{Z}$ the
sheaf of holomorphic $p$-forms and ${\mathscr{T}}_{Z}$ the sheaf of
holomorphic tangent vectors fields. For a filtered ${\mathscr{D}}_{Z}$-module
we mean a pair $({\mathcal{N}},F_{\bullet}{\mathcal{N}})$ where
${\mathcal{N}}$ is a coherent ${\mathscr{D}}_{Z}$-module and
$F_{\bullet}{\mathcal{N}}$ is a good filtration. Occasionally we will abuse
notations and say ${\mathcal{N}}$ also denotes the filtered
${\mathscr{D}}_{Z}$-module if the filtration is clear. Denote by
${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{Z}=\bigoplus_{\ell\in
Z}{\mathrm{g}}{\mathrm{r}}^{F}_{\ell}{\mathscr{D}}_{X}$ the associated graded
algebra and
${\mathrm{g}}{\mathrm{r}}^{F}{\mathcal{N}}=\bigoplus_{\ell\in{\mathbb{Z}}}{\mathrm{g}}{\mathrm{r}}^{F}_{\ell}{\mathcal{N}}$
the associated graded module. Note that
${\mathrm{g}}{\mathrm{r}}^{F}{\mathcal{N}}$ is a coherent
${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{Z}$-module. Let
$T^{*}Z=\mathrm{Spec}_{Z}{\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}$ be the
algebraic cotangent bundle and $T^{*}_{V}Z$ the geometric conormal bundle of a
subvariety $V$ in $Z$. The characteristic variety of ${\mathcal{N}}$ is the
support of ${\mathrm{g}}{\mathrm{r}}^{F}{\mathcal{N}}$ on $T^{*}Z$ and is
denoted by $char({\mathcal{N}})$. The characteristic cycle of ${\mathcal{N}}$
is the cycle associated to the coherent sheaf
${\mathrm{g}}{\mathrm{r}}^{F}{\mathcal{N}}$ on $T^{*}Z$ and is denoted by
$cc({\mathcal{N}})$. Neither the characteristic variety nor the characteristic
cycle depend on the choice of the filtration [HTT08]. For example, the
canonical bundle $\omega_{Z}$ is naturally a holonomic
${\mathscr{D}}_{Z}$-module with action

$\alpha.\xi=-d(\xi\righthalfcup\alpha)$

for local sections $\xi\in{\mathscr{T}}_{Z}$ and $\alpha\in\omega_{Z}$. It
also naturally has a good filtration

(2.1) $F_{\ell}\omega_{Z}=\left\\{\begin{aligned} \omega_{Z}&,&\ell\geq-n;\\\
0&,&\ell<-n.\end{aligned}\right.$

Then one can compute $cc(\omega_{Z})=[T^{*}_{Z}Z]$ which is the cycle of the
zero section of the cotangent bundle. We call ${\mathcal{N}}$ a holonomic
${\mathscr{D}}_{Z}$-module if $\dim char({\mathcal{N}})=n$. See more details
in [HTT08]. A Tate twist of filtered ${\mathscr{D}}_{Z}$-module is defined to
be ${\mathcal{N}}(-r)=({\mathcal{N}},F_{\bullet+r}{\mathcal{N}})$ for any
$r\in{\mathbb{Z}}$.

Denote by ${\mathbf{D}}^{b}(Z,{\mathbb{C}})$ the bounded derived category of
complexes with values in finite dimensional ${\mathbb{C}}$-vector spaces and
${\mathbf{D}}^{b}(Z,{\mathscr{D}})$ the bounded derived category of
${\mathscr{D}}_{Z}$-modules. Denote by ${\mathbf{D}}^{b}_{h}(Z,{\mathscr{D}})$
the full subcategory of ${\mathbf{D}}^{b}(Z,{\mathscr{D}})$ whose objects are
complexes with holonomic cohomologies. For a morphism $f:Z\rightarrow W$
between complex manifolds, denote by
$Rf_{*},Rf_{!}:{\mathbf{D}}^{b}(Z,{\mathbb{C}})\rightarrow{\mathbf{D}}^{b}(W,{\mathbb{C}})$
the derived pushforward and proper pushforward functors respectively and
$R^{k}f_{*},R^{k}f_{!}$ the $k$-th cohomology functors respectively. For any
${\mathcal{N}}^{\bullet}\in{\mathbf{D}}^{b}(Z,{\mathscr{D}})$, the pushforward
functor and the proper pushfoward functor
$f_{+},f_{\dagger}:{\mathbf{D}}^{b}(Z,{\mathscr{D}})\rightarrow{\mathbf{D}}^{b}(W,{\mathscr{D}})$
are by definition, respectively

$f_{+}{\mathcal{N}}^{\bullet}=Rf_{*}({\mathcal{N}}^{\bullet}\mathbin{\mathop{\otimes}\displaylimits_{{\mathscr{D}}_{Z}}^{L}}{\mathscr{D}}_{Z\rightarrow
W})\,\text{ and
}f_{\dagger}{\mathcal{N}}^{\bullet}=Rf_{!}({\mathcal{N}}^{\bullet}\mathbin{\mathop{\otimes}\displaylimits_{{\mathscr{D}}_{Z}}^{L}}{\mathscr{D}}_{Z\rightarrow
W}),$

where ${\mathscr{D}}_{Z\rightarrow W}=f^{*}{\mathscr{D}}_{W}$ is the transfer
module. In fact, the functor $f_{\dagger}$ preserves the holonomicty, i.e.,
$f_{\dagger}:{\mathbf{D}}^{b}_{h}(Z,{\mathscr{D}})\rightarrow{\mathbf{D}}^{b}_{h}(W,{\mathscr{D}})$
(see [HTT08]). Of course if $f$ is proper or proper on the support of
${\mathcal{N}}$ then $f_{+}=f_{\dagger}$. The de Rham complex of
${\mathcal{N}}$ is

${\mathrm{D}}{\mathrm{R}}_{Z}{\mathcal{N}}=_{\mathrm{def}}{\mathcal{N}}\otimes\bigwedge^{-\bullet}{\mathscr{T}}_{Z}=\\{{\mathcal{N}}\otimes\bigwedge^{n}{\mathscr{T}}_{Z}{\mathcal{N}}\rightarrow{\mathcal{N}}\otimes\bigwedge^{n-1}{\mathscr{T}}_{Z}\rightarrow\cdots\rightarrow{\mathcal{N}}\\}$

with ${\mathcal{N}}$ is in degree $0$. If without further indication, tensor
products are always taken over ${\mathscr{O}}$-modules. Some authors also call
it Spencer complex. The de Rham complex of $\omega_{Z}$

$\omega_{Z}\otimes\bigwedge^{-\bullet}{\mathscr{T}}_{Z}=\\{\omega_{Z}\otimes\bigwedge^{n}{\mathscr{T}}_{Z}\omega_{Z}\rightarrow\omega_{Z}\otimes\bigwedge^{n-1}{\mathscr{T}}_{Z}\rightarrow\cdots\rightarrow\omega_{Z}\\}$

is isomorphic to the usual de Rham complex
${\mathrm{D}}{\mathrm{R}}_{Z}{\mathscr{O}}_{Z}=\Omega^{n+\bullet}_{Z}$ of $Z$
under the isomorphisms

(2.2)
$\omega_{Z}\otimes\bigwedge^{p}{\mathscr{T}}_{Z}\rightarrow\Omega^{n-p}_{Z},\,\,\,\omega\otimes\partial_{J}\mapsto(-1)^{n-j_{1}+\cdots+n-j_{p}}dz_{{\bar{J}}},$

where $\partial_{J}$ is a local section of $\bigwedge^{p}{\mathscr{T}}_{Z}$,
$J$ is ordered index set and ${\bar{J}}$ is the complement with the natural
ordering, and $\omega=dz_{1}\wedge dz_{2}\wedge\cdots\wedge dz_{n}$. If
$F_{\bullet}{\mathcal{N}}$ is a good filtration, the de Rham complex is also
filtered:

$F_{\ell}{\mathrm{D}}{\mathrm{R}}_{Z}{\mathcal{N}}=F_{\ell+\bullet}{\mathcal{N}}\otimes\bigwedge^{-\bullet}{\mathscr{T}}_{Z}=\\{F_{\ell}{\mathcal{N}}\otimes\bigwedge^{n}{\mathscr{T}}_{Z}{\mathcal{N}}\rightarrow
F_{\ell+1}{\mathcal{N}}\otimes\bigwedge^{n-1}{\mathscr{T}}_{Z}\rightarrow\cdots\rightarrow
F_{\ell+n}{\mathcal{N}}\\}.$

The direct image functor and the de Rham functor are commute
$:Rf_{!}\circ{\mathrm{D}}{\mathrm{R}}_{Z}={\mathrm{D}}{\mathrm{R}}_{W}\circ
f_{\dagger}$ [MS, Corollary 4.4.4].

A sesquilinear pairing $S$ on ${\mathscr{D}}_{Z}$-module ${\mathcal{N}}$ is a
${\mathscr{D}}_{Z,\overline{Z}}$-module morphism
$S:{\mathcal{N}}\otimes_{\mathbb{C}}\overline{\mathcal{N}}\rightarrow\mathfrak{C}_{Z}$.
Here,
${\mathscr{D}}_{Z,\overline{Z}}={\mathscr{D}}_{Z}\otimes_{\mathbb{C}}{\mathscr{D}}_{\overline{Z}}$
for $\overline{{\mathscr{D}}_{Z}}$ is the sheaf antiholomorphic differential
operators, $\overline{\mathcal{N}}$ is the stupid conjugate of ${\mathcal{N}}$
as a $\overline{\mathscr{D}}_{Z}$-module and $\mathfrak{C}_{Z}$ is the sheaf
of currents on $Z$ with natural ${\mathscr{D}}_{Z,\overline{Z}}$-module
structure. We have the proper pushforward functor similarly as above on
${\mathscr{D}}_{Z,\overline{Z}}$-modules and also call it $f_{\dagger}$:

$f_{\dagger}(-)=_{\mathrm{def}}Rf_{!}(-\mathbin{\mathop{\otimes}\displaylimits_{{\mathscr{D}}_{Z,\overline{Z}}}^{L}}{\mathscr{D}}_{Z,\overline{Z}\rightarrow
W,\overline{W}}),$

where the transfer module ${\mathscr{D}}_{Z,\overline{Z}\rightarrow
W,\overline{W}}=_{\mathrm{def}}f^{*}{\mathscr{D}}_{W,\overline{W}}$. Because
of the natural morphism
$f_{\dagger}\mathfrak{C}_{Z}\rightarrow\mathfrak{C}_{W}$, we can pushforward
the sesquilinear pairing to get

${\mathscr{H}}^{0}f_{\dagger}S_{k}:{\mathscr{H}}^{k}f_{\dagger}{\mathcal{N}}\otimes_{\mathbb{C}}\overline{{\mathscr{H}}^{-k}f_{\dagger}{\mathcal{N}}}\rightarrow{\mathscr{H}}^{0}f_{\dagger}{\mathcal{N}}\otimes_{\mathbb{C}}\overline{{\mathcal{N}}}\rightarrow\mathfrak{C}_{W}.$

If $f$ is a closed embedding then
$f_{+}S:f_{+}{\mathcal{N}}\otimes_{\mathbb{C}}f_{+}\overline{\mathcal{N}}\rightarrow\mathfrak{C}_{W}$.
If $W$ is a point, then we have an induced pairing on the complex

$f_{\dagger}S:{\mathrm{D}}{\mathrm{R}}_{Z,\overline{Z}}{\mathcal{N}}\otimes_{\mathbb{C}}\overline{{\mathcal{N}}}\rightarrow{\mathrm{D}}{\mathrm{R}}_{Z,\overline{Z}}\mathfrak{C}_{Z}\simeq{\mathbb{C}}[2n],$

where
${\mathrm{D}}{\mathrm{R}}_{Z,\overline{Z}}{\mathcal{N}}\otimes_{\mathbb{C}}\overline{{\mathcal{N}}}\simeq{\mathrm{D}}{\mathrm{R}}_{Z}{\mathcal{N}}\otimes_{\mathbb{C}}\overline{{\mathrm{D}}{\mathrm{R}}_{Z}{\mathcal{N}}}$.
Taking cohomology at $0$-th degree yields, for each $k\in{\mathbb{Z}}$,

(2.3) $H^{k}_{c}(Z,{\mathrm{D}}{\mathrm{R}}_{Z}{\mathcal{N}})\otimes
H^{-k}_{c}(Z,\overline{{\mathrm{D}}{\mathrm{R}}_{Z}{\mathcal{N}}})\rightarrow
H^{0}_{c}(Z,{\mathrm{D}}{\mathrm{R}}_{Z,\overline{Z}}{\mathcal{N}}\otimes_{\mathbb{C}}\overline{{\mathcal{N}}})\rightarrow
H^{2n}_{c}(Z,{\mathbb{C}})\simeq{\mathbb{C}}.$

######  Example 2.1.

The ${\mathscr{D}}_{Z}$-module $\omega_{Z}$ carries a natural pairing
$S_{Z}:\omega_{Z}\otimes_{\mathbb{C}}\overline{\omega_{Z}}\rightarrow\mathfrak{C}_{Z}$,

(2.4) $\langle
S_{Z}(m^{\prime},m^{\prime\prime}),\eta\rangle=\frac{\varepsilon(n+1)}{(2\pi\sqrt{-1})^{n}}\int_{Z}\eta
m^{\prime}\wedge\overline{m^{\prime\prime}},$

for $m^{\prime},m^{\prime\prime}$ local sections of $\omega_{Z}$, $\eta$ a
test function on $Z$ and $\varepsilon(k)=(-1)^{\frac{k(k-1)}{2}}$. The
coefficient $\frac{\varepsilon(n+1)}{(2\pi\sqrt{-1})^{n}}$ in the definition
is chosen so that
$\frac{\varepsilon(n+1)}{(2\pi\sqrt{-1})^{n}}m\wedge\overline{m}=|m|^{2}$ is a
positive current for any local section $m$ of $\omega_{Z}$ and elimination the
choice of orenation (see more details in $\S$2.3). The pairing
$S_{Z}:\omega_{Z}\otimes_{\mathbb{C}}\overline{\omega_{Z}}\rightarrow\mathfrak{C}_{Z}$
yields a collection of pairings

$H^{k}_{c}(Z,{\mathrm{D}}{\mathrm{R}}_{Z}\omega_{Z})\otimes_{\mathbb{C}}\overline{H^{-k}_{c}(Z,{\mathrm{D}}{\mathrm{R}}_{Z}\omega_{Z})}\rightarrow{\mathbb{C}}.$

###  2.2. Logarithmic connections

If $D=\sum a_{i}D_{i}$ is a simple normal crossing divisor on $Z$ for
$a_{i}\geq 0$, denote by $\Omega_{Z}(\log D)$ the sheaf of meromorphic
differential $1$-forms with logarithmic poles along $D_{\mathrm{red}}=\sum
D_{i}$ and denote by $\Omega^{p}_{Z}(\log D)=\bigwedge^{p}\Omega_{Z}(\log D)$
the meromophic $p$-forms with logarithmic pole along $D$. Each
$\Omega^{p}_{Z}(\log D)$ is a locally free ${\mathscr{O}}_{Z}$-module.

In our convention, the de Rham complex of $Z$ is
${\mathrm{D}}{\mathrm{R}}_{Z}{\mathscr{O}}_{Z}$

$\Omega^{\bullet+n}_{Z}=\\{{\mathscr{O}}_{Z}\rightarrow\Omega_{Z}\rightarrow\Omega^{2}_{Z}\rightarrow\cdots\rightarrow\Omega^{n}_{Z}\\}[n].$

The log de Rham complex is

$\Omega^{\bullet+n}_{Z}(\log D)=\\{{\mathscr{O}}_{Z}\rightarrow\Omega_{Z}(\log
D)\rightarrow\Omega^{2}_{Z}(\log
D)\rightarrow\cdots\rightarrow\Omega^{n}_{Z}(\log D)\\}[n].$

We will follow the Koszul sign rule: for a chain complex $C^{\bullet}$ with
differential $d$, the shifted complex $C^{\bullet+n}=C^{\bullet}[n]$ equipped
with differential $(-1)^{n}d$. We define residue along $D_{i}$ by (see [EV92,
2.5])

${\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{D_{i}}:\Omega^{\bullet+n}_{Z}(\log
D)\rightarrow\Omega^{\bullet+\dim
D_{i}}_{D_{i}}(\log(D-D_{i})|_{D_{i}})\text{,
}\frac{dz_{i}}{z_{i}}\wedge\alpha\mapsto\alpha|_{D_{i}},$

where $z_{i}$ is the local defining equation of $D_{i}$ and
$\frac{dz_{i}}{z_{i}}\wedge\alpha$ is a local section of
$\Omega^{\bullet+n}_{Z}(\log D)$. It factors through

$\Omega^{\bullet+n}_{Z}(\log D)|_{D_{i}}\rightarrow\Omega^{\bullet+\dim
D_{i}}_{D_{i}}(\log(D-D_{i})|_{D_{i}}).$

By abuse of notations, we still call the above morphism
${\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{D_{i}}$. Let $D^{J}=\cap_{j\in J}D^{J}$
and $D_{J}=\sum_{j\in J}D_{j}$. Then we have a collection of residue maps, by
choosing an order on the indices and successively applying
${\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{D_{j}}$ for $j\in J$,

${\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{D^{J}}:\Omega^{\bullet+n}_{Z}(\log
D)\rightarrow\Omega^{\bullet+\dim D^{J}}_{D^{J}}(\log(D-D_{J})|_{D^{J}}).$

A log connection $\nabla$ with poles along $D$ on a coherent
${\mathscr{O}}_{Z}$-module ${\mathcal{F}}$ is a ${\mathbb{C}}$-linear morphism
$\nabla:{\mathcal{F}}\rightarrow\Omega_{Z}(\log D)\otimes{\mathcal{F}}$
satisfying the Leibniz rule $\nabla fs=df\otimes s+f\nabla s$ for $f$ local
section of ${\mathscr{O}}_{Z}$ and $s$ local section of ${\mathcal{F}}$. One
can extend standardly $\nabla$ to

${{\mathcal{F}}}$${\Omega_{Z}(\log
D)\otimes{\mathcal{F}}}$${\cdots}$${\Omega^{n}_{Z}(\log
D)\otimes{\mathcal{F}}.}$$\scriptstyle{\nabla}$$\scriptstyle{\nabla}$$\scriptstyle{\nabla}$$\scriptstyle{\nabla}$

If the above is a chain complex, i.e., $\nabla^{2}=0$ we say
$({\mathcal{F}},\nabla)$ is an integrable log connection. For any integrable
log connection $\nabla:{\mathcal{F}}\rightarrow\Omega_{Z}(\log
D)\otimes{\mathcal{F}}$, we call the morphism
${\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{D_{i}}\nabla:{\mathcal{F}}\rightarrow{\mathcal{F}}|_{D_{i}}$
induced by ${\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{D_{i}}:\Omega_{Z}(\log
D)\rightarrow{\mathscr{O}}_{D_{i}}$ its residue along $D_{i}$. Note that
${\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{D_{i}}$ is ${\mathscr{O}}_{Z}$-linear
and factors through again
${\mathcal{F}}|_{D_{i}}\rightarrow{\mathcal{F}}|_{D_{i}}$.

