weight filtration of the nilpotent operator $R$ is uniquely characterized by
the following two properties:

  * •

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

  * •

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

###  5.1. Strictness of $R$

Let $F_{\bullet}W_{r}{\mathcal{M}}=F_{\bullet}{\mathcal{M}}\cap
W_{r}{\mathcal{M}}$ be the induced filtration for every integer $r$. In fact,
the good filtration and the weight filtration interact nicely because of the
following theorem.

######  Theorem 5.1.

The power of $R$ is strict on $({\mathcal{M}},F_{\bullet}{\mathcal{M}})$,
i.e., $R^{a}F_{b}{\mathcal{M}}=F_{a+b}R^{a}{\mathcal{M}}$.

###### Proof.

The strictness is a local property; therefore, we can assume
${\mathcal{M}}={\mathscr{D}}_{X}/(t,D_{1},D_{2},...,D_{n}){\mathscr{D}}_{X}$
and $R$ is left multiplication by $z_{0}\partial_{0}$ on it, recalling that
$D_{i}=z_{i}\partial_{i}-z_{0}\partial_{0}$ for $1\leq i\leq k$ and
$D_{i}=\partial_{i}$ for $k+1\leq i\leq n$. It is clear that
$R^{a}F_{b}{\mathcal{M}}$ is contained in $F_{a+b}R^{a}{\mathcal{M}}$. It
suffices to show that for every $R^{a}P\in F_{a+b}{\mathcal{M}}$, we can find
an element $Q\in F_{b}{\mathcal{M}}$ such that $R^{a}P=R^{a}Q$. Assume $P\in
F_{\ell}{\mathcal{M}}$. If $\ell\leq b$ then there is nothing to prove. Thus,
we consider the situation that $\ell>b$. Then the class of $R^{a}P$ vanishes
in ${\mathrm{g}}{\mathrm{r}}^{F}_{a+\ell}{\mathcal{M}}$. In fact, we have the
following lemma:

######  Lemma 5.2.

Denote by $[R]$ the induced operator on
${\mathrm{g}}{\mathrm{r}}^{F}{\mathcal{M}}$. Then $\ker[R]^{r+1}$ is locally
generated by the classes of all degree $k-r$ monomials dividing
$t=z_{0}z_{1}\cdots z_{k}$.

We can easily check that monomials of degree $k-r$ dividing $t$ is in
$\ker[R]^{r+1}$. Indeed, it is already true that monomials of degree $k-r$
dividing $t$ is in $\ker R^{r+1}$. Without loss of generality, we only need to
check this for the monomial $z_{r+1}z_{r+2}\cdots z_{k}$:

$R^{r+1}z_{r+1}z_{r+2}\cdots z_{k}=z_{0}\partial_{0}z_{1}\partial_{1}\cdots
z_{r}\partial_{r}z_{r+1}z_{r+2}\cdots
z_{k}=t\partial_{0}\cdots\partial_{k}=0\in{\mathcal{M}}.$

We will prove the opposite direction after finishing the proof of the theorem.
Going back to the proof of the theorem, by the above lemma,

$P=\sum_{\begin{subarray}{c}J\subset I,\\\
\\#J=k-a+1\end{subarray}}z_{J}Q_{J}+Q_{\ell-1}$

where $z_{J}=\prod_{j\in J}z_{j}$, $Q_{J}\in F_{\ell}{\mathcal{M}}$ and
$Q_{\ell-1}\in F_{\ell-1}{\mathcal{M}}$. But $R^{a}$ kills the monomials
$z_{J}$ of degree $k-a+1$ dividing $t$. It follows that
$R^{a}P=R^{a}Q_{\ell-1}$. Iterating the procedure, we eventually find an
element $Q\in F_{b}{\mathcal{M}}$ such that $R^{a}P=R^{a}Q$ with $Q\in
F_{b}{\mathcal{M}}$. ∎

###### Proof of Lemma 5.2.

Note that we are over the commutative ring
${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}$. We proceed by induction on
$r$. Let $P\in{\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}$ be a
representative of an element in $\ker[R]^{r+1}$. When $r=0$, we have

$z_{0}\partial_{0}P=tQ_{0}+\sum^{n}_{i=1}D_{i}Q_{i}.$

Then
$tQ_{0}\in(\partial_{0},\partial_{1},...,\partial_{n}){\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}$.
Notice that $t,\partial_{0},\partial_{1},...,\partial_{n}$ is a regular
sequence over ${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}$. We have
$Q_{0}=\sum^{n}_{i=0}\partial_{i}Q^{\prime}_{i}$. This implies

$\displaystyle z_{0}\partial_{0}P$
$\displaystyle=\sum^{k}_{i=0}\frac{t}{z_{i}}z_{i}\partial_{i}Q^{\prime}_{i}+\sum^{n}_{j=k+1}t\partial_{j}Q^{\prime}_{j}+\sum^{n}_{i=1}D_{i}Q_{i}$
$\displaystyle=\sum^{k}_{i=0}\frac{t}{z_{i}}z_{0}\partial_{0}Q^{\prime}_{i}+\sum^{k}_{i=1}D_{i}(Q_{i}+\frac{t}{z_{i}}Q^{\prime}_{i})+\sum^{n}_{j=k+1}D_{j}(Q_{j}+tQ^{\prime}_{j}),$

from which we conclude that
$z_{0}\partial_{0}(P-\sum^{k}_{i=0}\frac{t}{z_{i}}Q^{\prime}_{i})\in(D_{1},D_{2},...,D_{n}){\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}$.
Because $z_{0}\partial_{0},D_{1},D_{2},...,D_{n}$ is again a regular sequence,
we see that
$P-\sum^{k}_{i=0}\frac{t}{z_{i}}Q^{\prime}_{i}\in(D_{1},D_{2},...,D_{n}){\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}$.
This concludes the base case for the induction.

Assume the statement is true for the cases when the exponent is less then
$r+1$. Let $z_{J}=\prod_{j\in J}z_{j}$. Now for $[P]\in\ker[R]^{r+1}$, we have
$[R][P]$ is in $\ker[R]^{r}$. By induction,

(5.21) $z_{0}\partial_{0}P=\sum_{\begin{subarray}{c}\\#J=k-r+1,\\\ J\subset
I\end{subarray}}z_{J}Q_{J}+\sum^{n}_{i=1}D_{i}Q_{i}.$

Fix an index subset $J$ of $I$ such that $\\#J=k-r+1$. Then $z_{J}Q_{J}$ is in
the submodule generated by $z_{i}$ for $i\in I\setminus J$ and $\partial_{j}$
for $j\in J$ and $k<j\leq n$ over
${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}$. Since $z_{i}$ for $i\in
I\setminus J$, $\partial_{j}$ for $j\in J$ and $k<j\leq n$ and $z_{J}$ form a
regular sequence, we have

$Q_{J}=\sum_{i\in I\setminus J}z_{i}Q^{\prime}_{i}+\sum_{j\in
J}\partial_{j}Q^{\prime}_{j}+\sum_{k<\ell\leq
n}\partial_{\ell}Q^{\prime}_{\ell}.$

Therefore, it follows that

$z_{J}Q_{J}=\sum_{i\in I\setminus J}z_{J}z_{i}Q^{\prime}_{i}+\sum_{j\in
J}\left(\frac{z_{J}}{z_{j}}z_{0}\partial_{0}Q^{\prime}_{j}+D_{j}\frac{z_{J}}{z_{j}}Q^{\prime}_{j}\right)+\sum_{k<\ell\leq
n}D_{\ell}z_{J}Q^{\prime}_{\ell}.$

Then substuiting in (5.21), we deduce that

$z_{0}\partial_{0}\left(P-\sum_{j\in
J}\frac{z_{J}}{z_{j}}Q^{\prime}_{j}\right)-\sum_{i\in I\setminus
J}z_{J}z_{i}Q^{\prime}_{i}$

is in the submodule generated by degree $k-r+1$ monomials dividing $t$ except
$z_{J}$, and $D_{1},D_{2},...,D_{n}$ over
${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}$. It follows that we can reduce
the monomials of degree $k-r+1$ dividing $t$ in the right-hand side equation
(5.21) one by one and at the last step, we get
$z_{0}\partial_{0}\left(P-P^{\prime}\right)-Q^{\prime}$, where $P^{\prime}$ is
a linear combination of degree $k-r$ monomials dividing $t$ and $Q^{\prime}$
is a linear combination of $k-r+2$ monomials dividing $t$, is in the submodule
generated by $D_{1},...,D_{n}$ over
${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}$. But $\ker[R]^{r-1}$ is
generated by classes represented by degree $k-r+2$ monomials dividing $t$ by
induction hypothesis. It says that the class of $P-P^{\prime}$ is in
$\ker[R]^{r}$ and by induction it is generated by degree $k-r+1$ monomials
dividing $t$. Therefore, $P$ is a linear combination of degree $k-r$ monomials
dividing $t$. This completes the proof. ∎

######  Corollary 5.3.

The $\ker R^{r+1}$ is also generated by degree $k-r$ monomials dividing $t$ if
one identifies ${\mathcal{M}}$ locally with
${\mathscr{D}}_{X}/(t,D_{1},D_{2},...,D_{n}){\mathscr{D}}_{X}$.

###### Proof.

It suffices to show that ${\mathrm{g}}{\mathrm{r}}^{F}\ker R^{r+1}$ is
generated by degree $k-r$ monomials dividing $t$. Notice that
${\mathrm{g}}{\mathrm{r}}^{F}\ker R^{r+1}$ is contained in $\ker[R]^{r+1}$,
since $[R]^{r+1}$ vanishes on ${\mathrm{g}}{\mathrm{r}}^{F}\ker R^{r+1}$. In
fact, we have ${\mathrm{g}}{\mathrm{r}}^{F}\ker R^{r+1}=\ker[R]^{r+1}$ because
degree $k-r$ monomials dividing $t$ are also in
${\mathrm{g}}{\mathrm{r}}^{F}\ker R^{r+1}$. ∎

###  5.2. The weight filtration

The results concerning the weight filtration and Lefschetz decomposition are
formal and we will work on the abstract setting.

######  Theorem 5.4.

Let $N:(\mathcal{G},F_{\bullet})\rightarrow(\mathcal{G},F_{\bullet+1})$ be a
nilpotent operator on a filtered ${\mathscr{D}}$-module
$({\mathcal{G}},F_{\bullet})$. Asume that every power of $N$ satisfies
strictness, i.e., $N^{a}F_{b}{\mathcal{G}}=F_{a+b}N^{a}{\mathcal{G}}$ for
$a\geq 0$ and $b\in{\mathbb{Z}}$. Then the induced operator
$N^{r}:F_{\ell}{\mathrm{g}}{\mathrm{r}}^{W}_{r}{\mathcal{G}}\rightarrow
F_{\ell+r}{\mathrm{g}}{\mathrm{r}}^{W}_{-r}{\mathcal{G}}$ is an isomorphism
for $r\geq 0$, where $W_{\bullet}$ is the weight filtration induced by $N$.