An integrable log connection is same as a left ${\mathscr{D}}_{Z}(\log
D)$-module, where ${\mathscr{D}}_{Z}(\log D)$ is the sub-algebra of
${\mathscr{D}}_{Z}$ generated locally by the differential operators $P$ such
that $P\cdot{\mathscr{I}}_{D}\subset{\mathscr{I}}_{D}$. Here, we denote by
${\mathscr{I}}_{D}$ the ideal sheaf of the normal crossing divisor $D$. Then
we can extend the definition of residues of a log connection as follows. The
sheaf ${\mathscr{O}}_{D_{i}}={\mathscr{O}}_{Z}/{\mathscr{I}}_{D_{i}}$
naturally has a left ${\mathscr{D}}_{Z}(\log D)$-module structure because
${\mathscr{I}}_{D_{i}}$ is also stable under by the ${\mathscr{D}}_{Z}(\log
D)$-action by the naive reason. Let ${\mathcal{F}}^{\bullet}$ be a complex of
integrable log connections. Then the complex

${\mathcal{F}}^{\bullet}\mathbin{\mathop{\otimes}\displaylimits_{{\mathscr{O}}_{Z}}^{L}}{\mathscr{O}}_{D_{i}}$

is a complex of ${\mathscr{D}}_{Z}(\log D)$-modules because taking tensor
products over ${\mathscr{O}}_{Z}$ is closed in the category of
${\mathscr{D}}_{Z}(\log D)$-modules and one can resolve either
${\mathcal{F}}^{\bullet}$ or ${\mathscr{O}}_{D_{i}}$ using locally
${\mathscr{D}}_{Z}(\log D)$-free resolutions. The $\ell$-th cohomology
${\mathscr{H}}^{\ell}({\mathcal{F}}^{\bullet}\otimes^{L}{\mathscr{O}}_{D_{i}})$
is indeed ${\mathscr{O}}_{D_{i}}$-module equipped with an integrable log
connection. The residue of of this log connection is
${\mathscr{O}}_{D_{i}}$-linear and is called the the $\ell$-th residue of the
complex ${\mathcal{F}}^{\bullet}$.

As in the case of ${\mathscr{D}}$-module, the sheaf $\omega_{Z}(\log
D)=\Omega^{n}_{Z}(\log D)$ carries a canonical right ${\mathscr{D}}_{Z}(\log
D)$-module structure and we have the left to right transformation
${\mathcal{F}}\mapsto\omega_{Z}(\log D)\otimes{\mathcal{F}}$ for any left
${\mathscr{D}}_{Z}(\log D)$-module ${\mathcal{F}}$. Moreover, we have the
following analog

######  Theorem 2.2.

The log de Rham complex of ${\mathscr{D}}_{Z}(\log D)$

$\left\\{{\mathscr{D}}_{Z}(\log D)\rightarrow\Omega_{Z}(\log
D)\otimes{\mathscr{D}}_{Z}(\log
D)\rightarrow\cdots\rightarrow\Omega^{n}_{Z}(\log
D)\otimes{\mathscr{D}}_{Z}(\log D)\right\\}[n]$

is a resolution of $\omega_{Z}(\log D)$ as right ${\mathscr{D}}_{Z}(\log
D)$-modules. The Spencer complex of ${\mathscr{D}}_{Z}(\log D)$

${\mathscr{D}}_{Z}(\log D)\otimes\bigwedge^{n}{\mathscr{T}}_{Z}(\log
D)\rightarrow{\mathscr{D}}_{Z}(\log
D)\otimes\bigwedge^{n-1}{\mathscr{T}}_{Z}(\log
D)\rightarrow\cdots\rightarrow{\mathscr{D}}_{Z}(\log D)$

is a resolution of ${\mathscr{O}}_{Z}$ as left ${\mathscr{D}}_{Z}(\log
D)$-modules.

For any integrable log connection ${\mathcal{F}}$, it induces a complex of
right ${\mathscr{D}}_{Z}$-modules,

(2.5) $\\{{\mathcal{F}}\otimes{\mathscr{D}}_{Z}\rightarrow\Omega_{Z}(\log
D)\otimes{\mathcal{F}}\otimes{\mathscr{D}}_{Z}\rightarrow\cdots\rightarrow\Omega^{n}_{Z}(\log
D)\otimes{\mathcal{F}}\otimes{\mathscr{D}}_{Z}\\}[n].$

In fact, it is nothing but the log de Rham complex of
${\mathcal{F}}\otimes{\mathscr{D}}_{Z}$ as a left ${\mathscr{D}}_{Z}(\log
D)$-module.

######  Lemma 2.3.

The log de Rham complex of ${\mathcal{F}}\otimes{\mathscr{D}}_{Z}$ is a
${\mathscr{D}}_{Z}$-module resolution of

$\omega_{Z}(\log
D)\otimes{\mathcal{F}}\mathbin{\mathop{\otimes}\displaylimits_{{\mathscr{D}}_{Z}(\log
D)}}{\mathscr{D}}_{Z}.$

###### Proof.

By the above theorem, we have

$\displaystyle\omega_{Z}(\log
D)\otimes{\mathcal{F}}\mathbin{\mathop{\otimes}\displaylimits_{{\mathscr{D}}_{Z}(\log
D)}}{\mathscr{D}}_{Z}$ $\displaystyle\simeq\omega_{Z}(\log
D)\otimes{\mathcal{F}}\mathbin{\mathop{\otimes}\displaylimits_{{\mathscr{D}}_{Z}(\log
D)}}\left({\mathscr{D}}_{Z}(\log
D)\otimes\bigwedge^{-\bullet}{\mathscr{T}}_{Z}(\log
D)\right)\otimes{\mathscr{D}}_{Z}$ $\displaystyle=\omega_{Z}(\log
D)\otimes{\mathcal{F}}\otimes\bigwedge^{-\bullet}{\mathscr{T}}_{Z}(\log
D)\otimes{\mathscr{D}}_{Z}$ $\displaystyle\simeq\Omega^{\bullet+n}_{Z}(\log
D)\otimes{\mathcal{F}}\otimes{\mathscr{D}}_{Z}.$

The last isomorphism follows from that the contraction $\omega_{Z}(\log
D)\otimes\bigwedge^{-\bullet}{\mathscr{T}}_{Z}(\log
D)\simeq\Omega^{\bullet+n}_{Z}(\log D)$. ∎

######  Example 2.4.

We will use the following fact: the complex of right
${\mathscr{D}}_{Z}$-modules

$\\{{\mathscr{D}}_{Z}\rightarrow\Omega_{Z}(\log
D)\otimes{\mathscr{D}}_{Z}\rightarrow\cdots\rightarrow\Omega^{n}_{Z}(\log
D)\otimes{\mathscr{D}}_{Z}\\}[n]$

is a filtered resolution of
$\omega_{Z}(*D)=\cup_{k\in{\mathbb{Z}}}\omega_{Z}(kD)$, equipped the induced
filtration by $\Omega^{n+\bullet}_{Z}(\log D)\otimes
F_{\ell+n+\bullet}{\mathscr{D}}_{Z}$. In fact, it is well-known that the
inclusion $\Omega^{n+\bullet}_{Z}(\log
D)\rightarrow\Omega^{n+\bullet}_{Z}(*D)$ is a filtered quasi-isomorphism
[Del71]. The inclusion extends to a filtered quasi-isomorphism
$\Omega^{n+\bullet}_{Z}(\log
D)\otimes{\mathscr{D}}_{Z}\rightarrow\Omega^{n+\bullet}_{Z}(*D)\otimes{\mathscr{D}}_{Z}$.
Since $\Omega^{n+\bullet}_{Z}(*D)\otimes{\mathscr{D}}_{Z}$ is a filtered
resolution of $\omega_{Z}(*D)$, we conclude the proof. It follows that, for
$f:Z\rightarrow W$,

$f_{\dagger}\omega_{Z}(*D)=Rf_{!}(\omega_{Z}(*D)\otimes^{L}_{{\mathscr{D}}_{Z}}{\mathscr{D}}_{Z\rightarrow
W})=Rf_{!}\Omega^{n+\bullet}_{Z}(\log D)\otimes{\mathscr{D}}_{W}.$

In particular, if $f$ is a closed embedding then $f_{!}=f_{+}$ is right exact
and $f_{\dagger}={\mathscr{H}}^{0}f_{\dagger}$, which means

$\\{{\mathscr{D}}_{W}\rightarrow f_{+}\Omega_{Z}(\log
D)\otimes{\mathscr{D}}_{W}\rightarrow\cdots\rightarrow
f_{+}\Omega^{n}_{Z}(\log D)\otimes{\mathscr{D}}_{W}\\}[n]$

is a resolution of $f_{\dagger}\omega_{Z}(*D)$. We put the induced filtration
to make it a filtered resolution and denote by

$f_{\dagger}(\omega_{Z}(*D),F_{\bullet}\omega_{Z}(*D))=(f_{\dagger}\omega_{Z}(*D),F_{\bullet}f_{\dagger}\omega_{Z}(*D)),$

or for simplicity just $f_{\dagger}\omega_{Z}(*D)$.

The ${\mathscr{D}}_{Z}$-module looks like
${\mathcal{L}}\otimes{\mathscr{D}}_{Z}$ for ${\mathcal{L}}$ is a
${\mathscr{O}}_{Z}$-module is called induced ${\mathscr{D}}_{Z}$-module. For
example, we have seen $\Omega^{\dim Z+\bullet}_{Z}\otimes{\mathscr{D}}_{Z}$
and $\Omega(\log D)^{\dim Z+\bullet}_{Z}\otimes{\mathscr{D}}_{Z}$ are
complexes of induced ${\mathscr{D}}_{Z}$-modules.

###  2.3. Polarized Hodge-Lefschetz structures

The goal of this subsection is to introduce polarized bigraded Hodge-Lefschetz
structures. The prototype of polarized Hodge-Lefschetz structures one should
keep in mind is the graded vector space consisting of cohomologies of a
compact Kähler manifold. Polarized bigraded Hodge-Lefschetz structures are the
degenerations of polarized Hodge-Lefschetz structures. We begin with the
convention on Hodge structures and we only consider complex Hodge structures.

A Hodge structure of weight $n$ is a finite dimensional vector space $V$ with
two decreasing filtrations $F^{\bullet}$ and $G^{\bullet}$ satisfying

$V=F^{p}\oplus G^{n+1-p},$

for each $p\in{\mathbb{Z}}$. Let $V^{p.q}=F^{p}\cap G^{q}$ for $p+q=n$. Then
the above definition is equivalent to

$V=\bigoplus_{p+q=n}V^{p,q}.$

A morphism of Hodge structures is just a morphism of vector spaces such that
it preserves the two filtrations. A polarization on the Hodge structure
$(V,F^{\bullet},G^{\bullet})$ is a non-degenerated hermitian pairing
$S:V\otimes_{\mathbb{C}}\overline{V}\rightarrow{\mathbb{C}}$ such that

  1. (1)

$F^{p}$ is orthogonal to $G^{n+1-p}$ with respect to $S$ for every
$p\in{\mathbb{Z}}$;

  2. (2)

$(-1)^{q}S(-,-)$ is hermitian inner product on $V^{p,q}$.

######  Remark 2.5.

A polarized Hodge structure of weight $n$ is completely determined by the
triple $(V,F_{\bullet}V,S)$ because

$G^{n+1-p}V=\left\\{a\in V:S(a,b)=0\text{ for all b in
}F^{p}V\right\\}=\overline{F^{p}V^{\perp_{S}}}.$

We will also call the triple $(V,F_{\bullet}V,S)$ a polarized Hodge structure.

######  Remark 2.6.

A Tate twist $(V,F^{\bullet},S)(r)$ on a polarized Hodge structure
$(V,F^{\bullet},S)$ is the triple $(V,F^{\bullet+r},(-1)^{r}S)$, for any
integer $r$.

Now let us move on to the geometric case. It is well-known that the $k$-th
cohomology group of a compact Kähler manifold $Z$ has Hodge decomposition

$H^{k}(Z,{\mathbb{C}})=\bigoplus_{p+q=k}H^{p,q}(Z)$

and thus it is a Hodge structure of weight $k$. Fix a choice of $\sqrt{-1}$.
Let $Z$ be a compact Kähler manifold of dimension $n$, and let $h$ be any
Kähler metric on $Z$. We denote the Kähler form by $\omega=-\mathrm{Im}\,h\in
A^{2}(Z,{\mathbb{R}})$ and denote its cohomology class by $[\omega]\in
H^{2}(Z,{\mathbb{R}})$; note that this depends on the choice of $\sqrt{-1}$
through the function $\mathrm{Im}:{\mathbb{C}}\rightarrow{\mathbb{R}}$. The
choice of $\sqrt{-1}$ endows the two-dimensional real vector space
${\mathbb{C}}$ with an orientation on $Z$. The induced orientation on $Z$ has
the property that

$\int_{Z}\frac{\omega^{n}}{n!}=\mathrm{vol}(Z)>0.$

The integral also depends on the orientation, hence on the choice of
$\sqrt{-1}$. To remove the dependence, instead of the usual integral, we
should use

$\frac{1}{(2\pi\sqrt{-1})^{n}}\int_{Z}:A^{2n}(Z,{\mathbb{C}})\rightarrow{\mathbb{C}}.$

Of course we still have

$\frac{1}{(2\pi\sqrt{-1})^{n}}\int_{Z}\frac{(2\pi\sqrt{-1}\omega)^{n}}{n!}=\mathrm{vol}(Z).$

Let ${L=[w]\wedge}$ be the Lefschetz operator for a Kähler class $[w]$. Then
for $k\leq\dim Z$ the primitive part

$P_{L}H^{k}(Z,{\mathbb{C}})=_{\mathrm{def}}\ker L^{\dim Z-k}\cap
H^{k}(X,{\mathbb{C}})$

is a polarized Hodge structure of weight $k$ with the polarization

$S(a,b)={\frac{\varepsilon(n-k+1)}{(2\pi\sqrt{-1})^{n}}}\int_{Z}(2\pi\sqrt{-1}L)^{n-k}a\wedge\overline{b},$

for $a,b\in P_{L}H^{k}(Z,{\mathbb{C}})$ because of the Hodge-Riemman bilinear
relation.

If we consider the cohomology groups all together, we will get the Hodge-
Lefschetz strcuture of central weight $n$. Denote by $(\mathsf{X,Y,H})$ the
${\mathfrak{s}}{\mathfrak{l}}_{2}({\mathbb{C}})$-triple, i.e.,

$[{\mathsf{X}},{\mathsf{Y}}]={\mathsf{H}},\,[{\mathsf{H}},{\mathsf{X}}]=2{\mathsf{X}},\,[{\mathsf{H}},{\mathsf{Y}}]=-2{\mathsf{Y}}.$

In the Lie group $\mathrm{SL_{2}}({\mathbb{C}})$, we have the Weil element
$\mathsf{w}=e^{\mathsf{X}}e^{-\mathsf{Y}}e^{\mathsf{X}}$ with the property
that ${\mathsf{w}}^{-1}=-{\mathsf{w}},$ and under the adjoint action of
$\mathrm{SL}_{2}(\mathbb{{\mathbb{C}}})$ on its Lie algebra, one has the
identities

$\mathsf{wHw^{-1}}=\mathsf{-H},\quad\mathsf{wXw^{-1}=-Y},\quad\mathsf{wYw^{-1}=-X}$

From this, one deduces that
$e^{\mathsf{X}}=\mathsf{w}e^{-\mathsf{X}}e^{\mathsf{Y}}=e^{\mathsf{Y}}\mathsf{w}e^{\mathsf{Y}}$.
Now $A^{\bullet}(Z)$ becomes a representation of
$\mathfrak{sl}_{2}(\mathbb{{\mathbb{C}}})$ if we set

${{\mathsf{X}}=2\pi\sqrt{-1}L}\quad\text{ and
}\quad{\mathsf{Y}}=(2\pi\sqrt{-1})^{-1}\Lambda$

and let ${\mathsf{H}}$ act as multiplication by $k-n$ on the subspace
$A^{k}(Z).$ The reason for this (non-standard) definition is that it makes the
representation not depend on the choice of $\sqrt{-1}$. It is easy to see how
${\mathsf{w}}$ acts on primitive forms. Suppose that $\alpha\in A^{n-k}(Z)$
satisfies ${\mathsf{Y}}\alpha=0.$ Then ${\mathsf{w}}\alpha\in A^{n+k}(Z).$ If
we now expand both sides of the identity

$e^{{\mathsf{X}}}\alpha=e^{{\mathsf{Y}}}{\mathsf{w}}e^{{\mathsf{Y}}}\alpha=e^{{\mathsf{Y}}}{\mathsf{w}}\alpha$

into power series, and then compare terms in degree $n+k,$ we get

${\mathsf{w}}\alpha=\frac{{\mathsf{X}}^{k}}{k!}\alpha.$

This formula is the reason for using ${\mathsf{w}}$ (instead of the otherwise
${\mathsf{w}}^{-1}$): there is no sign on the right-hand side.

A Hodge-Lefschetz structure is linear algebra data encoding both
representation theoretic and Hodge theoretic information. Recall that a finite
dimensional ${\mathfrak{s}}{\mathfrak{l}}_{2}({\mathbb{C}})$-representation is
a graded vector space $V=\bigoplus_{\ell\in{\mathbb{Z}}}V_{\ell}$ satisfying
the following three equivalent conditions.

  1. (1)

each graded piece $V_{\ell}$ is the $\ell$-eigenspace of ${\mathsf{H}}$;

  2. (2)

the morphism ${\mathsf{X}}^{\ell}:V_{-\ell}\rightarrow V_{\ell}$ is an
isomorphism for each $\ell\geq 0$;

  3. (3)

the morphism ${\mathsf{Y}}^{\ell}:V_{\ell}\rightarrow V_{-\ell}$ is an
isomorphism for each $\ell\geq 0$.

######  Example 2.7.

For any finite dimensional vector space $V$ together with a nilpotent operator
$N$, there exists a so-called monodromy filtration $W_{\bullet}$ uniquely
determined by the following two conditions

  * •

for each $\ell\in{\mathbb{Z}}$, $N:W_{\ell}\rightarrow W_{\ell-2}$;

  * •

the induced operator
$N^{\ell}:{\mathrm{g}}{\mathrm{r}}^{W}_{\ell}\rightarrow{\mathrm{g}}{\mathrm{r}}^{W}_{-\ell}$
is an isomorphism for each $\ell\geq 0$.