###### Proof.

It suffices to prove that for any $b\in F_{\ell+r}W_{-r}{\mathcal{G}}$, we
could find $a^{\prime}\in F_{\ell}W_{r}{\mathcal{G}}$ such that
$a=N^{r}a^{\prime}$. Because $W_{-r}{\mathcal{G}}\subset N^{r}{\mathcal{G}}$,
let $N^{r}a=b$ for some $a$. Then by strictness, there exists $a^{\prime}\in
F_{\ell}{\mathcal{G}}$ such that $N^{r}a^{\prime}=N^{r}a\in
W_{-r}{\mathcal{G}}$. It follows that $a^{\prime}\in W_{r}{\mathcal{G}}$.
Indeed, if $a^{\prime}\in W_{r+k}{\mathcal{G}}$ for some $k>0$ such that
$a^{\prime}\neq 0\in{\mathrm{g}}{\mathrm{r}}^{W}_{r+k}{\mathcal{G}}$. Then
$N^{r+k}a^{\prime}=0\in{\mathrm{g}}{\mathrm{r}}^{W}_{-r-k}{\mathcal{G}}$
because
$N^{r}a^{\prime}=0\in{\mathrm{g}}{\mathrm{r}}^{W}_{-r+k}{\mathcal{G}}$, from
which we conclude that $a^{\prime}\in F_{\ell}W_{r+k-1}{\mathcal{G}}$. Thus,
iterating the procedure, $a^{\prime}$ is actually in
$F_{\ell}W_{r}{\mathcal{G}}$. We conclude the proof.∎

Let
${\mathcal{P}}_{r}=_{\mathrm{def}}\ker\left(N^{r+1}:{\mathrm{g}}{\mathrm{r}}^{W}_{r}{\mathcal{G}}\rightarrow{\mathrm{g}}{\mathrm{r}}^{W}_{-r-2}{\mathcal{G}}\right)$
be the primitive part of ${\mathrm{g}}{\mathrm{r}}^{W}{\mathcal{G}}$, which
can be identified with

$\frac{\ker N^{r+1}}{\ker N^{r}+N\ker N^{r+2}}.$

See Example 2.7. Recall the Lefschetz decomposition:

${\mathrm{g}}{\mathrm{r}}^{W}_{r}{\mathcal{G}}=\bigoplus_{\begin{subarray}{c}\ell\geq
0,-\frac{r}{2}\end{subarray}}N^{\ell}{\mathcal{P}}_{r+2\ell}\text{ \,for any\,
}r\in{\mathbb{Z}}.$

There are two possible ways to define the filtration on ${\mathcal{P}}_{r}$:
first we have the natural filtration $F_{\ell}{\mathcal{P}}_{r}$ induced from
the inclusion
${\mathcal{P}}_{r}\rightarrow{\mathrm{g}}{\mathrm{r}}^{W}_{r}{\mathcal{G}}$
and second we can also define the filtration using

$\frac{F_{\ell}\ker N^{r+1}+\ker N^{r}+N\ker N^{r+2}}{\ker N^{r}+N\ker
N^{r+2}}.$

But indeed, the two different methods result in the same filtration because of
the strictness. Let $m\in F_{\ell}W_{r}+W_{r-1}$ such that $N^{r+1}m\in
W_{-r-3}$ so that represents a class in $F_{\ell}{\mathcal{P}}_{r}$. It
suffices to find an element in $F_{\ell}\ker N^{r+1}$ representing the same
class as $m$ in $F_{\ell}{\mathcal{P}}_{r}$. Let $m=m_{1}+m_{2}$ for $m_{1}\in
F_{\ell}W_{r}$ and $m_{2}\in W_{r-1}$. It follows that $N^{r+1}m_{1}\in
F_{\ell+r+1}W_{-r-3}$ because both $N^{r+1}m,N^{r+1}m_{2}\in W_{-r-3}$ and
$m_{1}\in F_{\ell}W_{r}$. Since $N^{r+3}:F_{\ell-2}W_{r+3}\rightarrow
F_{\ell+r+1}W_{-r-3}$ is surjective, there exists $x\in F_{\ell-2}W_{r+3}$
such that $N^{r+3}x=N^{r+1}m_{1}\in F_{\ell+r+1}W_{-r-3}$. See the proof of
the above theorem. It follows that $m_{1}-N^{2}x\in F_{\ell}\ker N^{r+1}$
represents the same element as $m$ in $F_{\ell}{\mathcal{P}}_{r}\subset
F_{\ell}{\mathrm{g}}{\mathrm{r}}^{W}_{r}$.

######  Corollary 5.5.

The Lefschetz decomposition of ${\mathrm{g}}{\mathrm{r}}^{W}{\mathcal{G}}$
respects filtrations, i.e.

$F_{\bullet}{\mathrm{g}}{\mathrm{r}}^{W}_{r}{\mathcal{G}}=\bigoplus_{\begin{subarray}{c}\ell\geq
0,-\frac{r}{2}\end{subarray}}N^{\ell}F_{\bullet-\ell}{\mathcal{P}}_{r+2\ell}\text{
\,for any\, }r\in{\mathbb{Z}}.$

Returning to our situation, it follows that:

######  Theorem 5.6.

The induced operator
$R^{r}:F_{\ell}{\mathrm{g}}{\mathrm{r}}^{W}_{r}{\mathcal{M}}\rightarrow
F_{\ell+r}{\mathrm{g}}{\mathrm{r}}^{W}_{-r}{\mathcal{M}}$ is an isomorphism.
Therefore, the Lefschetz decomposition of
${\mathrm{g}}{\mathrm{r}}^{W}{\mathcal{M}}$ respects filtrations, i.e.

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

###  5.3. Identifying the primitive part ${\mathcal{P}}_{r}$

Recall that $Y^{J}=\cap_{j\in J}Y_{j}$ for a subset $J$ of the index set $I$
and $\tilde{Y}^{(r+1)}$ is the disjoint union of $Y^{J}$ such that the
cardinality of $J$ is $r+1$. The morphism
$\tau^{(r+1)}:\tilde{Y}^{(r+1)}\rightarrow X$ is the natural morphism induced
by the closed embeddings $\tau^{J}:Y^{J}\rightarrow X$.

######  Theorem 5.7.

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

###### Proof.

Denote by $D^{J}$ the normal crossing divisor $Y^{J}\cap Y_{I\setminus J}$ on
$Y^{J}$. The residue morphism

${\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{\tilde{Y}^{(r+1)}}:\Omega^{\bullet+n+1}_{X}(\log
Y)|_{Y}\rightarrow\bigoplus_{\\#J=r+1}\Omega^{\bullet+n-r}_{Y^{J}}(\log
D^{J})$

extends to a morphism of complexes of filtered induced
${\mathscr{D}}_{X}$-modules

${\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{\tilde{Y}^{(r+1)}}:\Omega^{\bullet+n+1}_{X}(\log
Y)|_{Y}\otimes{\mathscr{D}}_{X}\rightarrow\bigoplus_{\\#J=r+1}\Omega^{\bullet+n-r}_{Y^{J}}(\log
D^{J})\otimes{\mathscr{D}}_{X}.$

Denote by ${\mathcal{H}}^{k}$ the $k$-th cohomology
${\mathscr{H}}^{k}\left(\Omega^{\bullet+n+1}_{X}(\log
Y)|_{Y}\otimes{\mathscr{D}}_{X}\right)$. Taking $0$-th cohomology of the above
yields, by Example 2.4

${\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{\tilde{Y}^{(r+1)}}:{\mathcal{H}}^{0}\rightarrow\bigoplus_{\\#J=r+1}\tau^{J}_{+}\omega_{Y^{J}}(*D^{J})(-r).$

Since the morphism $\frac{dt}{t}\wedge:\Omega^{\bullet+n}_{X/\Delta}(\log
Y)\rightarrow\Omega^{\bullet+n+1}_{X}(\log Y)$ also extends to the complexes
of induced ${\mathscr{D}}_{X}$-modules, we have a short exact sequence of
${\mathscr{D}}_{X}$-modules

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

Considering the associated long exact sequence

(5.22)
${0}$${{\mathcal{H}}^{-1}}$${{\mathcal{M}}}$${{\mathcal{M}}}$${{\mathcal{H}}^{0}}$${0}$$\scriptstyle{R}$$\scriptstyle{\frac{dt}{t}\wedge}$

we have the morphism
$\frac{dt}{t}\wedge:{\mathcal{M}}\rightarrow{\mathcal{H}}^{0}$ and it vanishes
on the image of $R$. To motivate the proof, let me do some local calculation.
Let
$\zeta=\frac{dz_{1}}{z_{1}}\wedge\frac{dz_{2}}{z_{2}}\wedge\cdots\wedge\frac{dz_{k}}{z_{k}}\wedge\cdots\wedge
dz_{n}$ represent a local frame of $\Omega^{n}_{X/\Delta}(\log Y)|_{Y}$. Then
a local section of ${\mathcal{M}}$ is represented by $\zeta\otimes P$ for $P$
a local section ${\mathscr{D}}_{X}$. Then
${\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{\tilde{Y}^{(r+1)}}\frac{dt}{t}\wedge\zeta\otimes
P$ is a section of $\bigoplus_{\\#J=r+1}\Omega^{n-r}_{Y^{J}}(\log
D^{J})\otimes{\mathscr{D}}_{X}$. Post-composing with the projection

$\bigoplus_{\\#J=r+1}\Omega^{n-r}_{Y^{J}}(\log
D^{J})\otimes{\mathscr{D}}_{X}\rightarrow\bigoplus_{\\#J=r+1}\tau^{J}_{+}\omega_{Y^{J}}(*D^{J})(-r),$

we make the morphism explicit:

${\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{\tilde{Y}^{(r+1)}}\circ\frac{dt}{t}\wedge:{\mathcal{M}}\rightarrow\bigoplus_{\\#J=r+1}\tau^{J}_{+}\omega_{Y^{J}}(*D^{J})(-r),\quad[\zeta\otimes
P]\mapsto[{\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{\tilde{Y}^{(r+1)}}\frac{dt}{t}\wedge\zeta\otimes
P].$

Let $\zeta\otimes z_{\overline{J}}P$ represent a class in $\ker R^{r+1}$ for
some fixed ordered index subset $J$ with $\\#J=r+1$, where
$z_{\overline{J}}=\prod_{j\in I\setminus J}z_{j}$ (Corollary 5.3). Its image
under the above morphism only contained in the component
$\tau^{J}_{+}\omega_{Y^{J}}(*D^{J})(-r)$ because $z_{\overline{J}}$ vanishes
on other components. Thus, the image is the class represented by

(5.23)
${\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{\tilde{Y}^{r+1}}\frac{dz_{0}}{z_{0}}\wedge\frac{dz_{1}}{z_{1}}\wedge\cdots\frac{dz_{k}}{z_{k}}\wedge\cdots\wedge
dz_{n}\otimes
z_{\overline{J}}P=\pm\frac{dz_{\overline{J}}}{z_{\overline{J}}}\wedge
dz_{k+1}\wedge\cdots dz_{n}\otimes
z_{\overline{J}}P\in\Omega^{n-r}_{Y^{J}}(\log D^{J})\otimes{\mathscr{D}}_{X},$

where $\frac{dz_{\overline{J}}}{z_{\overline{J}}}=\bigwedge_{j\in I\setminus
J}\frac{dz_{j}}{z_{j}}$ and the sign depends on the order of $J$. In fact,
from the calculation we see that the image does not have any pole along
$D^{J}$, so it is contained in the subsheaf consisting of classes represented
by $\Omega^{n-r}_{Y^{J}}\otimes{\mathscr{D}}_{X}$. This means that the class
of (5.23) in $\tau^{J}_{+}\omega_{Y^{J}}(*D^{J})(-r)$ is also contained in the
image of the inclusion

$\tau^{J}_{+}\omega_{Y^{J}}(-r)\rightarrow\tau^{J}_{+}\omega_{Y^{J}}(*D^{J})(-r),\quad[{dz_{\overline{J}}}\wedge
dz_{k+1}\wedge\cdots dz_{n}\otimes
P]\mapsto[\frac{dz_{\overline{J}}}{z_{\overline{J}}}\wedge
dz_{k+1}\wedge\cdots dz_{n}\otimes z_{\overline{J}}P].$

See Example 2.4. It follows that we obtain a factorization $\rho_{r}:\ker
R^{r+1}\rightarrow\tau^{(r+1)}_{+}\omega_{Y^{(r+1)}}(-r)$. In conclusion, we
have the following commutative diagram.

${\ker
R^{r+1}}$${\tau^{(r+1)}_{+}\omega_{Y^{(r+1)}}(-r)}$${{\mathcal{M}}}$${{\mathcal{H}}^{0}}$${\bigoplus_{\\#J=r+1}\tau^{J}_{+}\omega_{Y^{J}}(*D^{J})(-r)}$$\scriptstyle{\rho_{r}}$$\scriptstyle{\frac{dt}{t}\wedge}$$\scriptstyle{{\mathrm{R}}{\mathrm{e}}{\mathrm{s}}}$

For a local section $\zeta\otimes z_{K}P$ where $z_{K}=\prod_{i\in K}z_{i}$ a
monomial of degree $k-r+1$, representing a class in $\ker R^{r}$, its image
under $\rho_{r}$ is indeed zero because $z_{K}$ annihilates all
$\Omega^{n-r}_{Y^{J}}(\log D^{J})$ for index subset $J$ such that $\\#J=r+1$.
This implies the morphism $\rho_{r}$ kills $\ker R^{r}$. The morphism
$\rho_{r}$ also kills $R\ker R^{r+2}$, because by (5.22) $\frac{dt}{t}\wedge$
vanishes on the image of $R$. Thus it factors through

$\phi_{r}:{\mathcal{P}}_{r}=\frac{\ker R^{r+1}}{\ker R^{r}+R\ker
R^{r+2}}\rightarrow\tau^{(r+1)}_{+}\omega_{Y^{(r+1)}}(-r).$

The morphism $\phi_{r}$ is filtered surjective because for $dz_{\bar{J}}\wedge
dz_{k+1}\wedge\cdots\wedge dz_{n}\otimes P\in\Omega^{n-r}_{Y^{J}}\otimes
F_{\ell}{\mathscr{D}}_{X}$ representing a class in
$F_{\ell}\tau^{J}_{+}\omega_{Y^{J}}(-r)$ with $\\#J=r+1$, we can find a
lifting class represented by $\zeta\otimes z_{\overline{J}}P$ in $F_{\ell}\ker
R^{r+1}$. It follows that

$cc({\mathcal{P}}_{r})\geq
cc(\tau^{(r+1)}_{+}\omega_{Y^{(r+1)}})=\sum_{\\#J=r+1}\left[T^{*}_{Y^{J}}X\right].$

Summing up the inequalities gives

$\sum_{r\geq 0}(r+1)cc({\mathcal{P}}_{r})\geq\sum_{r\geq
0}(r+1)\sum_{\\#J=r+1}\left[T^{*}_{Y^{J}}X\right]=\sum_{J\subset
I}(\\#J)\left[T^{*}_{Y^{J}}X\right].$

On the other hand, by the Lefschetz decomposition and Theorem 4.7, we have

$\sum_{J\subset
I}(\\#J)\left[T^{*}_{Y^{J}}X\right]=cc({\mathcal{M}})=cc({\mathrm{g}}{\mathrm{r}}^{W}{\mathcal{M}})=\sum_{r\geq
0}(r+1)cc({\mathcal{P}}_{r}).$

Therefore, all inequalities must be equalities, i.e.
$cc({\mathcal{P}}_{r})=cc(\tau^{(r+1)}_{+}\omega_{\tilde{Y}^{(r+1)}})$. It
follows that $\phi_{r}$ is a filtered isomorphism [HTT08, Proposition 3.1.2].
∎

##  6\. Reduced case: Sesquilinear pairing on ${\mathcal{M}}$ and limiting
mixed Hodge structure

###  6.1. Sesquilinear pairing

We begin to construct the last data we need for the limiting mixed Hodge
structure – Sesquilinear pairing. In the sense that ${\mathcal{M}}$ is the
specialization of ${i_{X_{t}}}_{+}\omega_{X_{t}}$ for $t\neq 0$, the
sesquilinear
$S:{\mathcal{M}}\otimes_{\mathbb{C}}\overline{{\mathcal{M}}}\rightarrow\mathfrak{C}_{X}$
should also be the specialization of ${i_{X_{t}}}_{+}S_{X_{t}}$, where
$S_{X_{t}}$ is defined in $\S$2. Presumably one would expect that the pairing

$\displaystyle\langle S(\left[\zeta_{1}\otimes
P_{1}\right],\left[\zeta_{2}\otimes P_{2}\right]),\eta\rangle$
$\displaystyle=\lim_{t\rightarrow
0}\langle{i_{X_{t}}}_{+}S_{X_{t}}(\zeta_{1}\otimes P_{1},\zeta_{2}\otimes
P_{2}),\eta\rangle$ $\displaystyle=\lim_{t\rightarrow
0}\frac{\varepsilon(n+1)}{(2\pi\sqrt{-1})^{n}}\int_{X_{t}}P_{1}\overline{P_{2}}\eta\wedge\zeta_{1}\wedge\overline{\zeta_{2}}$

should work on ${\mathcal{M}}$ for $\zeta_{i}\otimes P_{i}$, $i=1,2$ sections
of $\Omega^{n}_{X/\Delta}\otimes{\mathscr{D}}_{X}$ over local chart $U$
representing classes of ${\mathcal{M}}$, and $\eta$ is a test function over
$U$. But one could check that the integral
$\int_{X_{t}}P_{1}\overline{P_{2}}\left(\eta\right)\zeta_{1}\overline{\wedge\zeta_{2}}$
could have order $(-\log|t|^{2})^{k}$ near the origin where $k+1$ is the
number of components that intersect in $U$, so the limit may not exist. To
avoid the issue, we use a Mellin transform device (see [Sab02, 4.E]): locally

$\displaystyle\langle S(\left[\zeta_{1}\otimes
P_{1}\right],\left[\zeta_{2}\otimes P_{2}\right]),\eta\rangle$
$\displaystyle=_{\mathrm{def}}{\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{s=0}\frac{\varepsilon(n+2)}{(2\pi\sqrt{-1})^{n+1}}\int_{X}|t|^{2s}P_{1}\overline{P_{2}}\eta\frac{dt}{t}\wedge\zeta_{1}\wedge\overline{\frac{dt}{t}\wedge\zeta_{2}}$
$\displaystyle={\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{s=0}\frac{\varepsilon(2)}{2\pi\sqrt{-1}}\int_{\Delta}|t|^{2s}\frac{dt}{t}\wedge\overline{\frac{dt}{t}}\left(\frac{\varepsilon(n+1)}{(2\pi\sqrt{-1})^{n}}\int_{X_{t}}P_{1}\overline{P_{2}}\eta\wedge\zeta_{1}\wedge\overline{\zeta_{2}}\right)$
$\displaystyle={\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{s=0}\frac{\varepsilon(2)}{2\pi\sqrt{-1}}\int_{\Delta}|t|^{2s}\frac{dt}{t}\wedge\overline{\frac{dt}{t}}\langle{i_{X_{t}}}_{+}S_{X_{t}}(\zeta_{1}\otimes
P_{1},\zeta_{2}\otimes P_{2}),\eta\rangle.$

The last expression in the definition in some extent explains that $S$ is the
specialization of ${i_{X_{t}}}_{+}S_{X_{t}}$ and the $0$-current
${\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{s=0}\frac{\varepsilon(2)}{2\pi\sqrt{-1}}\int_{\Delta}|t|^{2s}\frac{dt}{t}\wedge\overline{\frac{dt}{t}}$
is doing the job of renormalization of $\ {i_{X_{t}}}_{+}S_{X_{t}}$ for $t\neq
0$. In fact, for any test function $g$ on $\Delta$, we have

${\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{s=0}\frac{\varepsilon(2)}{2\pi\sqrt{-1}}\int_{\Delta}|t|^{2s}\frac{dt}{t}\wedge\overline{\frac{dt}{t}}g=g(0).$

We have not check that $S$ is well-defined, but let us do an example to see
how the Mellin transform works.

######  Example 6.1.

Suppose $Y$ is smooth, then $R$ is identical zero and
${\mathcal{M}}\simeq{i_{Y}}_{+}\omega_{Y}$, by Theorem 5.7. Thus, the pairing
$S$ should recover the natural pairing $S_{Y}$. In local coordinates $t=z_{0}$
and for any local sections $\zeta_{i}\otimes P_{i}=dz_{1}\wedge
dz_{2}\wedge\cdots\wedge dz_{n}\otimes P_{i}$ of $\Omega^{n}_{X/\Delta}(\log
Y)\otimes{\mathscr{D}}_{X}$, $i=1,2$ over local chart $U$,

$\displaystyle\langle S(\left[\zeta_{1}\otimes
P_{1}\right],\left[\zeta_{2}\otimes P_{2}\right]),\eta\rangle$
$\displaystyle={\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{s=0}\frac{\varepsilon(n+2)}{(2\pi\sqrt{-1})^{n+1}}\int_{X}|t|^{2s}P_{1}\overline{P_{2}}(\eta)\frac{dt}{t}\wedge\zeta_{1}\wedge\overline{\frac{dt}{t}\wedge\zeta_{2}}$
$\displaystyle={\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{s=0}\int_{X}|t|^{2s-2}P_{1}\overline{P_{2}}(\eta)\bigwedge^{n}_{i=0}\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge\overline{dz_{i}}$
$\displaystyle\text{ integration by parts on }t\text{ and }\bar{t}$
$\displaystyle={\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{s=0}\int_{X}\frac{|t|^{2s}}{s^{2}}\partial_{0}\overline{\partial_{0}}\left(P_{1}\overline{P_{2}}(\eta)\right)\bigwedge^{n}_{i=0}\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge\overline{dz_{i}}.$

Because the Laurent expansion of $s^{-2}|t|^{2s}$ is
$\sum^{\infty}_{\ell=0}\left(\log|t|^{2}\right)^{\ell}s^{\ell-2}$, the above
continuously equals to, by Poincaré-Lelong equation [GH14, Page 388]

$\displaystyle\int_{X}\log|t|^{2}\partial_{0}\overline{\partial_{0}}\left(P_{1}\overline{P_{2}}(\eta)\right)\bigwedge^{n}_{i=0}\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge\overline{dz_{i}}$
$\displaystyle=\int_{Y}P_{1}\overline{P_{2}}(\eta)\bigwedge^{n}_{i=1}\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge\overline{dz_{i}}$
$\displaystyle=\frac{\varepsilon(n+1)}{(2\pi\sqrt{1})^{n}}\int_{Y}P_{1}\overline{P_{2}}(\eta)\zeta_{1}\wedge\overline{\zeta_{2}}$
$\displaystyle=\langle{i_{Y}}_{+}S_{Y}([\zeta_{1}\otimes
P_{1}],[\zeta_{2}\otimes P_{2}]),\eta\rangle.$

We can take a cleaner point of view. In the case $Y$ is smooth, the form
$P_{1}\overline{P_{2}}(\eta)\zeta_{1}\overline{\wedge\zeta_{2}}$ is smooth in
the neighborhood of $Y$. It follows that ${i_{X_{t}}}_{+}S_{X_{t}}$ extends
smoothly to $t=0$ and the limit of ${i_{X_{t}}}_{+}S_{X_{t}}$ is exactly
${i_{Y}}_{+}S_{Y}$.

When $Y$ has several smooth irreducible components, the idea of computation is
similar to the above. Now we begin to establish the statements needed to
ensure $S$ is well-defined. For any test function $\eta$ over an arbitrary
open subset $U$ of $X$ and two sections $m_{1},m_{2}$ in
$H^{0}\left(U,\Omega^{n}_{X/\Delta}(\log Y)\otimes{\mathscr{D}}_{X}\right)$,
the $(2n+2)$-form $\frac{dt}{t}\wedge m_{1}\wedge\overline{\frac{dt}{t}\wedge
m_{2}}(\eta)$ is smooth away from $Y$ but with poles along $Y$ supported in
$U$. Locally, say $m_{i}=\zeta\otimes P_{i}$ for
$\zeta=\frac{dz_{1}}{z_{1}}\wedge\frac{dz_{2}}{z_{2}}\wedge\cdots\frac{dz_{k}}{z_{k}}\wedge
dz_{k+1}\wedge\cdots\wedge dz_{n}$ and $i=1,2$, the $(2n+2)$-form
$\frac{dt}{t}\wedge m_{1}\wedge\overline{\frac{dt}{t}\wedge m_{2}}(\eta)$ is
just
$P_{1}\overline{P_{2}}(\eta)\frac{dt}{t}\wedge\zeta\wedge\overline{\frac{dt}{t}\wedge\zeta}$.
Let $F(s)=F(s,m_{1},m_{2},\eta)$ be the meromorphic continuation via
integration by parts of the following function

$\frac{\varepsilon(n+2)}{(2\pi\sqrt{-1})^{n+1}}\int_{X}|t|^{2s}\frac{dt}{t}\wedge
m_{1}\wedge\overline{\frac{dt}{t}\wedge m_{2}}(\eta).$

The function $F(s)$ is holomorphic when $\mathrm{Re\,}s>0$ and has potential
poles at non-positive integers. Note that $F(s)$ is independent of local
coordinates. We are only interested in the polar part of the function $F(s)$
at $s=0$.

######  Theorem 6.2.

The polar part of $F(s)$ at $s=0$ only depends on the classes of $m_{1}$ and
$m_{2}$ in ${\mathcal{M}}$.

###### Proof.

Let $\left\\{\rho_{\lambda}\right\\}$ be a partition of unity of the open
covering $\left\\{U_{\lambda}\right\\}$ by local charts. Then

$F(s)=\sum_{\lambda}\frac{\varepsilon(n+2)}{(2\pi\sqrt{-1})^{n+1}}\int_{U_{\lambda}}|t|^{2s}\frac{dt}{t}\wedge
m_{1}\wedge\overline{\frac{dt}{t}\wedge m_{2}}(\rho_{\lambda}\eta).$

Since $\rho_{\lambda}\eta$ is a test function over $U_{\lambda}$, without loss
of generality, we can assume $U$ itself is a local chart. It follows that we
can assume that $m_{i}=\zeta\otimes P_{i}$ for $i=1,2$ and
$\zeta=\frac{dz_{1}}{z_{1}}\wedge\frac{dz_{2}}{z_{2}}\wedge\cdots\frac{dz_{k}}{z_{k}}\wedge
dz_{k+1}\wedge\cdots\wedge dz_{n}$. We begin with some properties of $F(s)$.

######  Lemma 6.3.

Under the assumption that $m_{i}=\zeta\otimes P_{i}$ for
$\zeta=\frac{dz_{1}}{z_{1}}\wedge\frac{dz_{2}}{z_{2}}\wedge\cdots\wedge\frac{dz_{k}}{z_{k}}\wedge\cdots\wedge
dz_{n}$ and for $i=1,2$, the followings are valid.