Let
${\mathrm{g}}{\mathrm{r}}^{W}=\bigoplus_{\ell\in{\mathbb{Z}}}{\mathrm{g}}{\mathrm{r}}^{W}_{\ell}$.
The $\ell$-th primitive part $P_{N}{\mathrm{g}}{\mathrm{r}}^{W}_{\ell}=\ker
N^{\ell+1}\cap{\mathrm{g}}{\mathrm{r}}^{W}_{\ell}$ consists of the classes of
generators of cyclic subspaces of $V$ of dimension $\ell$ as
${\mathbb{C}}[N]$-modules for $\ell\geq 0$. For each generator $v$, we have
$N^{\ell+1}v=0$ but $N^{\ell}v\neq 0$ and also $v$ is not a image of $N$.
Therefore, we have the identification

$P_{N}{\mathrm{g}}{\mathrm{r}}^{W}_{\ell}=\frac{\ker N^{\ell+1}}{\ker
N^{\ell}+\mathrm{im\,}N\cap\ker N^{\ell+1}}.$

Furthermore, we have the Lefschetz decomposition
${\mathrm{g}}{\mathrm{r}}^{W}_{\ell}=\bigoplus_{k\geq
0}N^{k}P_{N}V_{\ell+2k}$. Taking $N={\mathsf{Y}}$, the Lefschetz structure and
the grading uniquely determines the operator $X$ such that
$({\mathsf{X}},{\mathsf{Y}},{\mathsf{H}})$ is a
${\mathfrak{s}}{\mathfrak{l}}_{2}({\mathbb{C}})$-triple by the relation
${\mathsf{X}}{\mathsf{Y}}^{k}=k(\ell-k+1){\mathsf{Y}}^{k-1}$ on
$P_{N}{\mathrm{g}}{\mathrm{r}}^{W}_{\ell}$. Thus
${\mathrm{g}}{\mathrm{r}}^{W}$ naturally is a representation of
${\mathfrak{s}}{\mathfrak{l}}_{2}({\mathbb{C}})$.

By Hard Lefschetz theorem, for any compact Kähler manifold the vector space
$\bigoplus_{\ell\in{\mathbb{Z}}}H^{\dim Z+\ell}(Z,{\mathbb{C}})$ is a
representation of ${\mathfrak{s}}{\mathfrak{l}}_{2}({\mathbb{C}})$ by setting
${\mathsf{X}}=2\pi\sqrt{-1}L$ the Lefschetz operator,
${\mathsf{Y}}=(2\pi\sqrt{-1})^{-1}\Lambda$ the adjoint operator. But because
of the Lefschetz operator of is of type $(1,1)$, we actually have
${\mathsf{X}}:H^{k}(Z,{\mathbb{C}})\rightarrow H^{k+1}(Z,{\mathbb{C}})(1)$ is
a morphism of Hodge structures and ${\mathsf{X}}^{\ell}:H^{\dim
Z-\ell}(Z,{\mathbb{C}})\rightarrow H^{\dim Z+\ell}(Z,{\mathbb{C}})(\ell)$ is
an isomorphism of Hodge structures. This leads to the following definition: a
Hodge-Lefschetz structure of central weight $n$ is a
${\mathfrak{s}}{\mathfrak{l}}_{2}({\mathbb{C}})$-representation
$V=\bigoplus_{\ell\in{\mathbb{Z}}}V_{\ell}$ with two filtrations
$F^{\bullet}V$ and $G^{\bullet}V$ such that

  1. (1)

each graded piece $(V_{\ell},F^{\bullet}V_{\ell},G^{\bullet}V_{\ell})$ is a
Hodge structure of weight $n+\ell$;

  2. (2)

the operator
${\mathsf{X}}:(V_{\ell},F^{\bullet}V_{\ell},G^{\bullet}V_{\ell})\rightarrow(V_{\ell+2},F^{\bullet+1}V_{\ell+2},,G^{\bullet+1}V_{\ell+2})$
is a morphism of Hodge structures such that

${\mathsf{X}}^{\ell}:(V_{-\ell},F^{\bullet}V_{-\ell},G^{\bullet}V_{-\ell})\rightarrow(V_{\ell},F^{\bullet}V_{\ell},G^{\bullet}V_{\ell})(\ell)$

is an isomorphism of Hodge structures;

  3. (3)

the operator
${\mathsf{Y}}:(V_{\ell},F^{\bullet}V_{\ell},G^{\bullet}V_{\ell})\rightarrow(V_{\ell-2},F^{\bullet-1}V_{\ell-2},G^{\bullet-1}V_{\ell-2})$
is a morphism of Hodge structures such that

${\mathsf{Y}}^{\ell}:(V_{\ell},F^{\bullet}V_{\ell},G^{\bullet}V_{\ell})\rightarrow(V_{-\ell},F^{\bullet}V_{-\ell},G^{\bullet}V_{-\ell})(-\ell)$

is an isomorphism of Hodge structures.

It follows from the definition the primitive part $P_{X}V_{\ell}$ is a sub-
Hodge structure for each $\ell<0$. Let $V_{\ell}=H^{\dim
Z+\ell}(Z,{\mathbb{C}})$ and $V=\bigoplus_{\ell\in{\mathbb{Z}}}V_{\ell}$. It
follows that $V$ is a Hodge-Lefschetz structure of central weight $\dim Z$.
Hodge-Lefschetz structure interplays well with the Hodge-Riemann bilinear
relation. A polarization on a Hodge-Lefschetz structure $V$ of central weight
$n$ is a hermitian symmetric paring
$S:V\otimes_{\mathbb{C}}\overline{V}\rightarrow{\mathbb{C}}$ such that

  1. (1)

the restriction $S|_{V_{\ell}\otimes_{\mathbb{C}}\overline{V_{-k}}}$ is zero
for $\ell+k\neq 0$;

  2. (2)

$S({\mathsf{X}}-,-)=S(-,{\mathsf{X}}-)$ and
$S(-,{\mathsf{Y}}-)=S({\mathsf{Y}}-,-)$;

  3. (3)

$S_{-\ell}({\mathsf{X}}^{\ell}-,-)$ is a polarization on $P_{X}V_{-\ell}$, or
equivalently, $S_{\ell}\circ({\mathrm{i}}{\mathrm{d}\,}\otimes{\mathsf{w}})$
is a polarization on $V_{\ell}$ where
$S_{\ell}:V_{\ell}\otimes\overline{V_{-\ell}}\rightarrow{\mathbb{C}}$ is the
restriction of $S$.

Note that ${\mathsf{w}}:V_{k}\rightarrow V_{-k}(-k)$ is automatically an
isomorphism of Hodge structures (of weight $n+k$). We first prove an auxiliary
formula. Suppose that $a\in V_{-\ell}$ is primitive, in the sense that
${\mathsf{X}}^{\ell+1}a=0($ and $\ell\geq 0).$ Then ${\mathsf{Y}}a=0,$ and
from ${\mathsf{w}}e^{-{\mathsf{X}}}=e^{{\mathsf{X}}}e^{-{\mathsf{Y}}},$ we get
${\mathsf{w}}e^{-{\mathsf{X}}}a=e^{{\mathsf{X}}}a,$ and after expanding and
comparing terms in degree $\ell-2j,$ also

(2.6)
${\mathsf{w}}\frac{{\mathsf{X}}^{j}}{j!}a=(-1)^{j}\frac{{\mathsf{X}}^{\ell-j}}{(\ell-j)!}a$

since ${\mathsf{w}}^{2}$ acts on $V_{-\ell+2j}$ as
$(-1)^{-\ell+2j}=(-1)^{\ell},$ this formula is actually symmetric in $j$ and
$\ell-j,$.

######  Lemma 2.8.

If $V$ is a Hodge-Lefschetz structure, then ${\mathsf{w}}:V_{k}\rightarrow
V_{-k}(-k)$ is an isomorphism of Hodge structures.

###### Proof.

Any $a\in V_{k}$ has a unique Lefschetz decomposition

$a=\sum_{j\geq\max(k,0)}\frac{{\mathsf{X}}^{j}}{j!}a_{j}$

where $a_{j}\in V_{k-2j}$ satisfies $Ya_{j}=0.$ (We only need to consider
$j\geq k$ in the sum because ${\mathsf{X}}^{2j-k+1}a_{j}=0,$ which implies
that ${\mathsf{X}}^{j}a_{j}=0$ for $\left.j<k.\right)$ Suppose further that
$a\in V_{k}^{p,q},$ where $p+q=n+k.$ Then ${\mathsf{X}}^{i}a_{j}\in
V_{k+2i}^{p+i,q+i},$ and by descending induction on $j\geq\max(k,0),$ we
deduce that $a_{j}\in V_{k-2j}^{p-j,q-j}.$ In other words, the Lefschetz
decomposition holds in the category of Hodge structures.

We can now check what happens when we apply ${\mathsf{w}}$. Using (2.6), we
find that

${\mathsf{w}}a=\sum_{j\geq\max(k,0)}{\mathsf{w}}\frac{{\mathsf{X}}^{j}}{j!}a_{j}=\sum_{j\geq\max(k,0)}(-1)^{j}\frac{{\mathsf{X}}^{j-k}}{(j-k)!}a_{j}\in
V_{-k}^{p-k,q-k}$

and so ${\mathsf{w}}$ is a morphism of Hodge structures. The same calculation
shows that ${\mathsf{w}}^{-1}$ is also a morphism of Hodge structures. It
follows that ${\mathsf{w}}$ is an isomorphism of Hodge structures. ∎

The definition of polarized Hodge-Lefschetz structure of central weight $n$ is
redundant. In fact the definition is equivalent to a tuple
$(V,{\mathsf{X}},F^{\bullet},S)$ for
$V=\bigoplus_{\ell\in{\mathbb{Z}}}V_{\ell}$, $F^{\bullet}$ is a decreasing
filtration,
${\mathsf{X}}:(V_{\ell},F^{\bullet})\rightarrow(V_{\ell+2},F^{\bullet+1})$,
and $S$ is a Hermitian pairing such that

  1. (pHL1)

for each $\ell\geq 0$, ${\mathsf{X}}^{\ell}:F^{\bullet}V_{-\ell}\rightarrow
F^{\bullet+\ell}V_{\ell}$ is an isomorphism;

  2. (pHL2)

$S({\mathsf{X}}-,-)=S(-,{\mathsf{X}}-)$ and
$S|_{V_{\ell}\otimes_{\mathbb{C}}\overline{V_{-k}}}$ vanishes except for
$k=-\ell$;

  3. (pHL3)

the triple
$\left(P_{{\mathsf{X}}}V_{j},F_{\bullet},S\circ\left({\mathsf{X}}^{j}\circ{\mathrm{i}}{\mathrm{d}\,}\right)\right)$
is a porlarized Hodge structure of weight $n-j$.

The condition ${(\mathrm{pHL1})}$ in the above definition indicates the
Lefschetz decomposition respects the filtration $F^{\bullet}$. Therefore
${\mathsf{Y}}$ is determined uniquely and also filtered. The second condition
implies that $S({\mathsf{Y}}-,-)=S(-,{\mathsf{Y}}-)$. The third condition says
that $S\circ({\mathrm{i}}{\mathrm{d}\,}\otimes{\mathsf{w}})$ is non-degenerate
on $F^{p}V_{\ell}\otimes\overline{F^{p}V_{-\ell}}$. Therefore, we also get the
following concrete description of the Hodge structure on $V_{\ell}$: for
$p+q=n+\ell$

$\displaystyle V^{p,q}_{\ell}$ $\displaystyle=\\{a\in
F^{p}V_{\ell}:S_{\ell}(a,b)=0\text{ for all }b\in F^{p-\ell+1}V_{-\ell}\\},$
$\displaystyle G^{q}V_{\ell}$ $\displaystyle=\\{a\in
V_{\ell},S_{\ell}(a,b)=0\text{ for all }b\in F^{n-q+1}V_{-\ell}\\}.$

######  Example 2.9.

For a compact Kähler manifold $Z$ of dimension $n$, let
$V_{\ell}=H^{n+\ell}(Z,{\mathbb{C}})$ and
$V=\bigoplus_{\ell\in{\mathbb{Z}}}V_{\ell}$. Then $V$ together with
${\mathsf{X}}=2\pi\sqrt{-1}L$ and ${\mathsf{Y}}=(2\pi\sqrt{-1})^{-1}\Lambda$
and with the natural filtration is a Hodge-Lefschetz structure of central
weight $n$. By Hodge-Riemann bilinear relation, taking

(2.7)
$S_{\ell}(a,b)=\frac{\varepsilon(n+\ell+1)}{(2\pi\sqrt{-1})^{n}}\int_{Z}a\wedge\bar{b}=\varepsilon(\ell)(-1)^{\ell
n}\frac{\varepsilon(n+1)}{(2\pi\sqrt{-1})^{n}}\int_{Z}a\wedge\bar{b}$

for $a\in V_{\ell}$ and $b\in V_{-\ell}$ gives a polarization on $V$. The
polarized Hodge-Lefschetz structure $V$ is determined by the filtered
${\mathscr{D}}_{Z}$-module $\omega_{Z}$ together with the sesquilinear pairing
$S_{Z}$. The graded piece $V_{\ell}$ is just $\ell$-th hypercohomology of
${\mathrm{D}}{\mathrm{R}}_{Z}\omega_{Z}$ with induced filtration
$F^{\bullet}V_{\ell}$ given by the image of
$H^{\ell}(Z,F_{-\bullet}{\mathrm{D}}{\mathrm{R}}_{Z}\omega_{Z})$. And the
polarization $S_{k}$ is given by $\varepsilon(k)$ times the pairing

${H^{k}(Z,{\mathrm{D}}{\mathrm{R}}_{Z}\omega_{Z})\otimes\overline{H^{-k}(Z,{\mathrm{D}}{\mathrm{R}}_{Z}\omega_{Z})}}$${H^{0}(Z,{\mathrm{D}}{\mathrm{R}}_{Z,\overline{Z}}\omega_{Z}\otimes_{\mathbb{C}}\overline{\omega_{Z}})}$${H^{0}(Z,{\mathrm{D}}{\mathrm{R}}_{Z,\overline{Z}}\mathfrak{C}_{Z})\simeq{\mathbb{C}}}$$\scriptstyle{S_{Z}}$

We can work out the pairing explicitly. Note that we have a commutative
diagram

${{\mathrm{D}}{\mathrm{R}}_{Z,\overline{Z}}\omega_{Z}\otimes_{\mathbb{C}}\overline{\omega_{Z}}}$${{\mathrm{D}}{\mathrm{R}}_{Z,\overline{Z}}{\mathscr{O}}_{Z}\otimes_{\mathbb{C}}\overline{{\mathscr{O}}_{Z}}}$${{\mathrm{D}}{\mathrm{R}}_{Z,\overline{Z}}\mathfrak{C}_{Z}}$${{\mathrm{D}}{\mathrm{R}}_{Z,\overline{Z}}\mathfrak{Db}_{Z}}$$\scriptstyle{S}$$\scriptstyle{D}$

where the upper horizontal arrow is the isomorphism induced by (2.2) and
similarly the lower horizontal arrow is defined on the terms in degree $-k,$

$\mathfrak{C}_{Z}\otimes_{\mathscr{O}_{Z,\bar{Z}}}\bigwedge^{k}\mathscr{T}_{Z,\bar{Z}}\rightarrow\Omega_{Z,\bar{Z}}^{2n-k}\otimes_{\mathscr{O}_{Z,\bar{Z}}}\mathfrak{D}\mathfrak{b}_{Z}$

by the following rule: write a current locally as $D\omega\wedge\bar{\omega},$
with a distribution $D$ and denote by $\partial_{J}=\bigwedge_{J}\partial_{j}$
and $dx_{\bar{J}}=\bigwedge_{i\notin J}dx_{i}$ for an ordered index subset $J$
of $I$; then

(2.8)
$(D\omega\wedge\bar{\omega})\otimes\partial_{J}\wedge\bar{\partial}_{K}\mapsto(-1)^{\left(j_{1}+\cdots+j_{p}\right)+\left(k_{1}+\cdots+k_{q}\right)}(-1)^{nq}dx_{\bar{J}}\wedge\overline{dx}_{\bar{K}}\otimes
D$

where $\\#J=p$ and $\\#K=q,$ and $p+q=k.$ The sign factor is explained by the
number of swaps that are needed to move everything into the right place, which
is
$\left(2n-j_{1}\right)+\cdots+\left(2n-j_{p}\right)+\left(n-k_{1}\right)+\cdots+\left(n-k_{q}\right)$.
We can now derive a formula for the induced pairing

(2.9)
${\mathrm{D}}{\mathrm{R}}_{Z}{\mathscr{O}}_{Z}\otimes_{\mathbb{C}}\overline{{\mathrm{D}}{\mathrm{R}}_{Z}{\mathscr{O}}_{Z}}\rightarrow\mathrm{DR}_{Z,\bar{Z}}\mathfrak{D}\mathfrak{b}_{Z}.$

For the two local sections $\alpha=dx_{\bar{J}}$ and $\beta=dx_{\bar{K}}$,
under the isomorphism
${\mathrm{D}}{\mathrm{R}}_{Z}{\mathscr{O}}_{Z}\cong{\mathrm{D}}{\mathrm{R}}_{Z}\omega_{Z}$
in (2.2), the $(n-p)$-form $\alpha$ goes to

$(-1)^{np}(-1)^{j_{1}+\cdots+j_{p}}\cdot\omega\otimes\partial_{J}.$

and the $(n-q)$-form $\beta$ goes to

$(-1)^{nq}(-1)^{k_{1}+\cdots+k_{q}}\cdot\omega\otimes\partial_{K}.$

The pairing $S_{Z}$ on ${\mathrm{D}}{\mathrm{R}}_{Z}\omega_{Z}$ takes those
two sections to

(2.10)
$(-1)^{n(p+q)}(-1)^{\left(j_{1}+\cdots+j_{p}\right)+\left(k_{1}+\cdots+k_{q}\right)}S(\omega,\omega)\otimes\partial_{J}\wedge\bar{\partial}_{K}$

where $S_{Z}$ is defined in (2.4). Now
$S_{Z}(\omega,\omega)=D_{Z}\omega\wedge\bar{\omega},$ where $D$ is the
distribution

$D_{Z}=\frac{\varepsilon(n+1)}{(2\pi\sqrt{-1})^{n}}\int_{Z}$

Under the isomorphism in (2.8) the section (2.10) therefore goes to

$(-1)^{np}dx_{\bar{J}}\wedge\overline{dx}_{\bar{K}}\otimes
D_{Z}=(-1)^{n(\operatorname{deg}\alpha-n)}\alpha\wedge\bar{\beta}\otimes
D_{Z}$

The formula we have just derived also works for smooth forms, of course. In
other words, the same formula can be used to extend (2.9) to a pairing on the
de Rham complex of smooth forms. The resulting pairings on cohomology are,
assuming $Z$ is compact

(2.11)
$H^{n+k}(Z,\mathbb{C})\otimes\overline{H^{n-k}(Z,\mathbb{C})}\rightarrow\mathbb{C},\quad(\alpha,\beta)\mapsto(-1)^{n(\operatorname{deg}\alpha-n)}\frac{\varepsilon(n+1)}{(2\pi\sqrt{-1})^{n}}\int_{Z}\alpha\wedge\bar{\beta},$

which coincides with the pairing (2.7) precisely.