  1. (1)

the order of the pole of $F(s)$ at $s=0$ is at most $k+1$;

  2. (2)

if $P_{i}=tP^{\prime}_{i}$ for one of $i=1,2$, then $F(s)$ is holomorphic at
$s=0$;

  3. (3)

for $0\leq j\leq k$ we have,

$F(s,\zeta_{1}\otimes
P_{1},\zeta_{2}\otimes{z_{j}\partial_{j}}P_{2},\eta)=F(s,\zeta_{1}\otimes
z_{j}\partial_{j}P_{1},\zeta_{2}\otimes P_{2},\eta)=-sF(s,\zeta_{1}\otimes
P_{1},\zeta_{2}\otimes P_{2},\eta).$

###### Proof of the lemma.

The Laurent expansion of $F(s)$ at $s=0$ is

$\displaystyle F(s)$
$\displaystyle=\int_{X}|z_{I}|^{2s-2}P_{1}\overline{P_{2}}(\eta)\bigwedge^{n}_{i=0}(\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge{d\overline{z_{i}}}),\quad\text{where
}z_{I}=\prod_{i\in I}z_{i}$
$\displaystyle=\int_{X}\frac{|z_{I}|^{2s}}{s^{2k+2}}\partial_{I}\overline{\partial_{I}}P_{1}\overline{P_{2}}(\eta)\bigwedge^{n}_{i=0}(\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge{d\overline{z_{i}}}),\quad\text{where
}\partial_{I}=\prod^{k}_{i=0}\partial_{i}$
$\displaystyle=\sum^{\infty}_{\ell=0}\frac{s^{\ell-(2k+2)}}{\ell!}\int_{X}\left(\log|z_{I}|^{2}\right)^{\ell}\partial_{I}\overline{\partial_{I}}P_{1}\overline{P_{2}}(\eta)\bigwedge^{n}_{i=0}(\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge{d\overline{z_{i}}}).$

The order of the pole at $s=0$ is at most $k+1$: if $\ell<k+1$, the form

$\left(\log|z_{I}|^{2}\right)^{\ell}\partial_{I}\overline{\partial_{I}}P_{1}\overline{P_{2}}(\eta)\bigwedge^{n}_{i=0}(\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge{d\overline{z_{i}}})$

is actually exact because one of $a_{i}$’s must be $0$ in the expansion of
$\left(\log|z_{I}|^{2}\right)^{\ell}$ into a linear combination of
$\prod^{k}_{i=0}\left(\log|z_{i}|^{2}\right)^{a_{i}}$ with
$\sum^{k}_{i=0}a_{i}=\ell<k+1$. This proves $(1)$.

Suppose that $P_{1}=tP^{\prime}_{1}$. Then the function

$\displaystyle F(s)$
$\displaystyle=\int_{X}|z_{I}|^{2s-2}tP^{\prime}_{1}\overline{P_{2}}(\eta)\bigwedge^{n}_{i=0}(\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge{d\overline{z_{i}}}).$

is well-defined at $s=0$ because the form

$\frac{1}{\overline{z_{I}}}P^{\prime}_{1}\overline{P_{2}}(\eta)\bigwedge^{n}_{i=0}(\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge{\overline{dz_{i}}})$

is integrable. The same argument works for the case when
$P_{2}=tP^{\prime}_{2}$. This proves $(2)$.

Now we turn to the last statement

$\displaystyle F(s,\zeta\otimes
P_{1},\zeta\otimes\overline{z_{j}\partial_{j}}P_{2},\eta)$ $\displaystyle=$
$\displaystyle\frac{\varepsilon(n+2)}{(2\pi\sqrt{-1})^{n+1}}\int_{X}|t|^{2s}\overline{z_{j}\partial_{j}}(P_{1}\overline{P_{2}}\eta)\frac{dz_{0}}{z_{0}}\wedge\frac{dz_{1}}{z_{1}}\wedge\cdots\wedge
dz_{n}\wedge\overline{\frac{dz_{0}}{z_{0}}\wedge\frac{dz_{1}}{z_{1}}\wedge\cdots\wedge
dz_{n}}$ $\displaystyle=$
$\displaystyle\int_{X}|z_{I\setminus\\{j\\}}|^{2s-2}z^{s-1}_{j}\overline{z^{s}_{j}\partial_{j}}P_{1}\overline{P_{2}}\eta\bigwedge^{n}_{i=0}(\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge{\overline{dz_{i}}})$
$\displaystyle\text{integration by part on }dz_{j}=$
$\displaystyle-s\int_{X}|z_{I}|^{2s-2}P_{1}\overline{P_{2}}\eta\bigwedge^{n}_{i=0}(\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge{d\overline{z_{i}}})$
$\displaystyle=$ $\displaystyle-sF(s,\zeta\otimes P_{1},\zeta\otimes
P_{2},\eta).$

The same argument works for $F(s,\zeta\otimes
z_{j}\partial_{j}P_{1},\zeta\otimes P_{2},\eta)=-sF(s,\zeta\otimes
P_{1},\zeta\otimes P_{2},\eta).$ This proves $(3)$. ∎

Returning to the proof of the theorem, if one of $\zeta\otimes P_{i}$ is
$\frac{dz_{1}}{z_{1}}\wedge\frac{dz_{2}}{dz_{2}}\wedge\cdots\frac{dz_{k}}{z_{k}}\wedge
dz_{k+1}\wedge\cdots\wedge dz_{n}\otimes tP^{\prime}_{i}$, the above lemma
$(2)$ says $F(s)$ is holomorphic. If one of $\zeta\otimes P_{i}$ is
$\frac{dz_{1}}{z_{1}}\wedge\frac{dz_{2}}{dz_{2}}\wedge\cdots\frac{dz_{k}}{z_{k}}\wedge
dz_{k+1}\wedge\cdots\wedge dz_{n}\otimes D_{i}P$, then the $(3)$ above lemma
says $F(s)$ is in fact $0$. ∎

For any sections $\alpha,\beta\in{\mathcal{M}}$, let
$\left\\{\rho_{\lambda}\right\\}$ be a partition of unity of the open covering
$\left\\{U_{\lambda}\right\\}$ by local charts such that $\alpha,\beta$ lifts
to $\tilde{\alpha}_{\lambda},\tilde{\beta}_{\lambda}$ over $U_{\lambda}$ in
$\Omega^{n}_{X/\Delta}(\log Y)\otimes{\mathscr{D}}_{X}$. The above theorem
just says that the pairing
$S:{\mathcal{M}}\otimes_{\mathbb{C}}\overline{\mathcal{M}}\rightarrow\mathfrak{C}_{X}$
given by

$\langle
S(\alpha,\beta),\eta\rangle=_{\mathrm{def}}{\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{s=0}\sum_{\lambda}F(s,\tilde{\alpha}_{\lambda},\tilde{\beta}_{\lambda},\rho_{\lambda}\eta)$

is well-defined and does not depend on the choice of partition of unity. By
the above lemma we also have the following.

######  Corollary 6.4.

The operator $R$ is self-adjoint with respect to $S$, i.e.
$S\circ(R\otimes_{\mathbb{C}}{\mathrm{i}}{\mathrm{d}\,})=S\circ({\mathrm{i}}{\mathrm{d}\,}\otimes_{\mathbb{C}}R)$.

Because the self-adjointness, we have induced pairings on the graded quotient
$S_{r}:{\mathrm{g}}{\mathrm{r}}^{W}_{r}{\mathcal{M}}\otimes_{\mathbb{C}}\overline{{\mathrm{g}}{\mathrm{r}}^{W}_{-r}{\mathcal{M}}}\rightarrow\mathfrak{C}_{X}$
for every integer $r$. Denote by $P_{R}S_{r}$ the pairing

$S_{r}\circ({\mathrm{i}}{\mathrm{d}\,}\otimes_{\mathbb{C}}R^{r}):{\mathcal{P}}_{r}\otimes_{\mathbb{C}}\overline{{\mathcal{P}}_{r}}\rightarrow\mathfrak{C}_{X}.$

######  Theorem 6.5.

The isomorphism
$\phi_{r}:({\mathcal{P}}_{r},F_{\bullet}{\mathcal{P}}_{r})\rightarrow\tau^{(r+1)}_{+}\omega_{\tilde{Y}^{(r+1)}}(-r)$
in Theorem 5.7 respects the sesquilinear pairings up to a constant
$(-1)^{r}(r+1)!^{-1}$, i.e.

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

for any local sections $\alpha,\beta\in{\mathcal{P}}_{r}$.

###### Proof.

Because the problem is local, it suffices to prove the theorem for $\alpha$
and $\beta$ are represented by

$\frac{dz_{1}}{z_{1}}\wedge\frac{dz_{2}}{z_{2}}\wedge\cdots\frac{dz_{k}}{z_{k}}\wedge
dz_{k+1}\wedge\cdots\wedge dz_{n}\otimes z_{K_{i}}$

and $\\#K_{i}=k-r$ for $i=1,2$ over a local chart $U$ respectively. Recall
that $z_{K}=\prod_{j\in K}z_{j}$. Let $\eta$ be a test function over $U$. We
have

$\langle P_{R}S_{r}(\alpha,\beta),\eta\rangle=\langle
S(\alpha,R^{r}\beta),\eta\rangle={\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{s=0}(-s)^{r}\int_{X}|z_{I}|^{2s-2}z_{K_{1}}\overline{z_{K_{2}}}(\eta)\bigwedge^{n}_{i=0}(\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge{d\overline{z_{i}}}).$

If $\alpha\neq\beta$, the above is in fact zero. Indeed, for $v\in
K_{2}\setminus K_{1}$, by choosing $R^{r}=\prod_{i\in
I\setminus{K_{1}\setminus\\{v\\}}}{z_{i}\partial_{i}}$,

$\displaystyle\langle P_{R}S_{r}(\alpha,\beta),\eta\rangle$
$\displaystyle=\langle
S(R^{r}\alpha,\beta),\eta\rangle={\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{s=0}\int_{X}|z_{I}|^{2s-2}\frac{t}{z_{v}}\overline{z_{v}}\tilde{\eta}\bigwedge^{n}_{i=0}(\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge{d\overline{z_{i}}}),$

where
$\tilde{\eta}=\partial_{I\setminus(K_{1}\setminus\\{v\\})}{\overline{z_{K_{2}}}}{\left(\overline{z_{v}}\right)^{-1}}\eta$
is a smooth test function. The function

$\int_{X}|z_{I}|^{2s-2}\frac{t}{z_{v}}\overline{z_{v}}\tilde{\eta}\bigwedge^{n}_{i=0}(\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge{d\overline{z_{i}}})$

is holomorphic at $s=0$ because

$\frac{1}{\overline{z_{I}}}\frac{\overline{z_{v}}}{z_{v}}\tilde{\eta}\bigwedge^{n}_{i=0}(\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge{d\overline{z_{i}}})$

is integrable.

Therefore, we reduce the proof to the case when $\alpha=\beta$ represented by

$\frac{dz_{1}}{z_{1}}\wedge\frac{dz_{2}}{z_{2}}\wedge\cdots\wedge\frac{dz_{k}}{z_{k}}\wedge\cdots\wedge
dz_{n}\otimes z_{K}.$

We shall prove that

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

where $\overline{K}$ is the complement of $K$ in $I$. Without loss of
generality, we can assume $K=\\{r+1,r+2,...,k\\}$. Then

$\displaystyle P_{R}S_{r}(\alpha,{\alpha})$
$\displaystyle={\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{s=0}(-s)^{r}\int_{X}|z_{\overline{K}}|^{2s-2}\prod^{k}_{j=r+1}|z_{j}|^{2s}\eta\bigwedge^{n}_{i=0}(\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge{d\overline{z_{i}}})$
$\displaystyle=(-1)^{r}{\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{s=0}s^{-(r+2)}\int_{X}\prod^{k}_{i=0}|z_{i}|^{2s}\partial_{\bar{K}}\overline{\partial_{\bar{K}}}(\eta)\bigwedge^{n}_{i=0}(\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge{d\overline{z_{i}}})\text{,
where }\partial_{\bar{K}}=\prod^{r}_{i=0}\partial_{i}$
$\displaystyle=\frac{(-1)^{r}}{(r+1)!}\int_{X}\left(\log\prod^{k}_{i=0}|z_{i}|^{2}\right)^{r+1}\partial_{\bar{K}}\overline{\partial_{\bar{K}}}(\eta)\bigwedge^{n}_{i=0}(\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge{d\overline{z_{i}}})$
$\displaystyle(\star)$
$\displaystyle=\frac{(-1)^{r}}{(r+1)!}\int_{X}\prod^{r}_{i=0}\log|z_{i}|^{2}\partial_{\bar{K}}\overline{\partial_{\bar{K}}}(\eta)\bigwedge^{n}_{i=0}(\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge{d\overline{z_{i}}})$
$\displaystyle=\frac{(-1)^{r}}{(r+1)!}\int_{Y^{\overline{K}}}\eta\bigwedge^{n}_{i=r+1}(\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge{d\overline{z_{i}}})\quad\text{(Poincar\'{e}-Lelong
equation~{}\cite[cite]{[\@@bibref{}{GH}{}{}, Page 388]})}$
$\displaystyle=\frac{(-1)^{r}}{(r+1)!}\tau^{{\overline{K}}}_{+}S_{Y^{\overline{K}}}({\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{Y^{\overline{K}}}\frac{dt}{t}\wedge\alpha,{{\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{Y^{\overline{K}}}\frac{dt}{t}\wedge\alpha}).$

The equality $(\star)$ holds because if we expand
$\left(\log\prod^{k}_{i=0}|z_{i}|^{2}\right)^{r+1}$ as a linear combination of
$\prod^{k}_{i=0}\left(\log|z_{i}|^{2}\right)^{a_{i}}$ with
$\sum^{k}_{i=0}a_{i}=r+1$, the only possible non-exact form among

$\prod^{k}_{i=0}\left(\log|z_{i}|^{2}\right)^{a_{i}}\partial_{\bar{K}}\overline{\partial_{\bar{K}}}(\eta)\bigwedge^{n}_{i=0}(\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge{d\overline{z_{i}}}),$

is
$\left(\prod^{r}_{i=0}\log|z_{i}|^{2}\right)\partial_{\bar{K}}\overline{\partial_{\bar{K}}}(\eta)\bigwedge^{n}_{i=0}(\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge{d\overline{z_{i}}})$.
Note that while ${\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{Y^{\overline{K}}}$
depends on the order of the index sets $K$ and $I$, the pairing

$\frac{(-1)^{r}}{(r+1)!}\tau^{(r+1)}_{+}S_{{\tilde{Y}^{(r+1)}}}(\phi_{r}\alpha,{\phi_{r}\beta})=\frac{(-1)^{r}}{(r+1)!}\tau^{{\overline{K}}}_{+}S_{Y^{\overline{K}}}({\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{Y^{\overline{K}}}\frac{dt}{t}\wedge\alpha,{{\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{Y^{\overline{K}}}\frac{dt}{t}\wedge\alpha})$

does not because the sign will cancel out. We complete the proof. ∎

###  6.2. Constructure of the limiting mixed Hodge structure

We are going to show that the triple
$\left({\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}},F,W\right)$ gives a mixed
Hodge complex. Unlike the ${\mathbb{Q}}$-mixed Hodge complex considered by
Deligne [Del71], where the rational structure is a required input, we do not
have this piece of information in our situation. We will redo the Deligne’s
argument on mixed Hodge complex by sesquilinear pairings. It also worths to
point out that the sesqiuilinear pairing makes one check the first page weight
spectral sequence of ${\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}}$ is a polarzed
bigraded Hodge-Lefschetz structure easier than the case in [GNA90], where they
need to decompose the differential $d_{1}$ on the first page into a
combinatorial differential and a sum of Gysin morphisms.

We first set up the pairing on each page of the weight spectral sequence
abstractly. Let ${\mathcal{N}}$ be a holonomic ${\mathscr{D}}_{Z}$-module
equipped with a sesquilinear pairing
$S:{\mathcal{N}}\otimes_{\mathbb{C}}\overline{{\mathcal{N}}}\rightarrow\mathfrak{C}_{Z}$
on a complex manifold $Z$. Assume that $N$ has compact support. Let $N$ be a
nilpotent operator on ${\mathcal{N}}$ such that
$S\circ({\mathrm{i}}{\mathrm{d}\,}\otimes_{\mathbb{C}}N)=S\circ(N\otimes_{\mathbb{C}}{\mathrm{i}}{\mathrm{d}\,})$.
Let $W_{\bullet}{\mathcal{N}}$ be the monodromy filtration associated to $N$
on ${\mathcal{N}}$. Denote by $E^{i,j}_{r}$ be the weight spectral sequence
convergent to
${\mathrm{g}}{\mathrm{r}}^{W}_{-i}H^{i+j}(Z,{\mathrm{D}}{\mathrm{R}}_{Z}{\mathcal{N}})$
with
$E^{i,j}_{1}=H^{i+j}(Z,{\mathrm{g}}{\mathrm{r}}^{W}_{-i}{\mathrm{D}}{\mathrm{R}}_{Z}{\mathcal{N}})$.
By abuse of notation, denote by $S_{k}$ the induced pairing

$H^{k}(Z,{\mathrm{D}}{\mathrm{R}}_{Z}{\mathcal{N}})\otimes_{\mathbb{C}}\overline{H^{k}(Z,{\mathrm{D}}{\mathrm{R}}_{Z}{\mathcal{N}})}\rightarrow
H^{0}(Z,{\mathrm{D}}{\mathrm{R}}_{Z,\overline{Z}}{\mathcal{N}}\otimes_{\mathbb{C}}\overline{\mathcal{N}})\rightarrow
H^{0}_{c}(Z,{\mathrm{D}}{\mathrm{R}}_{Z,\overline{Z}}\mathfrak{C}_{Z})\simeq{\mathbb{C}}$

multiplying a sign factor $\varepsilon(k)$. Let $a$ be a local section of
$\left({\mathrm{D}}{\mathrm{R}}_{Z}{\mathcal{N}}\right)^{-j-1}$ and $b$ be a
local section of $\left({\mathrm{D}}{\mathrm{R}}_{Z}{\mathcal{N}}\right)^{i}$.
Then

$D(a\otimes_{\mathbb{C}}b)=da\otimes_{\mathbb{C}}b+(-1)^{-j-1}a\otimes_{\mathbb{C}}{db}$

for $D$ a differential on
${\mathrm{D}}{\mathrm{R}}_{Z,\overline{Z}}{\mathcal{N}}\otimes_{\mathbb{C}}\overline{\mathcal{N}}$.
Applying $S$, we find that

(6.24) $DS(a,b)=S(da,b)+(-1)^{-j-1}S(a,{db}).$

Since the differential $d$ is compatible with the weight filtration, we have
an induced pairing $E_{1}(S)_{k}$ on the first page $E^{i,j}_{1}$ of the
weight spectral sequence by the pairing

$H^{k}(Z,{\mathrm{g}}{\mathrm{r}}^{W}_{-i}{\mathrm{D}}{\mathrm{R}}_{Z}{\mathcal{N}})\otimes_{\mathbb{C}}\overline{H^{k}(Z,{\mathrm{g}}{\mathrm{r}}^{W}_{i}{\mathrm{D}}{\mathrm{R}}_{Z}{\mathcal{N}})}\rightarrow
H^{0}(Z,{\mathrm{D}}{\mathrm{R}}_{Z,\overline{Z}}{\mathrm{g}}{\mathrm{r}}^{W}_{-i}{\mathcal{N}}\otimes_{\mathbb{C}}\overline{{\mathrm{g}}{\mathrm{r}}^{W}_{i}{\mathcal{N}}})\rightarrow
H^{0}(Z,{\mathrm{D}}{\mathrm{R}}_{Z,\overline{Z}}\mathfrak{C}_{Z})$

multiplying a sign factor $\varepsilon(k)$. Then by equation (6.24) we obtain

$0=\varepsilon(-j)E_{1}(S)_{-j}(d_{1}a,b)+\varepsilon(-j-1)(-1)^{-j-1}E_{1}(S)_{-j-1}(a,d_{1}b),$

since $DSa\otimes_{C}\overline{b}$ is cohmologous to zero. Working out the
sign, the above is equivalent to

$E_{1}(S)_{-j}(d_{1}a,b)+E_{1}(S)_{-j-1}(a,d_{1}b)=0,$

i.e. the differential $d_{1}$ is skew-symmetrc with respect to $E_{1}(S)$. It
follows that we have an induced pairing on the second page:
$E_{2}(S)_{k}:E^{i,k-i}_{2}\otimes\overline{E^{-i,-k+i}_{2}}\rightarrow{\mathbb{C}}$
since $E_{2}=\ker d_{1}/\mathrm{Im\,}d_{1}$. Again, it follows from the
equation (6.24), the differential $d_{2}$ is skew-symmetric with respect to
$E_{2}(S)$. By an inductive argument, we get the induced pairing
$E_{r}(S):E_{r}\otimes\overline{E_{r}}\rightarrow{\mathbb{C}}$ on the $r$-th
page of the weight spectral sequence
$E_{r}\otimes\overline{E_{r}}\rightarrow{\mathbb{C}}$ such that $d_{r}$ is
skew-symmetric with respect to $E_{r}(S)$ for every $r\geq 1$.

Next, let $L=[\omega]\wedge$ be a Lefschetz operator for a Kähler class
$[\omega]\in H^{1}(Z,\Omega_{Z})\cap H^{2}(Z,{\mathbb{R}})$ on $Z$ which can
be thought as a morphism $L:{\mathbb{C}}\rightarrow{\mathbb{C}}[2]$ in
${\mathbf{D}}^{b}(Z,{\mathbb{C}})$ and so is ${\mathsf{X}}=2\pi\sqrt{-1}L$.
Therefore, we obtain a morphism
${\mathsf{X}}:{\mathrm{D}}{\mathrm{R}}_{Z}{\mathcal{N}}\rightarrow{\mathrm{D}}{\mathrm{R}}_{Z}{\mathcal{N}}[2]$.
Let us work out the relation between the sesquilinear pairing $S_{k}$ and the
operator ${\mathsf{X}}$. By funtorailty, we have the following commutative
diagram in ${\mathbf{D}}^{b}(Z,{\mathbb{C}})$.

${{\mathrm{D}}{\mathrm{R}}_{Z,\overline{Z}}{\mathcal{N}}\otimes_{\mathbb{C}}\overline{\mathcal{N}}}$${{\mathrm{D}}{\mathrm{R}}_{Z,\overline{Z}}\mathfrak{C}_{Z}}$${{\mathrm{D}}{\mathrm{R}}_{Z,\overline{Z}}\mathfrak{Db}_{Z}}$${{\mathcal{A}}^{\bullet}_{Z}\otimes\mathfrak{Db}_{Z}\left[2\dim
Z\right]}$${{\mathrm{D}}{\mathrm{R}}_{Z,\overline{Z}}{\mathcal{N}}\otimes_{\mathbb{C}}\overline{\mathcal{N}}\left[2\right]}$${{\mathrm{D}}{\mathrm{R}}_{Z,\overline{Z}}\mathfrak{C}_{Z}\left[2\right]}$${{\mathrm{D}}{\mathrm{R}}_{Z,\overline{Z}}\mathfrak{Db}_{Z}\left[2\right]}$${{\mathcal{A}}^{\bullet}_{Z}\otimes\mathfrak{Db}_{Z}\left[2\dim
Z+2\right]}$$\scriptstyle{S}$$\scriptstyle{{\mathsf{X}}\otimes_{\mathbb{C}}{\mathrm{i}}{\mathrm{d}\,}}$$\scriptstyle{\simeq}$$\scriptstyle{{\mathsf{X}}}$$\scriptstyle{\simeq}$$\scriptstyle{{\mathsf{X}}}$$\scriptstyle{{\mathsf{X}}}$$\scriptstyle{S[2]}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

Similarly, we have
$S[2]\circ({\mathrm{i}}{\mathrm{d}\,}\otimes_{\mathbb{C}}{{\mathsf{X}}})={\overline{\mathsf{X}}}S$.
It follows from ${\mathsf{X}}+\overline{\mathsf{X}}=0$ on
${\mathcal{A}}^{\bullet}_{Z}\otimes\mathfrak{Db}[2\dim Z]$ that

(6.25)
$\varepsilon(k)S_{k}({\mathsf{X}}-,-)+\varepsilon(k-2)S_{k-2}(-,-)=0,\quad\text{i.e.
}S_{k}({\mathsf{X}}-,-)=S_{k-2}(-,{\mathsf{X}}-).$

Returning to our situation, we begin to construct a polarized bigraded Hodge-
Lefschetz structure on

${\mathrm{g}}{\mathrm{r}}^{W}H^{\bullet}(X,{\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}}).$

Fix a Kähler class $[\omega]$ on $X$ and let
$L=[\omega]\wedge:{\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}}\rightarrow{\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}}[2]$
be the Lefschetz operator and ${\mathsf{X}}_{1}=2\pi\sqrt{-1}L$ as the
discussion above. Relabel the first page of the weight spectral sequence by

$V_{\ell,k}=H^{\ell}(X,{\mathrm{g}}{\mathrm{r}}^{W}_{k}{\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}})=\prescript{W}{}{E^{-k,\ell+k}_{1}}.$

Let $V=\bigoplus_{\ell,k\in{\mathbb{Z}}}V_{\ell,k}$ with filtration
$F_{\bullet}V$ induced by $F_{\bullet}{\mathcal{M}}$. Denote by $E_{i}(R)$ the
induced operator by $R$ on $\prescript{W}{}{E_{i}}$ and let
${\mathsf{Y}}_{2}=E_{1}(R)$. Denote by $S_{\ell,k}$ for
$\ell,k\in{\mathbb{Z}}$, the induced pairing on
$V_{\ell,k}\otimes\overline{V_{-\ell,-k}}$

$H^{\ell}(X,{\mathrm{g}}{\mathrm{r}}^{W}_{k}{\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}})\otimes\overline{H^{-\ell}(X,{\mathrm{g}}{\mathrm{r}}^{W}_{-k}{\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}}})\rightarrow
H^{0}(X,{\mathrm{D}}{\mathrm{R}}_{X,\overline{X}}{\mathrm{g}}{\mathrm{r}}^{W}_{k}{\mathcal{M}}\otimes_{\mathbb{C}}\overline{{\mathrm{g}}{\mathrm{r}}^{W}_{-k}{\mathcal{M}}})\rightarrow
H^{0}_{c}(X,{\mathrm{D}}{\mathrm{R}}_{X,\overline{X}}\mathfrak{C}_{X})\simeq{\mathbb{C}}.$

multiplying a sign factor $\varepsilon(\ell)$. Let $d_{1}$ be the differential
of $E_{1}$. In terms of relabeling, we have

$d_{1}:(V_{\ell,k},F_{\bullet}V_{\ell,k})\rightarrow(V_{\ell+1,k-1},F_{\bullet}V_{\ell+1,k-1}).$

######  Theorem 6.6.

The tuple $(V,{\mathsf{X}}_{1},{\mathsf{Y}}_{2},F_{\bullet}V,\bigoplus
S_{j,k},d_{1})$ gives a differential polarized bigraded Hodge-Lefschetz
structure of central weight $n$.