###  2.4. Polarized bigraded Hodge-Lefschetz structures

In the paper, what we really consider is the degeneration of “variation of
Hodge-Lefschetz structures” of a family of compact Kähler manifolds. As it
turns out the limit of the degeneration is a bigraded Hodge-Lefschetz
structure. We begin to define polarized bigraded Hodge-Lefschetz structures.
Similarly to the case of
${\mathfrak{s}}{\mathfrak{l}}_{2}({\mathbb{C}})$-representation, a
${\mathfrak{s}}{\mathfrak{l}}_{2}({\mathbb{C}})\times{\mathfrak{s}}{\mathfrak{l}}_{2}({\mathbb{C}})$-representation
is a bigraded vector space $V=\bigoplus_{\ell,k\in{\mathbb{Z}}}V_{\ell,k}$
satisfying the following three equivalent conditions:

  1. (1)

each bigraded piece $V_{\ell,k}$ is the $\ell$-th eigenspace of
${\mathsf{H}}_{1}$ and $k$-th eigenspace of ${\mathsf{H}}_{2}$;

  2. (2)

for each $\ell,k\in{\mathbb{Z}}$ we have
${\mathsf{X}}_{1}:V_{\ell,k}\rightarrow V_{\ell+2,k}$ and
${\mathsf{X}}_{2}:V_{\ell,k}\rightarrow V_{\ell,k+2}$ plus isomorphisms

${\mathsf{X}}^{\ell}_{1}:V_{-\ell,k}\rightarrow V_{\ell,k}\text{ and
}{\mathsf{X}}^{k}_{2}:V_{\ell,-k}\rightarrow V_{\ell,k};$

  3. (3)

for each $\ell,k\in{\mathbb{Z}}$ we have
${\mathsf{Y}}_{1}:V_{\ell,k}\rightarrow V_{\ell-2,k}$ and
${\mathsf{Y}}_{2}:V_{\ell,k}\rightarrow V_{\ell,k-2}$ plus the isomorphism

${\mathsf{Y}}^{\ell}_{1}:V_{\ell,k}\rightarrow V_{-\ell,k}\text{ and
}{\mathsf{Y}}^{k}_{2}:V_{\ell,k}\rightarrow V_{\ell,-k}.$

A bigraded Hodge-Lefschetz structure of central weight $n$ is a
${\mathfrak{s}}{\mathfrak{l}}_{2}({\mathbb{C}})\times{\mathfrak{s}}{\mathfrak{l}}_{2}({\mathbb{C}})$-representation
$V=\bigoplus_{\ell,k\in{\mathbb{Z}}}V_{\ell,k}$ with two filtrations
$F^{\bullet}V$ and $G^{\bullet}V$ such that

  1. (1)

the bifiltered vector space
$(V_{\ell,k},F^{\bullet}V_{\ell,k},G^{\bullet}V_{\ell,k})$ is a Hodge
structure of weight $n+\ell+k$;

  2. (2)

the two operators
${\mathsf{X}}_{1}:(V_{\ell,k},F^{\bullet},G^{\bullet})\rightarrow(V_{\ell+2,k},F^{\bullet+1},G^{\bullet+1})$
and
${\mathsf{X}}_{2}:(V_{\ell,k},F^{\bullet},G^{\bullet})\rightarrow(V_{\ell,k+2},F^{\bullet+1},G^{\bullet+1})$
are morphisms of Hodge structures such that

${\mathsf{X}}^{\ell}_{1}:(V_{-\ell,k},F^{\bullet},G^{\bullet})\rightarrow(V_{\ell,k},F^{\bullet},G^{\bullet})(\ell)\quad\text{and}\quad{\mathsf{X}}^{k}_{2}:(V_{\ell,-k},F^{\bullet},G^{\bullet})\rightarrow(V_{\ell,k},F^{\bullet},G^{\bullet})(k)$

are isomorphisms of Hodge structures.

  3. (3)

the two operators
${\mathsf{Y}}_{1}:(V_{\ell,k},F^{\bullet},G^{\bullet})\rightarrow(V_{\ell-2,k},F^{\bullet-1},G^{\bullet-1})$
and
${\mathsf{Y}}_{2}:(V_{\ell,k},F^{\bullet},G^{\bullet})\rightarrow(V_{\ell,k-2},F^{\bullet-1},G^{\bullet-1})$
are morphisms of Hodge structures such that

${\mathsf{Y}}^{\ell}_{1}:(V_{\ell,k},F^{\bullet},G^{\bullet})\rightarrow(V_{-\ell,k},F^{\bullet},G^{\bullet})(-\ell)\quad\text{and}\quad{\mathsf{Y}}^{k}_{2}:(V_{\ell,k},F^{\bullet},G^{\bullet})\rightarrow(V_{\ell,-k},F^{\bullet},G^{\bullet})(-k)$

are isomorphisms of Hodge structures.

A polarization on a bigraded Hodge-Lefschetz structure
$V=\bigoplus_{\ell,k\in{\mathbb{Z}}}V_{\ell,k}$ of central weight $n$ is a
hermitian symmetric pairing
$S:V\otimes_{\mathbb{C}}\overline{V}\rightarrow{\mathbb{C}}$ such that

  1. (1)

the restriction
$S|_{V_{\ell,k}\otimes_{\mathbb{C}}\overline{V_{i,j}}}:V_{\ell,k}\otimes_{\mathbb{C}}\overline{V_{i,j}}\rightarrow{\mathbb{C}}$
vanishies except for $\ell=-i$ and $k=-j$;

  2. (2)

$S({\mathsf{X}}_{1}-,-)=S(-,{\mathsf{X}}_{1}-)$ and
$S(-,{\mathsf{Y}}_{2}-)=S({\mathsf{Y}}_{2}-,-)$;

  3. (3)

$S_{\ell,k}({\mathsf{X}}^{\ell}_{1}-,(-{\mathsf{Y}}_{2})^{k}-)$ is a
polarization on the bi-primitive part
$P_{-\ell,k}=\ker{\mathsf{X}}^{\ell+1}_{1}\cap\ker{\mathsf{Y}}^{k+1}_{2}\cap
V_{-\ell,k}$, or equivalently,
$S_{\ell,k}(-,{\mathsf{w}}_{1}{\mathsf{w}}_{2}-)$ is a polarization on
$V_{\ell,k}$, where $S_{\ell,k}$ is the restriction of $S$ on
$V_{\ell,k}\otimes\overline{V_{-\ell,k}}$ and
${\mathsf{w}}_{i}=e^{{\mathsf{X}}_{i}}e^{-{\mathsf{Y}}_{i}}e^{{\mathsf{X}}_{i}}$
for $i=1,2$.

This is the practical definition because in the later application
${\mathsf{X}}_{1}$ will be the $2\pi\sqrt{-1}L$ and ${\mathsf{Y}}_{2}$ will
be, up to a scalar, the logarithmic of the monodromy for the degeneration.
Similiarly to the case of Hodge-Lefschetz structure, we have a simpler
definition.

######  Theorem 2.10.

A polarized bigraded Hodge-Lefschetz structure of central weight $n$ on a
filtered bigraded vector space $(V=\bigoplus_{\ell,k}V_{\ell,k},F^{\bullet}V)$
is uniquely determined by the following:

  1. $(\mathrm{pbHL1})$

for every $\ell,k\in{\mathbb{Z}}$ we have two operators
${\mathsf{X}}_{1}:(V_{\ell,k},F^{\bullet})\rightarrow(V_{\ell+2,k},F^{\bullet+1})$
and
${\mathsf{Y}}_{2}:(V_{\ell,k}F^{\bullet})\rightarrow(V_{\ell,k-2},F^{\bullet-1})$
such that

${\mathsf{X}}^{\ell}_{1}:F^{\bullet}V_{-\ell,k}\rightarrow
F^{\bullet+\ell}V_{\ell,k}\quad\text{and}\quad{\mathsf{Y}}^{k}_{2}:F^{\bullet}V_{\ell,k}\rightarrow
F^{\bullet-k}V_{\ell,-k}\text{ are isomorphisms};$

  2. $(\mathrm{pbHL2})$

a collection of Hermitian pairings
$S_{\ell,k}:V_{\ell,k}\otimes_{\mathbb{C}}\overline{V_{-\ell,-k}}\rightarrow{\mathbb{C}}$
such that

$S_{\ell,k}({\mathsf{X}}_{1}-,-)=S_{\ell+2,k}(-,{\mathsf{X}}_{1}-)\quad\text{and}\quad
S_{\ell,k}(-,{\mathsf{Y}}_{2}-)=S_{\ell,k-2}({\mathsf{Y}}_{2}-,-);$

  3. $(\mathrm{pbHL3})$

the triple
$\left(P_{-\ell,k},F^{\bullet}P_{-\ell,k},S\circ\left({\mathsf{X}}^{\ell}_{1}\otimes\left(-{\mathsf{Y}}_{2}\right)^{k}\right)\right)$
is a polarized Hodge structure of weight $n-\ell+k$ where
$F^{\bullet}P_{-\ell,k}=\ker{\mathsf{X}}^{\ell}_{1}\cap\ker{\mathsf{Y}}^{k}_{2}\cap
F^{\bullet}V_{-\ell,k}$ is the bi-primtive part.

Then the Hodge structure on $V_{j,k}$ can be described as: for $p+q=n+j+k$

$\displaystyle V^{p,q}_{j,k}$ $\displaystyle=\\{a\in
F^{p}V_{j,k}:S_{j,k}(a,b)=0\text{ for all }b\in F^{p-j-k+1}V_{-j-k}\\},$
$\displaystyle G^{q}V_{j,k}$ $\displaystyle=\\{a\in
V_{j,k}:S_{j,k}(a,b)=0\text{ for all }b\in F^{n-q+1}V_{-j,-k}\\}.$

The proof is simple and is left to the reader. Later when we construct the
limiting mixed Hodge structure, the polarized bigraded Hodge-Lefschetz
structure naturally comes up from the first page of weight spectral sequence
associated to a mixed Hodge complex. Modeled on the properties of the
differential of spectral sequence we give the following definition:

A differential of a polarized bigraded Hodge Lefschetz structure
$(V,F^{\bullet},{\mathsf{X}}_{1},{\mathsf{Y}}_{2},S)$ is a linear map
$d:V\rightarrow V$ such that

  1. (1)

$d:(V_{j,k},F^{\bullet})\rightarrow(V_{j+1,k-1},F^{\bullet})$ and $d^{2}=0$;

  2. (2)

$d$ is skew-symmetrc with respect to $S$, i.e., $S(d-,-)+S(-,d-)=0$;

  3. (3)

$[{\mathsf{X}}_{1},d]=0$ and $[{\mathsf{Y}}_{2},d]=0$.

######  Remark 2.11.

In fact, the above three conditions imply that $d$ is a morphism of Hodge
structures $d:V^{p,q}_{j,k}\rightarrow V^{p,q}_{j+1,k-1}$. A vector $a\in
G^{q}V_{j,k}$ means that $S(a,b)=0$ for all $b\in F^{n-q+1}V_{-j,-k}$. Then
$S(da,b)=S(a,db)=0$ for all $b\in F^{n-q+1}V_{-j-1,-k+1}$, indicating $da$
belongs to $G^{q}V_{j+1,k-1}$.

The main result of this subsection is the following version of Deligne’s
lemma, showed by Guillén and Navarro Aznar.

######  Theorem 2.12 ([GNA90, (4.5)]).

Tthe cohomology $\ker d/\mathrm{im}\,d$ of a polarized differential bigraded
Hodge-Lefschetz struture is again a polarized bigraded Hodge-Lefschetz
structure.

###### Proof.

Let $C:V\rightarrow V$ be the operator that acts as $(-1)^{q}$ on the subspace
$V_{j,k}^{p,q}$ in the Hodge decomposition of each $V_{j,k}.$ Since $d$ is a
morphism of Hodge structures, we have $[d,C]=0$. The fact that $S$ is a
polarization means that the Hermitian pairing

$h^{+}:V\otimes_{\mathbb{C}}\overline{V}\rightarrow\mathbb{C},\quad
h^{+}(a,b)=S\left(Ca,{\mathsf{w}}_{1}{\mathsf{w}}_{2}b\right)$

is positive-definite on $V$. Let $d^{*}$ be the adjoint of $d$ with respect to
$h^{+}$. Fix $a\in V_{j,k}$ and $b\in V_{j,k},$:

$\displaystyle h^{+}(da,b)$
$\displaystyle=S\left(Cda,{\mathsf{w}}_{1}{\mathsf{w}}_{2}b\right)=S\left(dCa,{\mathsf{w}}_{1}{\mathsf{w}}_{2}b\right)$
$\displaystyle=-S\left(Ca,d{\mathsf{w}}_{1}{\mathsf{w}}_{2}b\right)=-S\left(Ca,{\mathsf{w}}_{1}{\mathsf{w}}_{2}\cdot{\mathsf{w}}_{2}^{-1}{\mathsf{w}}_{1}^{-1}d{\mathsf{w}}_{1}{\mathsf{w}}_{2}\cdot
b\right)=h^{+}\left(a,d^{*}b\right),$

i.e. the adjoint
$d^{*}=-{\mathsf{w}}_{2}^{-1}{\mathsf{w}}_{1}^{-1}d{\mathsf{w}}_{1}{\mathsf{w}}_{2}$.

In addition to the two relations in the definition of differential

$\left[{\mathsf{X}}_{1},d\right]=0\quad\text{ and
}\quad\left[{\mathsf{Y}}_{2},d\right]=0$

we obtain from the grading another two relations

$\left[{\mathsf{H}}_{1},d\right]=d\quad\text{ and
}\quad\left[{\mathsf{H}}_{2},d\right]=-d.$

With respect to the
$\mathfrak{sl}_{2}(\mathbb{C})\times\mathfrak{sl}_{2}(\mathbb{C})$-action on
$\operatorname{End}_{\mathbb{C}}(V),$ the element $d$ therefore has weight
$(+1,-1),$ and is primitive with respect to the action by ${\mathsf{Y}}_{1}$
and ${\mathsf{X}}_{2}.$ Define

$d_{1}=\left[{\mathsf{Y}}_{1},d\right]\quad\text{ and }\quad
d_{2}=-\left[{\mathsf{X}}_{2},d\right].$

The reason for the minus sign is that we have
$\left[{\mathsf{Y}}_{2},d\right]=0$. Then $d_{1}$ has weight $(-1,-1),$ and is
primitive with respect to the action by ${\mathsf{X}}_{1}$ and
${\mathsf{X}}_{2};$ this gives

$\begin{array}[]{lll}{\left[{\mathsf{H}}_{1},d_{1}\right]=-d_{1},}&{\left[{\mathsf{X}}_{1},d_{1}\right]=d},&\left[{\mathsf{Y}}_{1},d_{1}\right]=0,\quad{\mathsf{w}}_{1}d_{1}{\mathsf{w}}_{1}^{-1}=d\\\
{\left[{\mathsf{H}}_{2},d_{1}\right]=-d_{1},}&&\left[{\mathsf{Y}}_{2},d_{1}\right]=0.\end{array}$

Similarly, $d_{2}$ has weight $(+1,+1),$ and therefore

$\begin{array}[]{ll}{\left[{\mathsf{H}}_{2},d_{2}\right]=d_{2},}&{\left[{\mathsf{X}}_{2},d_{2}\right]=0,\quad\left[{\mathsf{Y}}_{2},d_{2}\right]=-d,\quad{\mathsf{w}}_{2}d_{2}{\mathsf{w}}_{2}^{-1}=d}\\\
{\left[{\mathsf{H}}_{1},d_{2}\right]=d_{2},}&{\left[{\mathsf{X}}_{1},d_{2}\right]=0}.\end{array}$

Therefore,
$d^{*}=-\left[{\mathsf{Y}}_{1},d_{2}\right]=\left[{\mathsf{X}}_{2},d_{1}\right]\in\operatorname{End}_{\mathbb{C}}V$.
It has weight $(-1,+1),$ and is primitive with respect to ${\mathsf{X}}_{1}$
and ${\mathsf{Y}}_{2}$. From this, and the identities we already have, we
deduce the following set of relations:

$\begin{array}[]{l}{\left[{\mathsf{H}}_{1},d^{*}\right]=-d^{*},\quad\left[{\mathsf{X}}_{1},d^{*}\right]=d_{2},\quad\left[{\mathsf{Y}}_{1},d^{*}\right]=0,\quad{\mathsf{w}}_{1}d^{*}{\mathsf{w}}_{1}^{-1}=-d_{2}}\\\
{\left[{\mathsf{H}}_{2},d^{*}\right]=d^{*},\quad\left[{\mathsf{X}}_{2},d^{*}\right]=0,\quad\left[{\mathsf{Y}}_{2},d^{*}\right]=-d_{1},\quad{\mathsf{w}}_{2}d^{*}{\mathsf{w}}_{2}^{-1}=-d_{1}}.\end{array}$

We can check that the (formal) Laplace operator

$\Delta=dd^{*}+d^{*}d\in\operatorname{End}_{\mathbb{C}}(V)$

is invariant under the action of
$\mathfrak{sl}_{2}(\mathbb{C})\times\mathfrak{sl}_{2}(\mathbb{C}).$ For
example,

$\begin{array}[]{l}{\left[{\mathsf{X}}_{1},dd^{*}\right]={\mathsf{X}}_{1}dd^{*}-dd^{*}{\mathsf{X}}_{1}=d{\mathsf{X}}_{1}d^{*}-d\left({\mathsf{X}}_{1}d^{*}+d_{2}\right)=-dd_{2}}\\\
{\left[{\mathsf{X}}_{1},d^{*}d\right]={\mathsf{X}}_{1}d^{*}d-d^{*}d{\mathsf{X}}_{1}=\left(d^{*}{\mathsf{X}}_{1}-d_{2}\right)d-d^{*}{\mathsf{X}}_{1}d=-d_{2}d}\end{array}$

from which we conclude, using $d^{2}=0,$ that

$\left[{\mathsf{X}}_{1},\Delta\right]=-\left(dd_{2}+d_{2}d)=-(d\left(d{\mathsf{X}}_{2}-{\mathsf{X}}_{2}d\right)+\left(d{\mathsf{X}}_{2}-{\mathsf{X}}_{2}d\right)d\right)=0$