###### Proof.

Let us first check the conditions in Theorem 2.10 one by one. It is clear that
two operators ${\mathsf{X}}_{1},{\mathsf{Y}}_{2}$ are commute. Moreover, we
have
${\mathsf{Y}}_{2}:(V_{\ell,k},F_{\bullet}V_{\ell,k})\rightarrow(V_{\ell,k-2},F_{\bullet+1}V_{\ell,k-2})$
such that

${\mathsf{Y}}^{k}_{2}:F_{\bullet}V_{\ell,k}\rightarrow
F_{\bullet+k}V_{\ell,-k},$

is an isomorphism by Theorem 5.6. Denote by $P_{{\mathsf{Y}}_{2}}V_{-j,r}$ the
${{\mathsf{Y}}_{2}}$-primitive part $\ker{{\mathsf{Y}}^{r+1}_{2}}\cap
V_{-j,r}=H^{-j}(X,{\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{P}}_{r})$. It follows
from Theorem 5.7 that
$(P_{{\mathsf{Y}}_{2}}V_{-j,r},F_{\bullet}P_{{\mathsf{Y}}_{2}}V_{-j,r})$ is
filtered isomorphic to
$H^{-j}(\tilde{Y}^{(r+1)},{\mathrm{D}}{\mathrm{R}}_{\tilde{Y}^{(r+1)}}\omega_{{\tilde{Y}^{(r+1)}}})(-r)$
via $\phi_{r}$. Therefore,
${\mathsf{X}}_{1}F_{\bullet}P_{{\mathsf{Y}}_{2}}V_{-j,r}\subset
F_{\bullet-1}P_{{\mathsf{Y}}_{2}}V_{-j+2,r}$ and by Hard Lefschetz,

${\mathsf{X}}^{j}_{1}:F_{\bullet}P_{{\mathsf{Y}}_{2}}V_{-j,r}\rightarrow
F_{\bullet-j}P_{{\mathsf{Y}}_{2}}V_{j,r}$

is an isomorphism. It follows from the Lefschetz decomposition of
${\mathsf{Y}}_{2}$ that ${\mathsf{X}}^{j}_{1}:F_{\bullet}V_{-j,r}\rightarrow
F_{\bullet-j}V_{j,r}$ is an isomorphism. This proves $(\mathrm{pbHL1})$ in
Theorem 2.10. (pbHL2) follows from the equation (6.25).

Because the operator $R$ self-adjoin with respect to $S$ by Corollary 6.4, we
have $S_{j,r}(-,{\mathsf{Y}}_{2}-)=S_{j,r+2}({\mathsf{Y}}_{2}-,-)$. By Theorem
6.5, the morphism $\phi_{r}$ identifies
$P_{{\mathsf{Y}}_{2}}S_{-j,r}=_{\mathrm{def}}S_{-j,r}(-,{\mathsf{Y}}^{r}_{2}-)$
with $\frac{(-1)^{r}}{(r+1)!}S_{\tilde{Y}^{(r+1)},-j}$. Recall that

$S_{\tilde{Y}^{(r+1)},j}(a,b)={\frac{\varepsilon(n-r+j+1)}{(2\pi\sqrt{-1})^{n-r}}}\int_{\tilde{Y}^{(r+1)}}a\wedge\bar{b},\text{
for }a\in H^{n-r+j}(\tilde{Y}^{(r+1)})\text{ and }b\in
H^{n-r-j}(\tilde{Y}^{(r+1)}),$

and that $S_{\tilde{Y}^{(r+1)},j}({\mathsf{X}}^{j}_{1}-,-)$ is a polarization
on $H^{n-r-j}_{\mathrm{prim}}(\tilde{Y}^{(r+1)},{\mathbb{C}})$. The bi-
primitive part
$P_{-j,r}=\ker{\mathsf{X}}^{j}_{1}\cap\ker{\mathsf{Y}}^{r}_{2}\cap V_{-j,r}$
together with the induced filtration $F_{\bullet}P_{-j,r}$ and the
sesquilinear pairing $S_{j,r}({\mathsf{X}}^{j}_{1}-,(-{\mathsf{Y}})^{r}_{2}-)$
is identified with the polarized Hodge structure
$H^{n-r-j}_{\mathrm{prim}}(\tilde{Y}^{(r+1)},{\mathbb{C}})(-r)$ via
$\phi_{r}$. This proves (pbHL3).

It remains to prove that $d_{1}$ is a differential of the bigraded Hodge-
Lefschetz structure $V$. Clearly, we have

$[d_{1},{\mathsf{X}}_{1}]=[d_{1},{\mathsf{Y}}_{2}]=0$

because $d_{1}$ is induced by the differential of
${\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}}$ and $d_{1}$ preserves
$F_{\bullet}$. The differential $d_{1}$ is skew-symmetric with respect to
$\bigoplus_{j,r}S_{j,r}$ is formally follows the discussion at the beginning
of this subsection. Thus, we finished checking that $d_{1}$ is a differential.
∎

######  Corollary 6.7.

We have the following

  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 tuple
$\left(\bigoplus_{\ell\in{\mathbb{Z}}}{\mathrm{g}}{\mathrm{r}}^{W}H^{\ell}(X,{\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}}),F,{\mathsf{X}}_{1},{\mathsf{Y}}_{2}\right)$
together with the pairing induced by $\bigoplus S_{j,k}$ is a polarized
bigradged Hodge-Lefschetz structure of central weight $n$.

###### Proof.

We slightly modify the idea of cohomological mixed Hodge complex in [Del71]
for statement $(1)$ and $(2)$. I claim that the $k$-th weight spectral
sequence $V^{k}_{\ell,r}=_{\mathrm{def}}\prescript{W}{}{E_{k}}^{-r,\ell+r}$
together with the induced filtration $F_{\bullet}$ and the induced pairing
$S^{k}_{\ell,r}\circ\left({\mathrm{i}}{\mathrm{d}\,}\otimes{\mathsf{w}}\right):V^{k}_{\ell,r}\otimes\overline{V^{k}_{\ell,r}}\rightarrow{\mathbb{C}}$
is a polarized Hodge structure of weight $n+\ell+r$ and the differential
$d_{k}:V^{k}_{\ell,r}\rightarrow V^{k}_{\ell+1,r-k}$ is a morphism of Hodge
structures. Indeed, the differential $d_{k}$ is skew-symmetric with respect to
the sesquilinear pairing, i.e.
$S^{k}_{\ell,r}(d_{k}-,-)+S^{k}_{\ell+1,r-k}(-,d_{k}-)=0$. Therefore, if
$(-1)^{q}S^{k}_{\ell,r}\circ\left({\mathrm{i}}{\mathrm{d}\,}\otimes{\mathsf{w}}\right)$
for $q=n+\ell+r-p$ is a Hermitian inner product on

${\left(V^{k}_{\ell,r}\right)^{p,q}=\\{a\in
F^{p}V^{k}_{\ell,r}:S^{k}_{\ell,r}(a,b)=0\text{ for all }b\in
F^{p-\ell-r+1}V^{k}_{-\ell,-r}\\}}$

then
$(-1)^{q}S^{k+1}_{\ell,r}\circ\left({\mathrm{i}}{\mathrm{d}\,}\otimes{\mathsf{w}}\right)$
is also a Hermitian inner product on

$\left(V^{k+1}_{\ell,r}\right)^{p,q}=\\{a\in
F^{p}V^{k+1}_{\ell,r}:S^{k+1}_{\ell,r}(a,b)=0\text{ for all }b\in
F^{p-\ell-r+1}V^{k+1}_{-\ell,-r}\\}.$

In particular, we have the decomposition

$V^{k+1}_{\ell,r}=\bigoplus_{p+q=n+\ell+r}\left(V^{k+1}_{\ell,r}\right)^{p,q}$

and the morphism
$d_{k}:\left(V^{k}_{\ell,r}\right)^{p,q}\rightarrow\left(V^{k+1}_{\ell,r}\right)^{p,q}$
is compatible with the decomposition. See Remark 2.11. By induction the claim
is proved. It follows that $d_{k}$ vanishes for $k\geq 2$ by it is a morphism
of Hodge structures of different weights, which proves $(2)$.

Since each bigraded piece
$V_{\ell,r}=H^{\ell}\left(X,{\mathrm{g}}{\mathrm{r}}^{W}_{r}{\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}}\right)$
is pure Hodge structure of weight $n+r+\ell$, the two vector spaces
$H^{\ell}\left(X,{\mathrm{g}}{\mathrm{r}}^{F}{\mathrm{g}}{\mathrm{r}}^{W}_{r}{\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}}\right)$
and $V_{\ell,r}$ is isomorphic. Moreover, the isomorphism is compatible with
$d_{1}$, because $d_{1}$ respects $F_{\bullet}$ and

${\mathrm{g}}{\mathrm{r}}^{W}_{r}{\mathrm{g}}{\mathrm{r}}^{F}{\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}}={\mathrm{g}}{\mathrm{r}}^{F}{\mathrm{g}}{\mathrm{r}}^{W}_{r}{\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}}.$

Taking cohomology of $d_{1}$, we obtain that
${\mathrm{g}}{\mathrm{r}}^{W}_{r}H^{\ell}\left(X,{\mathrm{g}}{\mathrm{r}}^{F}{\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}}\right)$
is isomorphism to
${\mathrm{g}}{\mathrm{r}}^{W}_{r}H^{\ell}\left(X,{\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}}\right)$.
It follows from the dimension reason that
$H^{\ell}\left(X,{\mathrm{g}}{\mathrm{r}}^{F}{\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}}\right)$
is isomorphic to
$H^{\ell}\left(X,{\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}}\right)$, which is
exactly the degeneration of Hodge spectral sequence at
$\prescript{}{F}{E_{1}}$.

The statement $(3)$ follows from Theorem 2.12. ∎

The third statement in the above corollary ensures that the weight filtration
on the hypercohomology of ${\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}}$ is the
monodromy weight filtration of the nilpotent operator $R$, i.e.
$RW_{\bullet}H^{\ell}(X,{\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}})\subset
W_{\bullet-2}H^{\ell}(X,{\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}})(-1)$ and
$R^{r}:{\mathrm{g}}{\mathrm{r}}^{W}_{r}H^{\ell}(X,{\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}})\rightarrow{\mathrm{g}}{\mathrm{r}}^{W}_{-r}H^{\ell}(X,{\mathrm{D}}{\mathrm{R}}_{X}{\mathcal{M}})(-r)$
is a filtered isomorphism. We proved Theorem A for the case when $Y$ is
reduced.

##  7\. Non-reduced case: Generalized eigenspace ${\mathcal{M}}_{\alpha}$ and
the weight filtration

Now we move to the general situation. Recall that we have introduced the
notations: the index set $I$ consisting of indices of irreducible components
of $Y$ and $e_{i}$ is the multiplicity of $Y$ along the component $Y_{i}$.

###  7.1. The generalized eigen-modules ${\mathcal{M}}_{\alpha}$

We begin with studying the generalized eigen-modules $\ker(R-\alpha)^{\infty}$
of the morphism $R$ in the category of filtered ${\mathscr{D}}_{X}$-modules.
The generalized eigen-modules are naturally sub-modules of ${\mathcal{M}}$ and
one can put the induced filtration on it. However, this filtration does not
match with the expected weight of the mixed Hodge structure and is difficult
to study. Instead, we use the idea of Saito in [Sai90]: one regards the
generalized eigen-module as a sub-quotient of ${\mathcal{M}}$ and puts the
induced filtration on it. It turns out the filtration behaves nice. Now let us
begin to settle some definitions.

Define
${\mathcal{M}}_{\geq\alpha}=\ker\prod_{\lambda\geq\alpha}(R-\lambda)^{\infty}$,
${\mathcal{M}}_{>\alpha}=\ker\prod_{\lambda>\alpha}(R-\lambda)^{\infty}$ and
${\mathcal{M}}_{\alpha}={\mathcal{M}}_{\geq\alpha}/{\mathcal{M}}_{>\alpha}$.
Then ${\mathcal{M}}_{\alpha}$ is canonically isomorphic to the generalized
eigen-module $\ker\left(R-\alpha\right)^{\infty}$. Endow
${\mathcal{M}}_{\alpha}$ the filtration $F_{\bullet}{\mathcal{M}}_{\alpha}$
induced from $({\mathcal{M}},F_{\bullet}{\mathcal{M}})$,

$F_{\bullet}{\mathcal{M}}_{\alpha}=\frac{{\mathcal{M}}_{\geq\alpha}\cap
F_{\bullet}{\mathcal{M}}}{{\mathcal{M}}_{>\alpha}\cap
F_{\bullet}{\mathcal{M}}}.$

There are parallel definitions on the relative log de Rham complex. Denote by
$C^{\bullet}=\Omega^{\bullet+n}_{X/\Delta}(\log Y)\otimes{\mathscr{O}}_{Y}$
for simplicity. Define sub-complexes of $C^{\bullet}$ by

$C^{\bullet}_{\geq\alpha}=C^{\bullet}\otimes{\mathscr{O}}_{X}(-\left\lceil\alpha
Y\right\rceil),\quad
C^{\bullet}_{>\alpha}=C^{\bullet}\otimes{\mathscr{O}}_{X}(-\left\lfloor\alpha
Y\right\rfloor-Y_{{\mathrm{R}}{\mathrm{e}}{\mathrm{d}}})\quad\text{ and
}C^{\bullet}_{\alpha}=C^{\bullet}_{\geq\alpha}/C^{\bullet}_{>\alpha},$

where $Y_{{\mathrm{R}}{\mathrm{e}}{\mathrm{d}}}$ is the associated reduced
divisor of $Y$. Notice that if we let $I_{\alpha}$ be the subset of $I$
consisting of all $i$ such that $\alpha e_{i}$ is an integer, then

$C^{\bullet}_{\alpha}=C^{\bullet}_{\geq\alpha}\otimes{\mathscr{O}}_{Y_{I_{\alpha}}},\quad\text{where
}Y_{I_{\alpha}}=\sum_{i\in I_{\alpha}}Y_{i}.$

One can check $C^{\bullet}_{\alpha}$ is a generalized eigen-perverse sheaves
of the residue $[\nabla]$. Since ${\mathscr{O}}_{X}(-\lceil\alpha Y\rceil)$ is
preserved by relative log differential ${\mathscr{T}}_{X/\Delta}(-\log Y)$,
the multiplication by relative log differentials gives a morphism, recalling
that $D_{1},D_{2},...,D_{n}$ are local generators of
${\mathscr{T}}_{X/\Delta}(-\log Y)$ dual to the local generators
$\xi_{1},\xi_{2},...,\xi_{n}$ of $\Omega_{X/\Delta}(\log Y)$,

(7.26) ${\mathscr{O}}_{X}(-\lceil\alpha
Y\rceil)\otimes{\mathscr{D}}_{X}\rightarrow\Omega_{X/\Delta}(-\lceil\alpha
Y\rceil)\otimes{\mathscr{D}}_{X},\quad
z^{\lceil\alpha\mathbf{e}\rceil}_{I}\otimes P\mapsto\sum_{j}\xi_{j}\otimes
D_{j}z^{\lceil\alpha\mathbf{e}\rceil}_{I}\otimes P=\sum_{j}\xi_{j}\otimes
z^{\lceil\alpha\mathbf{e}\rceil}_{I}(D_{j}+\alpha_{j})\otimes P,$

where, using the multi-index notation,
$z^{\lceil\alpha\mathbf{e}\rceil}_{I}=\prod_{i\in I}z^{\lceil\alpha
e_{i}\rceil}_{i}$ denotes the local generator of
${\mathscr{O}}_{X}(-\lceil\alpha Y\rceil)$ and define
$\alpha_{i}=[D_{i},z^{\lceil\alpha\mathbf{e}\rceil}_{I}]/{z^{\lceil\alpha\mathbf{e}\rceil}_{I}}={\lceil\alpha
e_{i}\rceil}/{e_{i}}-{\lceil\alpha e_{0}\rceil}/{e_{0}}.$ The morphism extends
to a complex $\Omega^{{}n+\bullet}_{X/\Delta}(\log Y)(-\lceil\alpha
Y\rceil)\otimes{\mathscr{D}}_{X}$, which is a subcomplex of
$\Omega^{{}n+\bullet}_{X/\Delta}(\log Y)\otimes{\mathscr{D}}_{X}$ (see
(4.17)). Tensoring ${\mathscr{O}}_{Y}$ on the left gives
$C^{\bullet}_{\geq\alpha}\otimes{\mathscr{D}}_{X}$ by the above definition.
Further tensoring ${\mathscr{O}}_{Y_{I_{\alpha}}}$ on the left, we obtain the
complex of induced ${\mathscr{D}}_{X}$-modules
$C^{\bullet}_{\alpha}\otimes{\mathscr{D}}_{X}$ with the filtration defined by

$F_{\ell}\left(C^{\bullet}_{\alpha}\otimes{\mathscr{D}}_{X}\right)=C^{\bullet}_{\alpha}\otimes
F_{\ell+n+\bullet}{\mathscr{D}}_{X}.$

The following two theorems give the description of the generalized eigen-
modules in terms of complexes of the induced ${\mathscr{D}}_{X}$-modules.

######  Theorem 7.1.

The complex $C^{\bullet}_{\alpha}\otimes{\mathscr{D}}_{X}$ is filtered acyclic
and the characteristic cycle of the $0$-th cohomology is

$cc\left({\mathscr{H}}^{0}\left(C^{\bullet}_{\alpha}\otimes{\mathscr{D}}_{X}\right)\right)=\sum_{J\subset
I}\left(\\#I_{\alpha}\cap J\right)\left[T^{*}_{Y^{J}}X\right].$

###### Proof.

Similarly to the proof of Theorem 4.1 and Theorem 4.5, the associated graded
${\mathrm{g}}{\mathrm{r}}^{F}\left(C^{\bullet}_{\alpha}\otimes{\mathscr{D}}_{X}\right)$
locally is the Koszul complex of the regular sequence
$(t_{\alpha},D_{1},D_{2},...,D_{n})$ over
${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}$, where $t_{\alpha}=\prod_{i\in
I_{\alpha}}z_{i}$ is the defining equation of $Y_{I_{\alpha}}$. It follows
that
${\mathrm{g}}{\mathrm{r}}^{F}(C^{\bullet}_{\alpha}\otimes{\mathscr{D}}_{X})$
is acyclic and therefore, $C^{\bullet}_{\alpha}\otimes{\mathscr{D}}_{X}$ is
filtered acyclic. We also get that
${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{H}}^{0}(C^{\bullet}_{\alpha}\otimes{\mathscr{D}}_{X})$
is locally represented by

(7.27)
$\zeta_{\alpha}\otimes{\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}/(t_{\alpha},D_{1},D_{2},...,D_{n}){\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X},\quad\text{where
}\zeta_{\alpha}=z^{\lceil\alpha\mathbf{e}\rceil}_{I}\frac{dz_{1}}{z_{1}}\wedge\frac{dz_{2}}{z_{2}}\wedge\cdots\wedge\frac{dz_{k}}{z_{k}}\wedge
dz_{k+1}\wedge\cdots\wedge dz_{n}.$

As the calculation in Theorem 4.7, we get the characteristic cycle is
$\sum_{J\subset I}\left(\\#I_{\alpha}\cap
J\right)\left[T^{*}_{Y^{J}}X\right]$. ∎

######  Theorem 7.2.

There exists a canonical filtered isomorphism

(7.28)
${\psi_{\alpha}:\left({\mathscr{H}}^{0}\left(C^{\bullet}_{\alpha}\otimes{\mathscr{D}}_{X}\right),F_{\bullet}{\mathscr{H}}^{0}\left(C^{\bullet}_{\alpha}\otimes{\mathscr{D}}_{X}\right)\right)}$${({\mathcal{M}}_{\alpha},F_{\bullet}{\mathcal{M}}_{\alpha}).}$$\scriptstyle{\sim}$

In particular, the characteristic cycle
$cc({\mathcal{M}}_{\alpha})=\sum_{J\subset I}\left(\\#I_{\alpha}\cap
J\right)\left[T^{*}_{Y^{J}}X\right]$.

We first study ${\mathcal{M}}_{\geq\alpha}$ and ${\mathcal{M}}_{>\alpha}$
locally by pointing out their cyclic generator. In principal, this always can
be done because every holonomic ${\mathscr{D}}_{X}$-module locally is cyclic.

######  Lemma 7.3.

Locally, ${\mathcal{M}}_{\geq\alpha}$ is generated by
$z^{\lceil\alpha\mathbf{e}\rceil}_{I}$, and ${\mathcal{M}}_{>\alpha}$ is
generated by $z^{\lfloor\alpha\mathbf{e}\rfloor+\mathbf{1}}_{I}$ where
$\mathbf{1}=(1,1,...,1)\in{\mathbb{Z}}^{I}$.

###### Proof.

Let us first check that
$z^{\lceil\alpha\mathbf{e}\rceil}_{I}\in{\mathcal{M}}_{\geq\alpha}$. It
suffices to check that it is in

$\ker\prod_{i\in I}\prod^{e_{i}-1}_{j=\lceil\alpha
e_{i}\rceil}(R-\frac{j}{e_{i}}).$

This is follows from direct calculation:

$\displaystyle\prod_{i\in I}\prod^{e_{i}-1}_{j=\lceil\alpha
e_{i}\rceil}(R-\frac{j}{e_{i}})z^{\lceil\alpha\mathbf{e}\rceil}_{I}$
$\displaystyle=\prod_{i\in I}\prod^{e_{i}-1}_{j=\lceil\alpha
e_{i}\rceil}(R-\frac{j}{e_{i}})z^{\lceil\alpha e_{i}\rceil}_{i}=\prod_{i\in
I}\prod^{e_{i}-1}_{j=\lceil\alpha
e_{i}\rceil}(\frac{1}{e_{i}}z_{i}\partial_{i}-\frac{j}{e_{i}})z^{\lceil\alpha
e_{i}\rceil}_{i}$ $\displaystyle=\prod_{i\in I}\frac{1}{e^{e_{i}-\lceil\alpha
e_{i}\rceil}_{i}}z^{e_{i}}_{i}\partial^{e_{i}-\lceil\alpha
e_{i}\rceil}_{i}=t\prod_{i\in I}\frac{1}{e^{e_{i}-\lceil\alpha
e_{i}\rceil}_{i}}\partial^{e_{i}-\lceil\alpha
e_{i}\rceil}_{i}=0\in{\mathcal{M}}.$

Because $R$ satisfies the identity (4.19), ${\mathcal{M}}_{\geq\alpha}$ is
also equal to the image of $\prod_{i\in I}\prod^{\lceil\alpha
e_{i}\rceil-1}_{j=0}(R-\frac{j}{e_{i}})$. It follows from

$\prod_{i\in I}\prod^{\lceil\alpha
e_{i}\rceil-1}_{j=0}(R-\frac{j}{e_{i}})(1)=\prod_{i\in I}\prod^{\lceil\alpha
e_{i}\rceil-1}_{j=0}(\frac{1}{e_{i}}z_{i}\partial_{i}-\frac{j}{e_{i}})=z^{\lceil\alpha\mathbf{e}\rceil}_{I}\prod_{i\in
I}\frac{1}{e^{\lceil\alpha e_{i}\rceil}_{i}}\partial^{\lceil\alpha
e_{i}\rceil}_{i}$

that $z^{\lceil\alpha\mathbf{e}\rceil}_{I}\prod_{i\in I}\partial^{\lceil\alpha
e_{i}\rceil}_{i}$ generates ${\mathcal{M}}_{\geq\alpha}$. We deduce that
$z^{\lceil\alpha\mathbf{e}\rceil}_{I}$ generates ${\mathcal{M}}_{\geq\alpha}$.
The similar argument works for ${\mathcal{M}}_{>\alpha}$. ∎

###### Proof of Theorem 7.2.

It follows from the above lemma that ${\mathcal{M}}_{\alpha}$ is locally
isomorphic to