The other three commutators can be checked similarly. On the other hand,
$\Delta$ is also a morphism of Hodge structures: the reason is that

$d:V_{j,k}\rightarrow V_{j+1,k-1},\quad{\mathsf{Y}}_{1}:V_{j,k}\rightarrow
V_{j-2,k}(-1),\quad{\mathsf{X}}_{2}:V_{j,k}\rightarrow V_{j,k+2}(1)$

are all morphisms of Hodge structures, and $\Delta$ is obtained by composing
them in some order. It follows that $\ker\Delta\subseteq V$ is a bigraded
Hodge-Lefschetz structure, polarized by the restriction of $S$. Because of the
canonical isomorphism $\ker\Delta\simeq\ker d/\mathrm{im\,}d$ as bigraded
Hodge-Lefschetz structures, the induced pairing by $S$ on $\ker
d/\mathrm{im\,}d$ is also a polarization. This concludes the proof. ∎

##  3\. Log relative de Rham complex

Let $f:X\rightarrow\Delta$ be a proper holomorphic morphism smooth away from
the origin whose central fiber $Y$ is simple normal crossing but not
necessarily reduced. Assume $X$ is Kähler of dimension $n+1$ and $Y=\sum_{i\in
I}e_{i}Y_{i}$ where $Y_{i}$’s are smooth components and $I$ a finite index
set. Let $t$ be a parameter on $\Delta$ and $z_{0},z_{1},z_{2},...,z_{n}$ a
local coordinate system on $X$ such that $t=z^{e_{0}}_{0}z^{e_{1}}_{1}\cdots
z^{e_{k}}_{k}$ such that $e_{0},e_{1},...,e_{k}\geq 1$. Then we have
$\Omega_{\Delta}(\log 0)={\mathscr{O}}_{\Delta}\cdot\frac{dt}{t}$ and
$\Omega_{X}(\log Y)$ is locally generated by

$e_{0}\frac{dz_{0}}{z_{0}},e_{1}\frac{dz_{1}}{z_{1}},...,e_{k}\frac{dz_{k}}{z_{k}},dz_{k+1},dz_{k+2},...,dz_{n}$

over ${\mathscr{O}}_{X}$. Denote by $\xi_{0},\xi_{1},...,\xi_{n}$ the image of
the above generators in $\Omega_{X/\Delta}(\log Y)$, respectively. As a
quotient of $\Omega_{X}(\log Y)$, the sheaf $\Omega_{X/\Delta}(\log Y)$ is
generated by $\xi_{0},\xi_{1},...,\xi_{n}$, but under the relation

$\xi_{0}+\xi_{1}+\cdots+\xi_{n}=0\quad\text{because}\quad
f^{*}\frac{dt}{t}=e_{0}\frac{dz_{0}}{z_{0}}+e_{1}\frac{dz_{1}}{dz_{1}}+\cdots+e_{k}\frac{dz_{k}}{z_{k}}.$

Let ${\mathscr{T}}_{X/\Delta}(\log Y)$ be the dual bundle of
$\Omega_{X/\Delta}(\log Y)$. It is a subsheaf of ${\mathscr{T}}_{X}$,
generated by

(3.12) $D_{i}=\left\\{\begin{aligned}
\frac{1}{e_{i}}z_{i}\partial_{i}-\frac{1}{e_{0}}z_{0}\partial_{0}&,&1\leq
i\leq k\\\ \partial_{i}&,&i>k,\end{aligned}\right.$

where $\partial_{i}$ is the local section of ${\mathscr{T}}_{X}$ dual to
$dz_{i}$ in $\Omega_{X}$. It follows that $D_{1},D_{2},...,D_{n}$ is the dual
frame of $\xi_{1},\xi_{2},...,\xi_{n}$.

###  3.1. A “log connection”

We shall construct an operator in
${\mathrm{E}}{\mathrm{n}}{\mathrm{d}\,}_{{\mathbf{D}}^{b}(\Delta,{\mathbb{C}})}\left(Rf_{*}\Omega^{\bullet+n}_{X/\Delta}(\log
Y)\right)$ which should be regarded a “log connection”. Note that we have the
following short exact sequence of ${\mathscr{O}}_{X}$-modules

$0\rightarrow f^{*}\Omega_{\Delta}(\log
0)\otimes\Omega^{\bullet+n}_{X/\Delta}(\log
Y)\rightarrow\Omega^{\bullet+n+1}_{X}(\log
Y)\rightarrow\Omega^{\bullet+n+1}_{X/\Delta}(\log Y)\rightarrow 0.$

Under the identification $\frac{dt}{t}\wedge:{\mathscr{O}}_{X}\rightarrow
f^{*}\Omega_{\Delta}(\log 0)$, the above short exact sequence becomes

${0}$${\Omega^{\bullet+n}_{X/\Delta}(\log Y)}$${\Omega^{\bullet+n+1}_{X}(\log
Y)}$${\Omega^{\bullet+n+1}_{X/\Delta}(\log
Y)}$${0.}$$\scriptstyle{\frac{dt}{t}\wedge}$

Here, the morphism $\frac{dt}{t}\wedge:\Omega^{k}_{X/\Delta}(\log
Y)\rightarrow\Omega^{k+1}_{X}(\log Y)$ works as
$[\alpha]\mapsto\frac{dt}{t}\wedge\alpha$ which does not depend on the
representative of $[\alpha]$. Let
${\mathrm{C}}{\mathrm{o}}{\mathrm{n}}{\mathrm{e}}^{\bullet}=\Omega^{\bullet+n}_{X}(\log
Y)\oplus\Omega^{\bullet+n}_{X/\Delta}(\log Y)$ be the mapping cone of
$\frac{dt}{t}\wedge:\Omega^{\bullet+n-1}_{X/\Delta}(\log
Y)\rightarrow\Omega^{\bullet+n}_{X}(\log Y)$. In our convention, the
differential $\delta$ of the mapping cone works as
$\delta(\alpha,[\beta])=\left((-1)^{n}d\alpha+\frac{dt}{t}\wedge\beta,(-1)^{n}d[\beta]\right)$,
where $d$ is the usual exterior derivative on $\Omega^{{}\bullet}_{X}(\log Y)$
and by abuse of notation, also $d$ denotes the induced differential on
$\Omega^{{}\bullet}_{X/\Delta}(\log Y)$. Then we have the following diagram:

(3.13)
${{\mathrm{C}}{\mathrm{o}}{\mathrm{n}}{\mathrm{e}}^{\bullet}}$${\Omega^{\bullet+n}_{X/\Delta}(\log
Y)}$${\Omega^{\bullet+n}_{X/\Delta}(\log
Y)}$$\scriptstyle{q}$$\scriptstyle{p}$$\scriptstyle{p\circ q^{-1}}$

where
$q:{\mathrm{C}}{\mathrm{o}}{\mathrm{n}}{\mathrm{e}}^{\bullet}\rightarrow\Omega^{\bullet+n}_{X/\Delta}(\log
Y),(\alpha,[\beta])\mapsto[\alpha]$ is a quasi-isomorphism and $p$ is the
second projection. Therefore we have the morphism $p\circ q^{-1}$ in
${\mathrm{E}}{\mathrm{n}}{\mathrm{d}\,}_{{\mathbf{D}}^{b}(X,{\mathbb{C}})}\left(\Omega^{{}n+\bullet}_{X/\Delta}(\log
Y)\right)$. For any local section $g\in{\mathscr{O}}_{\Delta}$, the
multiplication by $g$ is an endomorphism of
$\Omega^{\bullet+n}_{X/\Delta}(\log Y)$ because it is
$f^{-1}{\mathscr{O}}_{\Delta}$-linear.

######  Lemma 3.1.

The operator $\nabla=(-1)^{n-1}p\circ q^{-1}$ satisfies
$[\nabla,g]=tg^{\prime}$ in
${\mathrm{E}}{\mathrm{n}}{\mathrm{d}\,}_{{\mathbf{D}}^{b}(X,{\mathbb{C}})}\left(\Omega^{\bullet+n}_{X/\Delta}(\log
Y)\right)$, where $g^{\prime}$ denotes the derivative of
$g\in{\mathscr{O}}_{\Delta}$.

###### Proof.

It is equivalent to show that $[p\circ q^{-1},g]=(-1)^{n}tg^{\prime}$. Define
$g(\alpha,[\beta])=(g\alpha,g[\beta]+(-1)^{n-1}tg^{\prime}[\alpha])$ for any
$(\alpha,[\beta])\in{\mathrm{C}}{\mathrm{o}}{\mathrm{n}}{\mathrm{e}}^{\bullet}$
and $g\in f^{-1}{\mathscr{O}}_{\Delta}$. We shall show that $g$ is an
endomorphism of ${\mathrm{C}}{\mathrm{o}}{\mathrm{n}}{\mathrm{e}}^{\bullet}$,
i.e., $g\delta(\alpha,[\beta])=\delta g(\alpha,[\beta])$. This follows from
that

$\displaystyle g\delta(\alpha,[\beta])$
$\displaystyle=g\left((-1)^{n}d\alpha+\frac{dt}{t}\wedge\beta,(-1)^{n}d[\beta]\right)$
$\displaystyle=\left((-1)^{n}gd\alpha+g\frac{dt}{t}\wedge\beta,(-1)^{n}gd[\beta]-tg^{\prime}d[\alpha]\right)$

and

$\displaystyle\delta g(\alpha,[\beta])$
$\displaystyle=\delta\left(g\alpha,g[\beta]+(-1)^{n-1}tg^{\prime}[\alpha]\right)$
$\displaystyle=\left((-1)^{n}dg\alpha+\frac{dt}{t}\wedge(g\beta+(-1)^{n-1}tg^{\prime}\alpha),(-1)^{n}d(g[\beta]+(-1)^{n-1}tg^{\prime}[\alpha])\right)$
$\displaystyle=\left((-1)^{n}gd\alpha+g\frac{dt}{t}\wedge\beta,(-1)^{n}gd[\beta]-tg^{\prime}d[\alpha]\right).$

It is easy to see that $g\circ q=q\circ g$ so that $q^{-1}\circ g=g\circ
q^{-1}$. Therefore,

$[p\circ q^{-1},g]=p\circ q^{-1}\circ g-g\circ p\circ q^{-1}=[p,g]\circ
q^{-1}$

But
$[p,g](\alpha,[\beta])=p(g\alpha,g[\beta]+(-1)^{n-1}tg^{\prime}[\alpha])-g[\beta]=(-1)^{n-1}tg^{\prime}[\alpha]$.
It follows that

$[p\circ q^{-1},g]\circ
q(\alpha,[\beta])=[p,g](\alpha,[\beta])=(-1)^{n-1}tg^{\prime}\circ
q(\alpha,[\beta]).$

By inverse $q$ we prove the statement. ∎

Because of the identification
$\frac{dt}{t}\wedge:{\mathscr{O}}_{\Delta}\rightarrow\Omega_{\Delta}(\log 0)$,
what we really get is a morphism in ${\mathbf{D}}^{b}(X,{\mathbb{C}})$

$\nabla:\Omega^{{}\bullet+n}_{X/\Delta}(\log Y)\rightarrow
f^{*}\Omega_{\Delta}(\log 0)\otimes\Omega^{{}\bullet+n}_{X/\Delta}(\log Y)$

such that $\nabla g=g\nabla+\frac{dt}{t}\otimes
tg^{\prime}\in{\mathrm{E}}{\mathrm{n}}{\mathrm{d}\,}_{{\mathbf{D}}^{b}(X,{\mathbb{C}})}(\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y))$ for any local section $g\in{\mathscr{O}}_{\Delta}$. Running the similar
construction, we obtain an induced ${\mathbb{C}}$-linear (in fact
$f^{-1}{\mathscr{O}}_{\Delta}$-linear) endomorphism $[\nabla]$ on
$\Omega^{\bullet+n}_{X/\Delta}(\log Y)|_{Y}$ in
${\mathbf{D}}^{b}(X,{\mathbb{C}})$ satisfying the following diagram.

${\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)}$${\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)}$${\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)|_{Y}}$${\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)[1]}$${\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)}$${\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)}$${\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)|_{Y}}$${\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)[1]}$$\scriptstyle{\nabla+1}$$\scriptstyle{t}$$\scriptstyle{\nabla}$$\scriptstyle{[\nabla]}$$\scriptstyle{\left(\nabla+1\right)[1]}$$\scriptstyle{t}$

Since $\Omega^{{}\bullet+n}_{X/\Delta}(\log Y)$ is
$f^{-1}{\mathscr{O}}_{\Delta}$-linear, each cohomolgy
$R^{k}f_{*}\Omega^{\bullet+n}_{X/\Delta}(\log Y)$ is a coherent
${\mathscr{O}}_{\Delta}$-module. Taking direct image, we get
${\mathbb{C}}$-linear morphisms between distinguished triangles in
${\mathbf{D}}^{b}_{\mathrm{coh}}(\Delta,{\mathscr{O}}_{\Delta})$:

(3.14) ${Rf_{*}\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)}$${Rf_{*}\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)}$${Rf_{*}\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)|_{Y}}$${Rf_{*}\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)[1]}$${Rf_{*}\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)}$${Rf_{*}\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)}$${Rf_{*}\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)|_{Y}}$${\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)[1]}$$\scriptstyle{Rf_{*}\nabla+1}$$\scriptstyle{t}$$\scriptstyle{Rf_{*}\nabla}$$\scriptstyle{Rf_{*}[\nabla]}$$\scriptstyle{Rf_{*}\left(\nabla+1\right)[1]}$$\scriptstyle{t}$

where the morphism

$Rf_{*}\nabla:Rf_{*}\Omega^{\bullet+n}_{X/\Delta}(\log Y)\rightarrow
Rf_{*}\Omega^{\bullet+n}_{X/\Delta}(\log Y)$

satisfies
$[Rf_{*}\nabla,g]=tg^{\prime}\in{\mathrm{E}}{\mathrm{n}}{\mathrm{d}\,}_{{\mathbf{D}}^{b}(\Delta,{\mathbb{C}})}\left(Rf_{*}\Omega^{\bullet+n}_{X/\Delta}(\log
Y)\right)$ for any local sections $g\in{\mathscr{O}}_{\Delta}$.

###  3.2. Residue

In the above situation, one should regard $Rf_{*}[\nabla]$ as the residue of
$Rf_{*}\nabla$. More generally, let ${\mathcal{F}}^{\bullet}$ be a complex of
${\mathscr{O}}_{\Delta}$-modules with a morphism
$\nabla\in{\mathrm{E}}{\mathrm{n}}{\mathrm{d}\,}_{{\mathbf{D}}^{b}(\Delta,{\mathbb{C}})}({\mathcal{F}}^{\bullet})$
such that $[\nabla,g]=tg^{\prime}$ for any $g\in{\mathscr{O}}_{\Delta}$. Let
$\mathcal{G}^{\bullet}$ be the mapping cone of
$t:{\mathcal{F}}^{\bullet}\rightarrow{\mathcal{F}}^{\bullet}$, which computes
to ${\mathcal{F}}^{\bullet}\otimes^{L}{\mathbb{C}}(0)$. Then by the axioms of
triangulated categories [HTT08], there exists an operator
$R\in{\mathrm{E}}{\mathrm{n}}{\mathrm{d}\,}_{{\mathbf{D}}^{b}(\Delta,{\mathbb{C}})}(\mathcal{G}^{\bullet})$
making the following diagram commute in
${\mathbf{D}}^{b}(\Delta,{\mathbb{C}})$.

${{\mathcal{F}}^{\bullet}}$${{\mathcal{F}}^{\bullet}}$${\mathcal{G}^{\bullet}={\mathcal{F}}^{\bullet}\otimes^{L}{\mathbb{C}}(0)}$${{\mathcal{F}}^{\bullet}[1]}$${{\mathcal{F}}^{\bullet}}$${{\mathcal{F}}^{\bullet}}$${\mathcal{G}^{\bullet}={\mathcal{F}}^{\bullet}\otimes^{L}{\mathbb{C}}(0)}$${{\mathcal{F}}^{\bullet}[1]}$$\scriptstyle{\nabla+1}$$\scriptstyle{t}$$\scriptstyle{\nabla}$$\scriptstyle{R}$$\scriptstyle{\left(\nabla+1\right)[1]}$$\scriptstyle{t}$

We call the operator $R$ a residue of $\nabla$. Note that the axioms of
triangulated categories cannot guarantee that the filling is unique. However,
the eigenvalues of $R_{\ell}$ only depends on $\nabla$, where $R_{\ell}$
denotes the induced operator on the cohomology
${\mathscr{H}}^{\ell}\left({\mathcal{F}}^{\bullet}\otimes^{L}{\mathbb{C}}(0)\right)$.
First, every object in ${\mathbf{D}}^{b}_{\mathrm{coh}}(\Delta,{\mathscr{O}})$
splits, meaning that
${\mathcal{F}}^{\bullet}\simeq\bigoplus_{\ell\in{\mathbb{Z}}}{\mathscr{H}}^{\ell}{\mathcal{F}}^{\bullet}[-\ell]$,
since there are no ${\mathrm{E}}{\mathrm{x}}{\mathrm{t}}^{i}$ for $i\geq 2$
between two coherent sheaves over a curve. It follows that the morphism
$\nabla$ breaks up into sum of morphism consisting of diagonal morphism
$\nabla_{\ell}:{\mathscr{H}}^{\ell}{\mathcal{F}}^{\bullet}[-\ell]\rightarrow{\mathscr{H}}^{\ell}{\mathcal{F}}^{\bullet}[-\ell]$
which is an actual log connection and off-diagonal morphism
${\mathscr{H}}^{\ell}{\mathcal{F}}^{\bullet}[-\ell]\rightarrow{\mathscr{H}}^{m}{\mathcal{F}}^{\bullet}[-m]$
but only for $\ell>m$. Thus the eigenvalues of $R_{\ell}$ are determined by
$\nabla_{\ell}$ and $\nabla_{\ell+1}$. When ${\mathcal{F}}^{\bullet}$ is a
locally free sheaf centered at degree zero and $\nabla$ is the usual log
connection. Then above definition coincides with the usual definition of the
residue of $\nabla$.

Returning to our case, the natural choice of a residue of $Rf_{*}\nabla$ is
$R=Rf_{*}[\nabla]$ because of the diagram (3.14): by the projection formula,
we have

$Rf_{*}\Omega^{\bullet+n}_{X/\Delta}(\log
Y)\mathbin{\mathop{\otimes}\displaylimits_{{\mathscr{O}}_{\Delta}}^{L}}{\mathbb{C}}(0)=Rf_{*}\left(\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)\mathbin{\mathop{\otimes}\displaylimits_{f^{-1}{\mathscr{O}}_{\Delta}}^{L}}f^{-1}{\mathbb{C}}(0)\right)=Rf_{*}\left(\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y)|_{Y}\right).$

Our main result concerning the relative log de Rham complex is the following.

######  Theorem 3.2.

The higher direct image $R^{\ell}f_{*}\Omega^{\bullet+n}_{X/\Delta}(\log Y)$
is locally free for each $\ell\in{\mathbb{Z}}$. Moreover, there exists a
canonical isomorphism for every $p\in\Delta$

$R^{\ell}f_{*}\Omega^{\bullet+n}_{X/\Delta}(\log
Y)\otimes{\mathbb{C}}(p)\simeq H^{\ell}(Y,\Omega^{\bullet+n}_{X/\Delta}(\log
Y)|_{X_{p}}),\quad\text{where }{\mathbb{C}}(p)\text{ is the residue filed at
}p.$

We first present two preliminary theorems.

######  Theorem 3.3.

The operator $R_{\ell}$ has eigenvalues in $[0,1)\cap{\mathbb{Q}}$ for each
$\ell\in{\mathbb{Z}}$.

###### Proof.

Later in $\S$4 (Theorem 4.19) we will show that in fact $[\nabla]$ satisfies
$p([\nabla])=0$ for

$p(\lambda)=\prod_{i\in I}\prod^{e_{i}-1}_{j=0}(\lambda-\frac{j}{e_{i}}).$

Hence so is $R^{\ell}f_{*}[\nabla]$ and this implies the eigenvalues are in
$[0,1)\cap{\mathbb{Q}}$.

Alternatively, by Grothendieck spectral sequence

$E^{p,q}_{2}=R^{p}f_{*}{\mathscr{H}}^{q}(\Omega^{\bullet+n}_{X/\Delta}(\log
Y)|_{Y})\Rightarrow R^{p+q}f_{*}(\Omega^{\bullet+n}_{X/\Delta}(\log Y)|_{Y}),$

it suffices to show that the induced operator
$R^{p}f_{*}{\mathscr{H}}^{q}[\nabla]$ on
$R^{p}f_{*}{\mathscr{H}}^{q}\Omega^{\bullet+n}_{X/\Delta}(\log Y)|_{Y}$ has
eigenvalues in $[0,1)\cap{\mathbb{Q}}$ for each $q\in{\mathbb{Z}}$ since
$E^{p,q}_{\infty}$ is a sub-quotient of $E^{p,q}_{2}$. The following is proved
by Steenbrink [Ste76, Proposition 1.13]:

######  Lemma 3.4.

The stalk of ${\mathscr{H}}^{q}\Omega^{\bullet+n}_{X/\Delta}(\log Y)|_{Y}$ at
a point $u$ is generated by the germs
$(t^{\frac{a}{e}}\xi_{i_{1}}\wedge\xi_{i_{2}}\wedge\cdots\xi_{i_{q+n}})_{u}$
for all $0\leq a<e$ and all $0\leq i_{1},i_{2},...,i_{q+n}\leq n$ over the
ring ${\mathbb{C}}\\{t^{\frac{1}{e}}\\}/t{\mathbb{C}}\\{t^{\frac{1}{e}}\\}$
where $e$ is the $\gcd$ of $e_{0},e_{1},...,e_{k}$ and
${\mathbb{C}}\\{t^{\frac{1}{e}}\\}$ is the ring of convergent power series
with the variable $t^{\frac{1}{e}}$.

We will elaborate the proof of the lemma later. Temporarily admitting the
lemma, then

${\mathscr{H}}^{q}[\nabla]_{u}(t^{\frac{a}{e}}\xi_{i_{1}}\wedge\xi_{i_{2}}\wedge\cdots\xi_{i_{q+n}})_{u}=(\frac{a}{e}t^{\frac{a}{e}}\xi_{i_{1}}\wedge\xi_{i_{2}}\wedge\cdots\wedge\xi_{i_{q+n}})_{u},$

meaning that the eigenvalues of ${\mathscr{H}}^{q}[\nabla]$ are
$0,\frac{1}{e},\frac{2}{e},...,\frac{e-1}{e}\in[0,1)\cap{\mathbb{Q}}$ in a
neighborhood of $u$. This implies that there exists an open neighborhood $U$
containing $u$ and a polynomial $p_{U}(\lambda)$ whose roots are in
$[0,1)\cap{\mathbb{Q}}$ such that
$p_{U}\left({\mathscr{H}}^{q}[\nabla]\right)=0$ over $U$. By the properness of
$Y$, we can take a finite open covering $\mathcal{U}=\\{U_{i}\\}$ of $Y$ such
that
$p({\mathscr{H}}^{q}[\nabla])=\prod_{i}p_{U_{i}}({\mathscr{H}}^{q}[\nabla])=0$.
It follows that $p(R^{p}f_{*}{\mathscr{H}}^{q}[\nabla])=0$, meaning
eigenvalues of $R^{p}f_{*}{\mathscr{H}}^{q}[\nabla]$ in
$[0,1)\cap{\mathbb{Q}}$. ∎

###### Proof of Lemma 3.4.

We will actually prove the original statement of [Ste76, Proposition 1.13]
that, in the same notations as in the lemma, the stalk at a point $u$ of
${\mathscr{H}}^{q}\Omega^{\bullet+n}_{X/\Delta}(\log Y)$ is generated by germs

$\left(t^{\frac{a}{e}}\xi_{i_{1}}\wedge\xi_{i_{2}}\wedge\cdots\xi_{i_{q+n}}\right)_{u}$

for all $a\in{\mathbb{Z}}_{\geq 0}$ and all tuples $0\leq
i_{1},i_{2},...,i_{q+n}\leq n$ over ${\mathbb{C}}\\{t^{\frac{1}{e}}\\}$ The
lemma is a direct corollary.

The complex of stalks $\Omega^{\bullet+n}_{X/\Delta}(\log Y)_{u}$ can be
identified with the Kozul complex of operators $D_{1},D_{2},...,D_{n}$ on
${\mathscr{O}}_{X,u}$ putting in degree $-n,-n+1,...,0$. Define
$G^{j}\Omega^{\ell}_{X/\Delta}(\log Y)_{u}$ to be the submodules of
$\Omega^{\ell}_{X/\Delta}(\log Y)_{u}$ spanned by the germs

$\xi_{i_{1}}\wedge\xi_{i_{2}}\wedge\cdots\wedge\xi_{i_{\ell}}\quad\text{ such
that }\\#\\{m:i_{m}\leq k\\}\geq j.$

Then $\\{G^{\ell}\Omega^{\bullet}_{X/\Delta}(\log
Y)_{u}\\}_{\ell\in{\mathbb{Z}}}$ is a decreasing filtration of
$\Omega^{\bullet}_{X/\Delta}(\log Y)_{u}$. The associated spectral sequence
has
$E^{r,\bullet}_{0}={\mathrm{g}}{\mathrm{r}}^{r}_{G}\Omega^{r+\bullet}_{X/\Delta}(\log
Y)_{u}$. Notice that
${\mathrm{g}}{\mathrm{r}}^{r}_{G}\Omega^{r+\bullet}_{X/\Delta}(\log Y)_{u}$
can be identified with direct sums of Koszul complex of operators
$D_{k+1},D_{k+2},...,D_{n}$ on ${\mathscr{O}}_{X,u}$, so
$E^{r,\ell}_{1}=H^{r+\ell}({\mathrm{g}}{\mathrm{r}}^{r}_{G}\Omega^{\bullet}_{X/\Delta}(\log
Y))=0$ for $\ell\neq 0$ and $E^{r,0}_{1}$ is spanned by germs

$\xi_{i_{1}}\wedge\xi_{i_{2}}\wedge\cdots\wedge\xi_{i_{\ell}}\text{ such that
}\\#\\{i_{m}\leq k\\}=j$

over ${\mathbb{C}}\\{z_{0},z_{1},...,z_{k}\\}$, thanks to the usual Poincaré
lemma. Consequently, the spectral sequence degenerates at $E_{2}$ with
$E^{r,0}_{2}={\mathscr{H}}^{r}(\Omega^{\bullet}_{X/\Delta}(\log Y))_{u}$. Now
$E^{\bullet,0}_{1}$ is the Koszul complex of operators $D_{1},D_{2},...,D_{k}$
on ${\mathbb{C}}\\{z_{0},z_{1},...,z_{k}\\}$. Because each $D_{i}$ for $0\leq
i\leq k$ is a homogenous differential operator, $E_{2}$ can be computed
monomial by monomial.

For simplicity let
$\xi_{i_{1},i_{2},...,i_{r}}=\xi_{i_{1}}\wedge\xi_{i_{2}}\wedge\cdots\wedge\xi_{i_{r}}$.
Now I claim that a cocycle

$v=\sum_{i_{1}<i_{2},...<i_{r}}c_{i_{1},i_{2},...,i_{r}}z^{a_{0}}_{0}z^{a_{1}}_{1}\cdots
z^{a_{k}}_{k}\xi_{i_{1},i_{2},..,i_{r}}\in E^{r,0}_{1}$

is cohomologous to zero if $A_{j}:={a_{j}}/{e_{j}}-{a_{0}}/{e_{0}}\neq 0$ for
some $1\leq j\leq k$. Note that $D_{j}(z^{a_{0}}_{0}z^{a_{1}}_{1}\cdots
z^{a_{k}}_{k})=A_{j}z^{a_{0}}_{0}z^{a_{1}}_{1}\cdots z^{a_{k}}_{k}$ for every
$1\leq j\leq k$. Since $v$ is a cocycle, the coefficients satisfy

(3.15)
$\sum^{r}_{\ell=1}(-1)^{\ell}c_{i_{1},i_{2},...,\hat{i_{\ell}},...,i_{r+1}}A_{i_{\ell}}=0.$

Assume that not all $A_{j}$’s are zero for $1\leq j\leq k$ then $A=\sum
A^{2}_{i}$ is non-zereo. Then the number

$d_{i_{1},i_{2},...,i_{r-1}}=\sum^{k}_{\alpha=1}\frac{A_{\alpha}}{A}c_{\alpha,i_{1},i_{2},...,i_{r-1}}.$

is well-defined. Here we extend standardly that
$c_{\sigma(i_{1}),\sigma(i_{2}),..,\sigma(i_{r})}={\mathrm{s}}{\mathrm{i}}{\mathrm{g}}{\mathrm{n}}(\sigma)c_{i_{1},i_{2},...,i_{r}}$
for any permutation $\sigma$. Then the element

$\sum_{i_{1}<i_{2}<...<i_{r-1}}d_{i_{1},i_{2},...,i_{r-1}}z^{a_{0}}_{0}z^{a_{1}}_{1}\cdots
z^{a_{k}}_{k}\xi_{i_{1},i_{2},...,i_{r-1}}$

in $E^{r-1,0}_{1}$ has coboundary

$\displaystyle\sum^{k}_{\alpha=1}\sum_{i_{1}<...<i_{r-1}}A_{\alpha}d_{i_{1},i_{2},...,i_{r-1}}z^{a_{0}}_{0}z^{a_{1}}_{1}\cdots
z^{a_{k}}_{k}\xi_{\alpha,i_{1},i_{2},...,i_{r-1}}$ $\displaystyle=$
$\displaystyle\sum_{i_{1}<...<i_{r}}\sum^{r}_{\ell=1}(-1)^{\ell}A_{i_{\ell}}d_{i_{1},i_{2},...,\hat{i_{\ell}},...,i_{r}}z^{a_{0}}_{0}z^{a_{1}}_{1}\cdots
z^{a_{k}}_{k}\xi_{i_{1},i_{2},...,i_{r}}$ $\displaystyle=$
$\displaystyle\sum_{i_{1}<...<i_{r}}\sum^{k}_{\alpha=1}\sum^{r}_{\ell=1}(-1)^{\ell}\frac{A_{i_{\ell}}A_{\alpha}}{A}c_{\alpha,i_{1},i_{2},...,\hat{i_{\ell}},...,i_{r}}z^{a_{0}}_{0}z^{a_{1}}_{1}\cdots
z^{a_{k}}_{k}\xi_{i_{1},i_{2},...,i_{r}}$ $\displaystyle\text{
applying~{}\eqref{cocycle} }=$
$\displaystyle\sum_{i_{1}<...<i_{r}}\sum^{k}_{\alpha=1}\frac{A^{2}_{\alpha}}{A}c_{,i_{1},i_{2},...,i_{r}}z^{a_{0}}_{0}z^{a_{1}}_{1}\cdots
z^{a_{k}}_{k}\xi_{i_{1},i_{2},...,i_{r}}=v.$

We conclude the claim. Therefore, $E^{r,0}_{2}$ is generated over
${\mathbb{C}}$ by $z^{a_{0}}_{0}z^{a_{1}}_{1}\cdots
z^{a_{k}}_{k}\xi_{i_{1},i_{2},..,i_{r}}$ with

$D_{i}(z^{a_{0}}_{0}z^{a_{1}}_{1}\cdots z^{a_{k}}_{k})=0.$

That is, $z^{a_{0}}_{0}z^{a_{1}}_{1}\cdots z^{a_{k}}_{k}=t^{a/e}$ for some
$a$. Hence, we conclude the lemma. ∎

######  Theorem 3.5.

Let ${\mathcal{F}}^{\bullet}$ be a complex of ${\mathscr{O}}_{\Delta}$-modules
with coherent cohomologies, equipped with a log connection, i.e an operator

$\nabla\in{\mathrm{E}}{\mathrm{n}}{\mathrm{d}\,}_{{\mathbf{D}}^{b}(\Delta,{\mathbb{C}})}\left({\mathcal{F}}^{\bullet}\right)\quad\text{such
that }[\nabla,g]=tg^{\prime}$

for ant local holomorphic function $g$ where $g^{\prime}$ is the derivative of
$g$. Assume that the residue $R_{\ell}$ of $\nabla$ defined in the beginning
of this subsection acting on each cohomology
${\mathscr{H}}^{\ell}\left({\mathcal{F}}^{\bullet}\otimes^{L}{\mathbb{C}}(0)\right)$
has eigenvalues in $[0,1)$. Then every
${\mathscr{H}}^{\ell}({\mathcal{F}}^{\bullet})$ is locally free.

###### Proof.

By the definition of residue, we have the morphism of distinguished triangles

${{\mathcal{F}}^{\bullet}}$${{\mathcal{F}}^{\bullet}}$${\mathcal{{\mathcal{F}}}^{\bullet}\otimes^{L}{\mathbb{C}}(0)}$${{\mathcal{F}}^{\bullet}[1]}$${{\mathcal{F}}^{\bullet}}$${{\mathcal{F}}^{\bullet}}$${\mathcal{{\mathcal{F}}}^{\bullet}\otimes^{L}{\mathbb{C}}(0)}$${{\mathcal{F}}^{\bullet}[1]}$$\scriptstyle{\nabla+1}$$\scriptstyle{t}$$\scriptstyle{\nabla}$$\scriptstyle{R}$$\scriptstyle{\left(\nabla+1\right)[1]}$$\scriptstyle{t}$

in ${\mathbf{D}}^{b}(\Delta,{\mathbb{C}})$. Taking cohomologies gives

(3.16)
${\cdots}$${{\mathscr{H}}^{\ell-1}\left({\mathcal{F}}^{\bullet}\mathbin{\mathop{\otimes}\displaylimits^{L}}{\mathbb{C}}(0)\right)}$${{\mathscr{H}}^{\ell}\left({\mathcal{F}}^{\bullet}\right)}$${{\mathscr{H}}^{\ell}({\mathcal{F}}^{\bullet})}$${{\mathscr{H}}^{\ell}\left({\mathcal{F}}^{\bullet}\mathbin{\mathop{\otimes}\displaylimits^{L}}{\mathbb{C}}(0)\right)}$${\cdots}$${\cdots}$${{\mathscr{H}}^{\ell-1}\left({\mathcal{F}}^{\bullet}\mathbin{\mathop{\otimes}\displaylimits^{L}}{\mathbb{C}}(0)\right)}$${{\mathscr{H}}^{\ell}({\mathcal{F}}^{\bullet})}$${{\mathscr{H}}^{\ell}({\mathcal{F}}^{\bullet})}$${{\mathscr{H}}^{\ell}\left({\mathcal{F}}^{\bullet}\mathbin{\mathop{\otimes}\displaylimits^{L}}{\mathbb{C}}(0)\right)}$${\cdots}$$\scriptstyle{R_{\ell}}$$\scriptstyle{t}$$\scriptstyle{\nabla+1}$$\scriptstyle{\nabla}$$\scriptstyle{R_{\ell+1}}$$\scriptstyle{t}$

For simplicity, fix $\ell$ and let
${\mathscr{H}}={\mathscr{H}}^{\ell}({\mathcal{F}}^{\bullet})$ and denote by
$\ker t$ the kernel of the morphism $t:{\mathscr{H}}\rightarrow{\mathscr{H}}$.
It suffices to prove that $\ker t$ is trivial on ${\mathscr{H}}$. We are going
to show that $\ker t$ is a subset of $t^{k}{\mathscr{H}}$ for all $k\geq 0$
and thus, by Krull’s theorem $\ker t$ is zero.

It follows from the diagram (3.16) that $\nabla+1$ on $\ker t$ and $\nabla$ on
${\mathscr{H}}/t{\mathscr{H}}$ have eigenvalues in $[0,1)$. Therefore, there
exists a polynomial $b_{1}(s)\in{\mathbb{C}}[s]$ with roots in $[0,1)$ such
that

$b_{1}(\nabla){\mathscr{H}}\subset t{\mathscr{H}},$

and another a polynomial $b_{2}(s)\in{\mathbb{C}}[s]$ with eigenvalues in
$[0,1)$ such that

$b_{2}(\nabla+1)\ker t=0.$

Suppoe $v$ is an element in $\ker t\cap t^{k}{\mathscr{H}}$ for some $k\geq
0$. It follows that $v=t^{k}v_{1}$ for some $v_{1}\in{\mathscr{H}}$. Because
the roots of $b_{1}(s-k)$ are bigger then the roots of $b_{2}(s+1)$, the two
polynomials $b_{1}(s-k)$ and $b_{2}(s+1)$ are relative prime. We deduce that
there exist $p(s),q(s)\in{\mathbb{C}}[s]$ such that

$1=p(s)b_{1}(s-k)+q(s)b_{2}(s+1).$

Therefore, combining the fact that $b_{2}(\nabla+1)v$ vanishes,

$v=p(\nabla)b_{1}(\nabla-k)v+q(\nabla)b_{2}(\nabla+1)v=p(\nabla)b_{1}(\nabla-k)t^{k}v_{1}.$

Because of the identity $(\nabla-k)t^{k}=t^{k}\nabla$, the above is equivalent
to

$v=t^{k}p(\nabla+k)b_{1}(\nabla)v_{1}.$

Because $b_{1}(\nabla)v_{1}=tv_{2}$ for some $v_{2}\in{\mathscr{H}}$,
substituting in the last equality yields

$v=t^{k}p(\nabla+k)b_{1}(\nabla)v_{1}=t^{k}p(\nabla+k)b_{1}(\nabla)tv_{2}=t^{k+1}p(\nabla+k+1)b_{1}(\nabla+1)v_{2}\in
t^{k+1}{\mathscr{H}}.$

We proved that $v$ is also an element in $t^{k+1}{\mathscr{H}}$. By induction
and Krull’s theorem we conclude the proof. ∎

Now we can immediately finish

###### Proof of Theorem 3.2.

The complex $Rf_{*}\Omega^{\bullet+n}_{X/\Delta}(\log Y)$ with $Rf_{*}\nabla$
satisfies the condition of Theorem 3.5. Therefore, each cohomology
$R^{\ell}f_{*}\Omega^{\bullet+n}_{X/\Delta}(\log Y)$ is locally free. The
second statement in the theorem follows from the the locally freeness of
$R^{\ell}f_{*}\Omega^{\bullet+n}_{X/\Delta}(\log Y)$ plus the Grauert’s base
change theorem. ∎

##  4\. Transfer to ${\mathscr{D}}-$modules

Lemma 3.4 implies the restriction of the relative log de Rham complex on $Y$
is semi-perverse. Indeed, it is even perverse, showed in [Ste76, $\S$2].
Therefore, there should be a regular holonomic ${\mathscr{D}}$-module whose de
Rham complex is the restriction of the relative log de Rham complex on $Y$, in
the view of Riemann-Hilbert correspondence established by Kashiwara [Kas84]
and Mebkhout [Meb84]. The stupid filtration should also translates to a
coherent filtration from Hodge theoretic point of view. Then the endomorphism
$[\nabla]$ in the derived category can be captured by an endomorphism of a
${\mathscr{D}}$-module. This enable us to study the relation between the
filtration and $[\nabla]$ much easier and cleaner. In this section, we will
construct the filtered ${\mathscr{D}}$-module and the endomorphism.

###  4.1. Construction of filtered holonomic ${\mathscr{D}}_{X}$-modules

Since ${\mathscr{T}}_{X/\Delta}(\log)$ is a subsheaf of ${\mathscr{T}}_{X}$,
the multiplication by sections in ${\mathscr{T}}_{X/\Delta}(\log Y)$ induces a
morphism ${\mathscr{D}}_{X}\rightarrow\Omega_{X/\Delta}(\log
Y)\otimes{\mathscr{D}}_{X}$, with $P\mapsto\sum^{k}_{i=1}\xi_{i}\otimes
D_{i}P$ locally. The morphism extends to a filtered complex of
${\mathscr{D}}_{X}$-modules

(4.17) $\Omega^{n+\bullet}_{X/\Delta}(\log
Y)\otimes{\mathscr{D}}_{X}=\\{{\mathscr{D}}_{X}\rightarrow\Omega_{X/\Delta}(\log
Y)\otimes{\mathscr{D}}_{X}\rightarrow\cdots\rightarrow\Omega^{n}_{X/\Delta}(\log
Y)\otimes{\mathscr{D}}_{X}\\}[n]$

with filtration $F_{\ell}\left(\Omega^{n+\bullet}_{X/\Delta}(\log
Y)\otimes{\mathscr{D}}_{X}\right)$ given by

$\Omega^{{}n+\bullet}_{X/\Delta}(\log Y)\otimes
F_{\ell+n+\bullet}{\mathscr{D}}_{X}=\\{F_{\ell}{\mathscr{D}}_{X}\rightarrow\Omega_{X/\Delta}(\log
Y)\otimes
F_{\ell+1}{\mathscr{D}}_{X}\rightarrow\cdots\rightarrow\Omega^{n}_{X/\Delta}(\log
Y)\otimes F_{\ell+n}{\mathscr{D}}_{X}\\}[n].$

Let $\tilde{\mathcal{M}}$ be the $0$-th cohomology of
$\Omega^{n+\bullet}_{X/\Delta}(\log Y)\otimes{\mathscr{D}}_{X}$ and
$F_{\ell}{\mathcal{M}}$ be the ${\mathscr{O}}_{X}$-submodule induced by the
the filtration $F_{\ell}\left(\Omega^{n+\bullet}_{X/\Delta}(\log
Y)\otimes{\mathscr{D}}_{X}\right)$.

######  Theorem 4.1.

The complex $\Omega^{n+\bullet}_{X/\Delta}(\log Y)\otimes{\mathscr{D}}_{X}$ is
a filtered resolution of a filtered ${\mathscr{D}}_{X}$-module
$(\tilde{\mathcal{M}},F_{\bullet}\tilde{\mathcal{M}})$.

###### Proof.

Notice that
${\mathrm{g}}{\mathrm{r}}^{F}\left(\Omega^{n+\bullet}_{X/\Delta}(\log
Y)\otimes{\mathscr{D}}_{X}\right)=\Omega^{n+\bullet}_{X/\Delta}(\log
Y)\otimes{\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}$, can be identified
locally with the Koszul complex associated to the regular sequence
$D_{1},D_{2},...,D_{n}$ over the ring
${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}$. It follows that
$\Omega^{n+\bullet}_{X/\Delta}(\log
Y)\otimes{\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}$ is acyclic. Therefore,
each graded peace
${\mathrm{g}}{\mathrm{r}}^{F}_{\ell}\left(\Omega^{n+\bullet}_{X/\Delta}(\log
Y)\otimes{\mathscr{D}}_{X}\right)$ is acyclic. We deduce inductively that
$F_{\ell}\left(\Omega^{n+\bullet}_{X/\Delta}(\log
Y)\otimes{\mathscr{D}}_{X}\right)$ is also acyclic; this can be seen from the
long exact sequence associated to the short exact sequence

$0\rightarrow F_{\ell-1}\left(\Omega^{n+\bullet}_{X/\Delta}(\log
Y)\otimes{\mathscr{D}}_{X}\right)\rightarrow
F_{\ell}\left(\Omega^{n+\bullet}_{X/\Delta}(\log
Y)\otimes{\mathscr{D}}_{X}\right)\rightarrow{\mathrm{g}}{\mathrm{r}}^{F}_{\ell}\left(\Omega^{n+\bullet}_{X/\Delta}(\log
Y)\otimes{\mathscr{D}}_{X}\right)\rightarrow 0.$

Taking direct limit, we conclude that $\Omega^{n+\bullet}_{X/\Delta}(\log
Y)\otimes{\mathscr{D}}_{X}$ is a resolution of $\tilde{\mathcal{M}}$. The long
exact sequence also implies the $0$-th cohomology of
$F_{\ell}\left(\Omega^{n+\bullet}_{X/\Delta}(\log
Y)\otimes{\mathscr{D}}_{X}\right)$ is isomorphic to
$F_{\ell}\tilde{\mathcal{M}}$. This completes the proof. ∎

######  Remark 4.2.

Note that $\Omega^{n+\bullet}_{X/\Delta}(\log Y)\otimes{\mathscr{D}}_{X}$ is a
complex of $(f^{-1}{\mathscr{O}}_{\Delta},{\mathscr{D}}_{X})$-bimodules
because $\Omega^{n+\bullet}_{X/\Delta}(\log Y)$ is
$f^{-1}{\mathscr{O}}_{\Delta}$-linear. It follows that $\tilde{\mathcal{M}}$
is also a $(f^{-1}{\mathscr{O}}_{\Delta},{\mathscr{D}}_{X})$-bimodule. Note we
have two different actions of $t$ on $\tilde{\mathcal{M}}$ due to the bimodule
structure. We usually use the left multiplication by $t$. One can think of
$\tilde{\mathcal{M}}$ as a flat family assembling the ${\mathscr{D}}$-module
${i_{X_{p}}}_{+}\omega_{X_{p}}$ of the smooth fibers $X_{p}$ for $p\in\Delta$
and a specialization ${\mathcal{M}}=\tilde{\mathcal{M}}/t\tilde{\mathcal{M}}$
because using the left $f^{-1}{\mathscr{O}}_{\Delta}$ structure, we have
filtered isomorphisms

${\mathbb{C}}(p)\mathbin{\mathop{\otimes}\displaylimits}\tilde{\mathcal{M}}\simeq{\mathbb{C}}(p)\mathbin{\mathop{\otimes}\displaylimits}\Omega^{n+\bullet}_{X/\Delta}(\log
Y)\otimes{\mathscr{D}}_{X}\simeq\Omega^{n+\bullet}_{X/\Delta}(\log
Y)|_{X_{p}}\otimes{\mathscr{D}}_{X}\simeq{i_{X_{p}}}_{*}\Omega^{n+\bullet}_{X_{p}}\otimes{\mathscr{D}}_{X}\simeq{i_{X_{p}}}_{+}\omega_{X_{p}},$

where $i_{X_{p}}:X_{p}\rightarrow X$ is the closed embedding of the smooth
fiber $X_{p}$.

######  Remark 4.3.

The theorem also says by choosing the local trivialization
$\xi_{1}\wedge\xi_{2}\wedge\cdots\wedge\xi_{n}$ of $\Omega^{n}_{X/\Delta}(\log
Y)$, $\tilde{\mathcal{M}}$ can be identified locally with
${\mathscr{D}}_{X}/(D_{1},D_{2},...,D_{n}){\mathscr{D}}_{X}$ and
${\mathrm{g}}{\mathrm{r}}^{F}\tilde{\mathcal{M}}$ can be identified locally
with
${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}/(D_{1},D_{2},...,D_{n}){\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}$.

######  Remark 4.4.

Let ${\mathscr{D}}_{X/\Delta}(\log Y)$ be the subalgebra of
${\mathscr{D}}_{X}$ generated by ${\mathscr{T}}_{X/\Delta}(\log Y)$. One can
show that $\tilde{\mathcal{M}}$ is nothing but

$\omega_{X/\Delta}(\log
Y)\mathbin{\mathop{\otimes}\displaylimits_{{\mathscr{D}}_{X/\Delta}(\log
Y)}}{\mathscr{D}}_{X}.$

And the filtration $F_{\bullet}\tilde{\mathcal{M}}$ is induced from
$F_{\bullet}\omega_{X/\Delta}(\log Y)$, where $F_{\ell}\omega_{X/\Delta}(\log
Y)$ is $\omega_{X/\Delta}(\log Y)$ for $\ell\geq-n$ and is zero otherwise. To
keep the proof elementary, we avoid talking about
${\mathscr{D}}_{X/\Delta}(\log Y)$-modules.

######  Theorem 4.5.

The complex $\Omega^{n+\bullet}_{X/\Delta}(\log
Y)|_{Y}\otimes{\mathscr{D}}_{X}$ is a filtered resolution of a filtered
holonimic ${\mathscr{D}}_{X}$-module
$({\mathcal{M}},F_{\bullet}{\mathcal{M}})$.

###### Proof.

Because of the bimodule structure, we have $\Omega^{n+\bullet}_{X/\Delta}(\log
Y)|_{Y}\otimes{\mathscr{D}}_{X}$ is the cokernel of the left multiplication by
$t$ on $\Omega^{n+\bullet}_{X/\Delta}(\log Y)\otimes{\mathscr{D}}_{X}$.
Therefore, the first part of the statement is equivalent to
$t:\tilde{\mathcal{M}}\rightarrow\tilde{\mathcal{M}}$ is injective. It
suffices to prove that
$t:{\mathrm{g}}{\mathrm{r}}^{F}\tilde{\mathcal{M}}\rightarrow{\mathrm{g}}{\mathrm{r}}^{F}\tilde{\mathcal{M}}$
is injective because the multiplication by $t$ is a filtered morphism. But
this follows from $t,D_{1},D_{2},...,D_{n}$ is a regular sequence over the
ring ${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}$. It also follows that
${\mathrm{g}}{\mathrm{r}}^{F}{\mathcal{M}}$ is isomorphic locally to
${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}/(t,D_{1},D_{2},...,D_{n}){\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}$.
This means the characteristic variety of ${\mathcal{M}}$ is cut out by
$t,D_{1},D_{2},...,D_{n}\in{\mathscr{O}}_{T^{*}X}$ and thus, the
characteristic variety is of dimension $n+1$. This proves the holonomicity of
${\mathcal{M}}$. ∎

######  Remark 4.6.

Similarly to the case of $\tilde{\mathcal{M}}$, the ${\mathscr{D}}_{X}$-module
${\mathcal{M}}$ is just

$\omega_{X/\Delta}(\log
Y)|_{Y}\mathbin{\mathop{\otimes}\displaylimits_{{\mathscr{D}}_{X/\Delta}(\log
Y)}}{\mathscr{D}}_{X}$

with the filtration $F_{\bullet}{\mathcal{M}}$ induced by
$\left(F_{\bullet}\omega_{X/\Delta}(\log Y)\right)|_{Y}$.

###  4.2. Properties of ${\mathcal{M}}$

We first calculate the characteristic cycle of ${\mathcal{M}}$ which is
important for later when we identifying the primitive part of
${\mathrm{g}}{\mathrm{r}}^{W}{\mathcal{M}}$. Then we prove that the de Rham
complex of ${\mathcal{M}}$ with the induced filtration recover
$\Omega^{{}\bullet+n}_{X/\Delta}(\log Y)|_{Y}$ with the stupid filtration.
Lastly, we translate the operator
$[\nabla]\in{\mathrm{E}}{\mathrm{n}}{\mathrm{d}\,}_{{\mathbf{D}}^{b}(X,{\mathbb{C}})}(\Omega^{{}\bullet+n}_{X/\Delta}(\log
Y))|_{Y}$ to an operator $R$ on ${\mathcal{M}}$

######  Theorem 4.7.

The characteristic cycle of ${\mathcal{M}}$ is

$cc({\mathcal{M}})=\sum_{J\subset I}\sum_{j\in
J}e_{j}\left[T^{*}_{Y^{J}}X\right],$

where $[T^{*}_{Y^{J}}X]$ is the cycle of the conormal bundle of $Y^{J}$ in
$T^{*}X$ and $e_{i}$ is the multiplicity of $Y$ along each component $Y_{i}$
for $i\in I$.

###### Proof.

The statement is local and we identify ${\mathcal{M}}$ with
${\mathscr{D}}_{X}/(t,D_{1},D_{2},...,D_{n})$. We first describe the
characteristic variety of ${\mathcal{M}}$. The support of
${\mathrm{g}}{\mathrm{r}}^{F}{\mathcal{M}}$ as a sheaf on $T^{*}X$ is defined
by the radical of the ideal
$(t,D_{1},D_{2},...,D_{n}){\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}$. In
fact, $z_{i}\partial_{i}$ for $0\leq i\leq k$ is in the radical because

$\left(z_{i}\partial_{i}\right)^{e_{0}+e_{1}+\cdots+e_{k}}\equiv(z_{0}\partial_{0})^{e_{0}}(z_{1}\partial_{1})^{e_{1}}\cdots(z_{k}\partial_{k})^{e_{k}}\equiv
t\partial^{e_{0}}_{0}\partial^{e_{1}}_{1}\cdots\partial^{e_{k}}_{k}\equiv
0\,\mathrm{mod}\,(t,D_{1},D_{2},...,D_{n}){\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}.$

Therefore, $char({\mathcal{M}})$ is cut out by
$t_{\mathrm{red}},z_{0}\partial_{0},z_{1}\partial_{1},...,z_{k}\partial_{k},\partial_{k+1},...,\partial_{n}$,
where $t_{\mathrm{red}}=z_{0}z_{1}\cdots z_{k}$. It follows that
$char({\mathcal{M}})=\bigcup_{J\subset I}T^{*}_{Y^{J}}X$.

Denote by $\mathfrak{p}(Z)$ the prime ideal defining a integral subvariety
$Z$. Let $m_{J}$ be the length of
${\mathrm{g}}{\mathrm{r}}^{F}{\mathcal{M}}_{\mathfrak{p}(T^{*}_{Y^{J}}X)}$ as
an Artinian
${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X,\mathfrak{p}}$-module . Then
$cc({\mathcal{M}})=\sum_{J\in I}m_{J}\left[T^{*}_{Y^{J}}X\right]$. For
simplicity let us assume $J=\\{0,1,2,..,\mu\\}$ and by abuse of notation we
also the prime ideal $\mathfrak{p}=\mathfrak{p}(T^{*}_{Y^{J}}X)$ of the
variety $T^{*}_{Y^{J}}X$ is locally generated by
$z_{0},z_{1},...,z_{\mu},\partial_{\mu+1},\partial_{\mu+2},...,\partial_{n}$
over ${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}$ in some local coordinate
system. Notice that

${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X,\mathfrak{p}}/(t,D_{1},D_{2},...,D_{n}){\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X,\mathfrak{p}}={\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X,\mathfrak{p}}/(D^{\prime}_{0},D^{\prime}_{1},...,D^{\prime}_{n}){\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X,\mathfrak{p}}$

where

(4.18) $D^{\prime}_{i}=\left\\{\begin{aligned}
z^{e_{0}+e_{1}+\cdots+e_{\mu}}_{0}&,&\text{ for }i=0\\\
\frac{1}{e_{i}}z_{i}-\frac{1}{e_{0}}z_{0}\frac{\partial_{0}}{\partial_{i}}&,&\text{
for }1\leq i\leq\mu\\\
\frac{1}{e_{i}}\partial_{i}-\frac{1}{e_{0}}z_{0}\frac{\partial_{0}}{z_{i}}&,&\text{
for }\mu+1\leq i\leq k\\\ \partial_{i}&,&\text{ for }i>k,\end{aligned}\right.$

because
$\partial_{0},\partial_{1},...,\partial_{\mu},z_{\mu+1},z_{\mu+2},...,z_{k}$
are invertible in
${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X,\mathfrak{p}}$. Therefore,
${\mathrm{g}}{\mathrm{r}}^{F}{\mathcal{M}}_{\mathfrak{p}}$ can be identifies
with

${\mathbb{C}}\\{z_{0}\\}/(z^{e_{0}+e_{1}+\cdots+e_{\mu}}_{0}).$

Then
$m_{J}=\dim_{\mathbb{C}}{\mathbb{C}}\\{z_{0}\\}/(z^{e_{0}+e_{1}+\cdots+e_{\mu}}_{0})=\sum_{j\in
J}e_{j}$. This completes the computation. ∎

######  Remark 4.8.

The above theorem verifies that $cc({\mathcal{M}})=\lim_{p\rightarrow
0}cc({i_{p}}_{+}\omega_{X_{p}})=\lim_{p\rightarrow
0}\left[T^{*}_{X_{p}}X\right]$ as cycles in algebraic cotangent space $T^{*}X$
for $p\in\Delta^{*}$ where $i_{p}:X_{p}\rightarrow X$ the closed embedding of
the smooth fiber. In fact, one can show that
${\mathbb{C}}(p)\otimes{\mathrm{g}}{\mathrm{r}}^{F}\tilde{\mathcal{M}}$, using
the left $f^{-1}{\mathscr{O}}_{\Delta}$-module structure, is isomorphic to
${\mathrm{g}}{\mathrm{r}}^{F}{i_{p}}_{+}\omega_{X_{p}}$ as in Remark 4.2.
Refer to [Gin86] for general results about the characteristic cycles of
specializations of holonomic ${\mathscr{D}}$-modules.

######  Corollary 4.9.

The de Rham complex ${\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}}$ together with
filtration $F_{\bullet}{\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}}$ is
isomorphic to $\Omega^{n+\bullet}_{X/\Delta}(\log Y)|_{Y}$ with the stupid
filtration in the derived category of filtered complexes of sheaves of
${\mathbb{C}}$-vector spaces.

###### Proof.

We have showed that $F_{\ell}\left(\Omega^{n+\bullet}_{X/\Delta}(\log
Y)\otimes{\mathscr{D}}_{X}\right)$ is a resolution of $F_{\ell}{\mathcal{M}}$.
Therefore, the total complex of
$F_{\ell+*}\left(\Omega^{n+\bullet}_{X/\Delta}(\log
Y)\otimes{\mathscr{D}}_{X}\right)\otimes\bigwedge^{-*}{\mathscr{T}}_{X}$ is
quasi-isomorphic to
$F_{\ell+*}{\mathcal{M}}\otimes\bigwedge^{-*}{\mathscr{T}}_{X}$, which is
exactly $F_{\ell}{\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}}$. It remains to
show the total complex also quasi-isomorphic to
$F_{\ell}\Omega^{n+\bullet}_{X/\Delta}(\log Y)$. This follows from that

$\displaystyle F_{\ell+*}\left(\Omega^{n+\bullet}_{X/\Delta}(\log
Y)\otimes{\mathscr{D}}_{X}\right)\otimes\bigwedge^{-*}{\mathscr{T}}_{X}$
$\displaystyle=\Omega^{n+\bullet}_{X/\Delta}(\log Y)\otimes
F_{\ell+n+\bullet}\left({\mathscr{D}}_{X}\otimes\bigwedge^{-*}{\mathscr{T}}_{X}\right)$
$\displaystyle\simeq\Omega^{n+\bullet}_{X/\Delta}(\log Y)\otimes
F_{\ell+n+\bullet}{\mathscr{O}}_{X}$
$\displaystyle=F_{\ell}\Omega^{n+\bullet}_{X/\Delta}(\log Y).$

Here, $F_{\ell}{\mathscr{O}}_{X}={\mathscr{O}}_{X}$ for $\ell\geq 0$ and
otherwise it is zero. ∎

######  Theorem 4.10.

The endomorphism
$\nabla\in{\mathrm{E}}{\mathrm{n}}{\mathrm{d}\,}_{{\mathbf{D}}^{b}(X,{\mathbb{C}})}\Omega^{n+\bullet}_{X/\Delta}(\log
Y)$ in Lemma 3.1 transfers to a filtered morphism

$\nabla:(\tilde{\mathcal{M}},F_{\bullet}\tilde{\mathcal{M}})\rightarrow(\tilde{\mathcal{M}},F_{\bullet+1}\tilde{\mathcal{M}}),\quad\left[[\alpha]\otimes
P\right]\mapsto(\frac{dt}{t}\wedge)^{-1}\left\\{d(\alpha\otimes P)\right\\}$

where $\alpha\in\Omega^{n}_{X}(\log Y)$ and $P\in{\mathscr{D}}_{X}$ so that
$[\alpha]\otimes P\in\Omega^{{}n}_{X/\Delta}(\log Y)\otimes{\mathscr{D}}_{X}$.
Moreover, restriction on $Y$ yields a filtered morphism

$R:({\mathcal{M}},F_{\bullet}{\mathcal{M}})\rightarrow({\mathcal{M}},F_{\bullet+1}{\mathcal{M}})$

such that

(4.19) $\prod_{i\in I}\prod^{e_{i}-1}_{j=0}(R-\frac{j}{e_{i}})=0.$

###### Proof.

The morphism $\frac{dt}{t}\wedge:\Omega^{n+\bullet}_{X/\Delta}(\log
Y)\rightarrow\Omega^{n+1+\bullet}_{X}(\log Y)$ extends to the corresponding
complexes of induced ${\mathscr{D}}_{X}$-modules

$\frac{dt}{t}\wedge:\Omega^{n+\bullet}_{X/\Delta}(\log
Y)\otimes{\mathscr{D}}_{X}\rightarrow\Omega^{n+1+\bullet}_{X}(\log
Y)\otimes{\mathscr{D}}_{X}.$

Let
${\mathrm{C}}{\mathrm{o}}{\mathrm{n}}{\mathrm{e}}^{\bullet}\otimes{\mathscr{D}}_{X}$
be the mapping cone of the above morphism. We get a diagram of complexes of
${\mathscr{D}}_{X}$-modules similarly to (3.13) and taking $0$-th cohomology
we get the following.

(4.20)
${{\mathscr{H}}^{0}\left({\mathrm{C}}{\mathrm{o}}{\mathrm{n}}{\mathrm{e}}^{\bullet}\otimes{\mathscr{D}}_{X}\right)}$${\tilde{\mathcal{M}}}$${\tilde{\mathcal{M}}}$$\scriptstyle{q}$$\scriptstyle{p}$$\scriptstyle{p\circ
q^{-1}}$

where abuse of notation, still denote by $p$ and $q$ the induced morphisms
from diagram (3.13). Now $q$ is an isomorphism of ${\mathscr{D}}_{X}$-modules.
Let $[\alpha\otimes P,[\beta]\otimes Q]$ be a class in
${\mathscr{H}}^{0}({\mathrm{C}}{\mathrm{o}}{\mathrm{n}}{\mathrm{e}}^{\bullet}\otimes{\mathscr{D}}_{X})$
for any $\alpha\otimes P\in\Omega^{n}_{X}(\log Y)\otimes{\mathscr{D}}_{X}$ and
$[\beta]\otimes Q\in\Omega^{n}_{X/\Delta}(\log Y)\otimes{\mathscr{D}}_{X}$.
Then

$\delta\left(\alpha\otimes P,[\beta]\otimes
Q\right)=\left((-1)^{n}d(\alpha\otimes P)+\frac{dt}{t}\wedge\beta\otimes
Q,(-1)^{n}d([\beta]\otimes Q)\right)=0.$

Here, the sign factor $(-1)^{n}$ shows up due to we follow the Koszul sign
rule. Because $\frac{dt}{t}\wedge:\Omega^{n}_{X/\Delta}(\log
Y)\rightarrow\Omega^{n+1}_{X}(\log Y)$ is an isomorphism, we have

$[\beta]\otimes Q=(-1)^{n-1}(\frac{dt}{t}\wedge)^{-1}\\{d(\alpha\otimes
P)\\}.$

Therefore,
$q^{-1}:\tilde{\mathcal{M}}\rightarrow{\mathscr{H}}^{0}({\mathrm{C}}{\mathrm{o}}{\mathrm{n}}{\mathrm{e}}^{\bullet}\otimes{\mathscr{D}}_{X})$
is given by $[[\alpha]\otimes P]\mapsto[\alpha\otimes
P,(-1)^{n}(\frac{dt}{t}\wedge)^{-1}\\{d(\alpha\otimes P)\\}]$. Then we have

$\nabla=(-1)^{n-1}p\circ q^{-1}:[[\alpha]\otimes
P]\mapsto(\frac{dt}{t}\wedge)^{-1}\\{d(\alpha\otimes P)\\}.$

Restricting to $Y$ we have the induced operator $R$ on ${\mathcal{M}}$. If
$\alpha=\xi_{1}\wedge\xi_{2}\wedge\cdots\wedge\xi_{n}$ then

$\displaystyle R[\xi_{1}\wedge\xi_{2}\wedge\cdots\wedge\xi_{n}\otimes P]$
$\displaystyle=(\frac{dt}{t}\wedge)^{-1}\left(d\left(e_{1}\frac{dz_{1}}{z_{1}}\wedge
e_{2}\frac{dz_{2}}{z_{2}}\wedge\cdots\wedge dz_{n}\otimes P\right)\right)$
$\displaystyle=(\frac{dt}{t}\wedge)^{-1}\left(e_{0}\frac{dz_{0}}{z_{0}}\wedge
e_{1}\frac{dz_{1}}{z_{1}}\wedge e_{2}\frac{dz_{2}}{z_{2}}\wedge\cdots\wedge
dz_{n}\otimes\frac{1}{e_{0}}z_{0}\partial_{0}P\right)$
$\displaystyle=\left[\xi_{1}\wedge\xi_{2}\wedge\cdots\wedge\xi_{n}\otimes\frac{1}{e_{0}}z_{0}\partial_{0}P\right].$

We see that $RF_{\bullet}{\mathcal{M}}\subset F_{\bullet+1}{\mathcal{M}}$. The
reason for $\nabla F_{\bullet}\tilde{\mathcal{M}}\subset
F_{\bullet+1}\tilde{\mathcal{M}}$ is similar. To prove the last statement, we
work locally and identify ${\mathcal{M}}$ with
${\mathscr{D}}_{X}/(t,D_{1},...,D_{n})$ via the local trivialization
$\xi_{1}\wedge\xi_{2}\wedge\cdots\wedge\xi_{n}$ of $\Omega^{n}_{X/\Delta}(\log
Y)$. Then for $P\in{\mathscr{D}}_{X}$,
$R[P]=[\frac{1}{e_{0}}z_{0}\partial_{0}P]$. In fact, because of the relation
$D_{1},D_{2},...,D_{n}$, the left multiplication by
$\frac{1}{e_{0}}z_{0}\partial_{0}$ on ${\mathcal{M}}$ is the same as the
multiplication by $\frac{1}{e_{i}}z_{i}\partial_{i}$ for $1\leq i\leq k$. It
follows from the identity

$(z\partial)(z\partial-1)\cdots(z\partial-\ell)=z^{\ell+1}\partial^{\ell+1}$

for any $\ell\geq 0$ that

$\displaystyle\prod_{i\in I}\prod^{e_{i}-1}_{j=0}(R-\frac{j}{e_{i}})[P]$
$\displaystyle=\prod_{i\in
I}\prod^{e_{i}-1}_{j=0}(\frac{1}{e_{i}}z_{i}\partial_{i}-\frac{j}{e_{i}})[P]=\prod_{i\in
I}\frac{1}{e^{e_{i}}_{i}}z^{e_{i}}_{i}\partial^{e_{i}}_{i}[P]=t\prod_{i\in
I}\frac{1}{e_{i}^{e_{i}}}\partial^{e_{i}}_{i}[P]$
$\displaystyle=0\in{{\mathscr{D}}_{X}}/{(D_{1},D_{2},...,D_{n},t){\mathscr{D}}_{X}}.$

This completes the proof. ∎

######  Remark 4.11.

Note that $\nabla:\tilde{\mathcal{M}}\rightarrow\tilde{\mathcal{M}}$ is also
can be identified with the left multiplication by
$\frac{1}{e_{i}}z_{i}\partial_{i}$ for $i\leq k$, by choosing the
trivialization of $\Omega^{n}_{X/\Delta}(\log Y)$, because of the relations
$D_{i}=\frac{1}{e_{i}}z_{i}\partial_{i}-\frac{1}{e_{0}}z_{0}\partial_{0}$ for
$1\leq i\leq k$. This means for any function $g\in
f^{-1}{\mathscr{O}}_{\Delta}$, we have $[\nabla,g]=tg^{\prime}$ where $t$ and
$g$ are local sections of $f^{-1}{\mathscr{O}}_{\Delta}$ acting on the left of
$\tilde{\mathcal{M}}$. This makes $\tilde{\mathcal{M}}$ a
$(f^{-1}{\mathscr{D}}_{\Delta}(\log 0),{\mathscr{D}}_{X})$-bimodule. Using
Godement resolution, the direct image
$Rf_{*}{\mathrm{D}}{\mathrm{R}}_{X}\tilde{\mathcal{M}}$ is a complex of left
${\mathscr{D}}_{\Delta}(\log 0)$-modules. Similarly, as we already saw in the
proof, locally the morphism $R:{\mathcal{M}}\rightarrow{\mathcal{M}}$ can be
identified with left multiplication by $\frac{1}{e_{i}}z_{i}\partial_{i}$ for
$0\leq i\leq k$, meaning $[R,g]=tg^{\prime}=0$ for $g$ local sections of
$f^{-1}{\mathscr{O}}_{\Delta}$ acting left on ${\mathcal{M}}$.

######  Remark 4.12.

The ${\mathscr{D}}_{X}$-module ${\mathcal{M}}$ is even regular holonomic. Even
though it is irrelevant for our purpose, we can also check ${\mathcal{M}}$ is
regular using the definition. Recall that a holonomic right
${\mathscr{D}}_{Z}$-module ${\mathcal{N}}$ is called regular if there exists a
good filtration $F_{\bullet}{\mathcal{N}}$ such that for any
$\sigma\in{\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{Z}$ vanishing on the
charateristic variety of ${\mathcal{N}}$ one has
${\mathrm{g}}{\mathrm{r}}^{F}{\mathcal{N}}\sigma=0$. In the case of
${\mathcal{M}}$, define locally

$G_{\ell}{\mathcal{M}}=\sum_{r,k\geq
0}R^{k}F_{\ell+r}{\mathcal{M}}t^{r}_{\mathrm{red}}$

where $t_{\mathrm{red}}=z_{0}z_{1}\cdots z_{k}$. This is a finite sum because
${\mathcal{M}}$ is supported on $t=0$ and $R$ has a characteristic polynomial.
It follows that $G_{\bullet}$ is a good filtration for ${\mathcal{M}}$. I
claim that $G_{\bullet}{\mathcal{M}}$ gives the filtration in the definition
of the regularity. Since the characteristic variety of ${\mathcal{M}}$ is
locally cut out by
$t_{\mathrm{red}},z_{0}\partial_{0},...,z_{k}\partial_{k},\partial_{k+1},...,\partial_{n}$
(see Theorem 4.7) it suffices to check that
$G_{\ell}{\mathcal{M}}t_{\mathrm{red}}\subset G_{\ell-1}{\mathcal{M}}$,
$G_{\ell}{\mathcal{M}}z_{i}\partial_{i}\subset G_{\ell}{\mathcal{M}}$ for
$0\leq i\leq k$ and $G_{\ell}{\mathcal{M}}\partial_{i}\subset
G_{\ell}{\mathcal{M}}$ for $k+1\leq i\leq n$. It is clear that
$G_{\ell}{\mathcal{M}}t_{\mathrm{red}}\subset G_{\ell-1}{\mathcal{M}}$. Due to
locally
${\mathrm{g}}{\mathrm{r}}^{F}{\mathcal{M}}={\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}/(t,D_{1},D_{2},...,D_{n}){\mathrm{g}}{\mathrm{r}}^{F}{\mathcal{M}}$,
it follows that ${\mathrm{g}}{\mathrm{r}}^{F}{\mathcal{M}}D_{i}=0$ for $1\leq
i\leq n$. In particular,
${\mathrm{g}}{\mathrm{r}}^{F}{\mathcal{M}}\partial_{i}=0$ for $k+1\leq i\leq
n$, i.e. $F_{\ell}{\mathcal{M}}\partial_{i}\subset F_{\ell}{\mathcal{M}}$ for
$k+1\leq i\leq n$. Therefore, for $k+1\leq i\leq n$, because
$\left[t_{\mathrm{red}},\partial_{i}\right]=0$,

$G_{\ell}{\mathcal{M}}\partial_{i}=\sum_{r,k\geq
0}R^{k}F_{\ell+r}{\mathcal{M}}t^{r}_{\mathrm{red}}\partial_{i}\subset\sum_{r,k\geq
0}R^{k}F_{\ell+r}{\mathcal{M}}t^{r}_{\mathrm{red}}=G_{\ell}{\mathcal{M}}.$

Since
$[t^{r}_{\mathrm{red}},z_{i}\partial_{i}]=\left(z_{i}\partial_{i}-r\right)t^{r}_{\mathrm{red}}$,
and $[z_{i}\partial_{i},F_{\ell}{\mathscr{D}}_{X}]\subset
F_{\ell}{\mathscr{D}}_{X}$, we have

$R^{k}F_{\ell+r}{\mathcal{M}}t^{r}_{\mathrm{red}}z_{i}\partial_{i}=R^{k}F_{\ell+r}{\mathcal{M}}\left(z_{i}\partial_{i}-r\right)t^{r}_{\mathrm{red}}\subset
R^{k}\left(z_{i}\partial_{i}F_{\ell+r}{\mathcal{M}}+F_{\ell+r}{\mathcal{M}}\right)t^{r}_{\mathrm{red}}.$

But locally $R$ has the same effect as the left multiplication by one of
$\frac{1}{e_{i}}z_{i}\partial_{i}$ for $0\leq i\leq k$. Hence,

$R^{k}\left(z_{i}\partial_{i}F_{\ell+r}{\mathcal{M}}+F_{\ell+r}{\mathcal{M}}\right)t^{r}_{\mathrm{red}}=R^{k+1}F_{\ell+r}{\mathcal{M}}t^{r}_{\mathrm{red}}+R^{k}F_{\ell+r}{\mathcal{M}}t^{r}_{\mathrm{red}}.$

It follows that $G_{\ell}{\mathcal{M}}z_{i}\partial_{i}\subset
G_{\ell}{\mathcal{M}}$ for $0\leq i\leq k$.

In fact, later we will see that ${\mathcal{M}}$ is an extensions of regular
holonomic ${\mathscr{D}}_{X}$-modules which will again prove that
${\mathcal{M}}$ is regular (see Theorem 5.7 for the reduced case and Theorem
7.13 for the general case).

##  5\. Reduced case: Strictness and the weight filtration

We begin to study the weight filtration $W_{\bullet}{\mathcal{M}}$ induced $R$
on ${\mathcal{M}}$. For simplicity to state the results and illustrate the
ideas, we assume $Y$ is reduced in $\S$5 and $\S$6. The general case will be
treated in $\S$7 and $\S$8. Since $Y$ is reduced, the multiplicity $e_{i}$ of
irreducible component $Y_{i}$ is $1$ and $R$ is nilpotent. Recall that the