$\zeta\otimes{\left(z^{\lceil\alpha\mathbf{e}\rceil}_{I},D_{1},D_{2},...,D_{n}\right){\mathscr{D}}_{X}}/{\left(z^{\lfloor\alpha\mathbf{e}\rfloor+\mathbf{1}}_{I},D_{1},D_{2},...,D_{n}\right){\mathscr{D}}_{X}}$

where
$\zeta=\frac{dz_{1}}{z_{1}}\wedge\frac{dz_{2}}{z_{2}}\wedge\cdots\wedge\frac{dz_{k}}{z_{k}}\wedge\cdots\wedge
dz_{n}$ so that $\zeta_{\alpha}=z^{\lceil\alpha\mathbf{e}\rceil}_{I}\zeta$.
Since ${\mathscr{H}}^{0}(C^{\bullet}_{\alpha}\otimes{\mathscr{D}}_{X})$ by
(7.26) is locally isomorphic to

$\zeta_{\alpha}\otimes{\mathscr{D}}_{X}/(t_{\alpha},D_{1}+\alpha_{1},D_{2}+\alpha_{2},...,D_{n}+\alpha_{n}){\mathscr{D}}_{X},$

the multiplication
${\mathscr{H}}^{0}(C^{\bullet}_{\alpha}\otimes{\mathscr{D}}_{X})\rightarrow{\mathcal{M}}_{\alpha},\,\zeta_{\alpha}\otimes
P\mapsto\zeta\otimes z^{\lceil\alpha\mathbf{e}\rceil}_{I}P$ is well-defined,
does not depend on the coordinate and therefore, gives a filtered morphism

${\psi_{\alpha}:({\mathscr{H}}^{0}(C^{\bullet}_{\alpha}\otimes{\mathscr{D}}_{X}),F_{\bullet}{\mathscr{H}}^{0}(C^{\bullet}_{\alpha}\otimes{\mathscr{D}}_{X}))}$${({\mathcal{M}}_{\alpha},F_{\bullet}{\mathcal{M}}_{\alpha}).}$

The surjectivity is clear from the local description. It follows that
$cc\left({\mathscr{H}}^{0}\left(C^{\bullet}_{\alpha}\otimes{\mathscr{D}}_{X}\right)\right)\geq
cc({\mathcal{M}}_{\alpha})$. Summing over all the rational numbers $\alpha$ in
$[0,1)$ gives

$\sum_{\alpha}cc\left({\mathscr{H}}^{0}\left(C^{\bullet}_{\alpha}\otimes{\mathscr{D}}_{X}\right)\right)\geq\sum_{\alpha}cc({\mathcal{M}}_{\alpha})=cc({\mathcal{M}}).$

On the other hand, by Theorem 4.5 and Theorem 7.1, the
${\mathscr{D}}_{X}$-module ${\mathcal{M}}$ is also successive extensions of
${\mathscr{H}}^{0}\left(C^{\bullet}_{\alpha}\otimes{\mathscr{D}}_{X}\right)$
for $\alpha\in{\mathbb{Q}}\cap[0,1)$. Thus,

$\sum_{\alpha}cc\left({\mathscr{H}}^{0}\left(C^{\bullet}_{\alpha}\otimes{\mathscr{D}}_{X}\right)\right)=cc({\mathcal{M}}).$

This forces that $\psi_{\alpha}$ must be isomorphism and therefore, filtered
injective.

It remains to show that

(7.29)
${F_{\ell}\psi_{\alpha}:F_{\ell}{\mathscr{H}}^{0}(C^{\bullet}_{\alpha}\otimes{\mathscr{D}}_{X})}$${F_{\ell}{\mathcal{M}}_{\alpha},}$

is sujective. Suppose that
$z^{\lceil\alpha\mathbf{e}\rceil}_{I}P\in{\mathscr{D}}_{X}$ is a
representative of a class in $F_{\ell}{\mathcal{M}}_{\alpha}$. Then we can
write

$z^{\lceil\alpha\mathbf{e}\rceil}_{I}P=P^{\prime}+\sum^{n}_{i=1}D_{i}Q_{i}+z^{\lfloor\alpha\mathbf{e}\rfloor+\mathbf{1}}_{I}T$

for $P^{\prime}\in F_{\ell+n}{\mathscr{D}}_{X}$ and
$T,Q_{i}\in{\mathscr{D}}_{X}$. It follows that

$z^{\lceil\alpha\mathbf{e}\rceil}_{I}(P-t_{\alpha}T)=P^{\prime}+\sum^{n}_{i=1}D_{i}Q_{i}$

By the regular sequence argument of Theorem 4.5, we can assume that
$P-t_{\alpha}T$ is in $F_{\ell+n}{\mathscr{D}}_{X}$. Then the class
represented by $P-t_{\alpha}T$ in
${\mathscr{H}}^{0}(C^{\bullet}_{\alpha}\otimes{\mathscr{D}}_{X})$ is actually
in $F_{\ell}{\mathscr{H}}^{0}(C^{\bullet}_{\alpha}\otimes{\mathscr{D}}_{X})$
by the local formula. Therefore, we find a lifting represented by $P$ in
$F_{\ell}{\mathscr{H}}^{0}(C^{\bullet}_{\alpha}\otimes{\mathscr{D}}_{X})$ of
the class of $z^{\lceil\alpha\mathbf{e}\rceil}_{I}P$ in
$F_{\ell}{\mathcal{M}}_{\alpha}$. We conclude the proof. ∎

Without loss of generality, we can assume by abuse of notation that locally
$I_{\alpha}=\\{0,1,...,\mu\\}$ so that $t_{\alpha}=z_{0}z_{1}\cdots z_{\mu}$.
Let $R_{\alpha}$ be the induced operator $(R-\alpha)$ on
$({\mathcal{M}}_{\alpha},F_{\bullet}{\mathcal{M}}_{\alpha})$. One easily gets
a nice local formula of $R_{\alpha}$:

######  Corollary 7.4.

The endormorphism $R_{\alpha}$ of ${\mathcal{M}}_{\alpha}$ acts locally as
$\psi_{\alpha}\circ({\mathrm{i}}{\mathrm{d}\,}\otimes\frac{1}{e_{j}}z_{j}\partial_{j})\circ(\psi_{\alpha})^{-1}$
for any $j\in I_{\alpha}$.

###### Proof.

Because $R-\alpha$ acts on the left hand side of the identification (7.27) by
the left multiplication by $\frac{1}{e_{0}}z_{0}\partial_{0}-\alpha$, the
statement follows from

$\displaystyle R_{\alpha}\left[\zeta\otimes
z^{\lceil\alpha\mathbf{e}\rceil}_{I}\right]=$
$\displaystyle\left[\zeta\otimes\left(\frac{1}{e_{j}}z_{j}\partial_{j}-\alpha\right)\left(z^{\lceil\alpha\mathbf{e}\rceil}_{I}\right)\right]$
$\displaystyle=$
$\displaystyle\left[\zeta\otimes\left(\left(\frac{1}{e_{j}}\lceil\alpha
e_{j}\rceil-\alpha\right)z^{\lceil\alpha\mathbf{e}\rceil}_{I}+z^{\lceil\alpha\mathbf{e}\rceil}_{I}\left(\frac{1}{e_{j}}z_{j}\partial_{j}\right)\right)\right]$
$\displaystyle=$ $\displaystyle\psi_{\alpha}{\left[\zeta
z^{\lceil\alpha\mathbf{e}\rceil}_{I}\otimes\left(\frac{1}{e_{j}}z_{j}\partial_{j}\right)\right]}=\psi_{\alpha}\circ\left({\mathrm{i}}{\mathrm{d}\,}\otimes\frac{1}{e_{j}}z_{j}\partial_{j}\right)\circ\psi_{\alpha}^{-1}\left[\zeta_{\alpha}\otimes
1\right].$

This completes the proof. ∎

By the local formula of $R_{\alpha}$, it is obvious that
$R_{\alpha}:({\mathcal{M}}_{\alpha},F_{\bullet}{\mathcal{M}}_{\alpha})\rightarrow({\mathcal{M}}_{\alpha},F_{\bullet+1}{\mathcal{M}}_{\alpha})$
is a filtered morphism.

###  7.2. Striness of $R_{\alpha}$

Similar to the reduced case, every power of $R_{\alpha}$ is strict.

######  Theorem 7.5.

The power of the endomorphism $R_{\alpha}$ on
$({\mathcal{M}}_{\alpha},F_{\bullet}{\mathcal{M}}_{\alpha})$ is strict:

(7.30)
$R_{\alpha}^{a}F_{b}{\mathcal{M}}_{\alpha}=F_{a+b}R_{\alpha}^{a}{\mathcal{M}}_{\alpha},\quad\text{for
any }a\in{\mathbb{Z}}_{\geq 0}\text{ and }b\in{\mathbb{Z}}.$

Let $[R_{\alpha}]$ be the endomorphism on
${\mathrm{g}}{\mathrm{r}}^{F}{\mathcal{M}}_{\alpha}$ induced by $R_{\alpha}$.
To prove the above theorem, we need the following statement on
$\ker{\left[R_{\alpha}\right]}\subset{\mathrm{g}}{\mathrm{r}}^{F}{\mathcal{M}}_{\alpha}$.

######  Lemma 7.6.

$\ker{[R_{\alpha}]}^{r+1}$ is locally generated by monomials of degree $\mu-r$
that divid $t_{\alpha}$.

###### Proof of Theorem 7.5.

Temporarily admitting this lemma, let $R_{\alpha}^{r+1}m$ be an element in
$F_{\ell+r+1}{\mathcal{M}}_{\alpha}$. Assume that $m\in
F_{k}{\mathcal{M}}_{\alpha}$. If $k>\ell$ then the projection of
$R_{\alpha}^{r+1}m$ vanishes in
${\mathrm{g}}{\mathrm{r}}^{F}_{k+r+1}{\mathcal{M}}_{\alpha}$. It follows from
the lemma that $m$ can be written as

$m=\sum_{{\begin{subarray}{c}\\#J=\mu-r,\\\ J\subset
I_{\alpha}\end{subarray}}}z_{J}m_{J}+\sum^{n}_{i=1}D_{i}Q_{i}+m^{\prime},\quad\text{
for }\,z_{J}=\prod_{j\in J}z_{j}$

where $Q_{i},m^{\prime}\in F_{k-1}{\mathcal{M}}_{\alpha}$. Because for every
$J\subset I_{\alpha}$ of cardinality $r+1$ we can arrange

$R^{r+1}_{\alpha}z_{J}=\prod_{j\in I_{\alpha}\setminus
J}\frac{1}{e_{j}}z_{j}\partial_{j}z_{J}=t_{\alpha}\prod_{j\in
I_{\alpha}}\frac{1}{e_{j}}\partial_{j}=0\in{\mathcal{M}}_{\alpha}$

it follows that $R_{\alpha}^{r+1}m$ is equal to,

$\displaystyle\sum_{{\begin{subarray}{c}\\#J=\mu-r,\\\ J\subset
I_{\alpha}\end{subarray}}}R_{\alpha}^{r+1}z_{J}m_{J}+R_{\alpha}^{r+1}(\sum^{n}_{i=1}D_{i}Q_{i}+m^{\prime})=$
$\displaystyle\sum_{{\begin{subarray}{c}\\#J=\mu-r,\\\ J\subset
I_{\alpha}\end{subarray}}}t_{\alpha}m^{\prime}_{J}+\sum^{n}_{i=1}(D_{i}+\alpha)R_{\alpha}^{r+1}Q_{i}+R_{\alpha}^{r+1}(m^{\prime}-\sum^{n}_{i=1}\alpha
Q_{i})$ $\displaystyle=$ $\displaystyle
R_{\alpha}^{r+1}(m^{\prime}-\sum^{n}_{i=1}\alpha
Q_{i})\in{\mathcal{M}}_{\alpha}.$

But now $m^{\prime}-\sum^{n}_{i=1}\alpha Q_{i}\in
F_{k-1}{\mathcal{M}}_{\alpha}$. Iterating the above argument one can find
$\tilde{m}\in F_{\ell}{\mathcal{M}}_{\alpha}$ such that

$R_{\alpha}^{r+1}m=R_{\alpha}^{r+1}\tilde{m}.$

This completes the proof of the theorem. ∎

###### Proof of the lemma.

The proof is essentially the same as the reduced case. Note that we are now
working over the commutative ring
${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}$. We prove by induction on $r$.
Let $P\in{\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}$ represent an element
of $\ker[R_{\alpha}]^{r+1}$. When $r=0$, we have

(7.31)
$\frac{1}{e_{0}}z_{0}\partial_{0}P=t_{\alpha}Q_{0}+\sum^{n}_{i=1}D_{i}Q_{i}\,\,\text{
recalling that }\,t_{\alpha}=z_{0}z_{1}\cdots z_{\mu}.$

Then $t_{\alpha}Q_{0}$ is in the ideal generated by
$\partial_{0},\partial_{1},...,\partial_{\mu},z_{\mu+1}\partial_{\mu+1},z_{\mu+2}\partial_{\mu+2},...,z_{k}\partial_{k},\partial_{k+1},...\partial_{n}$
over ${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}$. Because $t_{\alpha}$
together with
$\partial_{0},\partial_{1},...,\partial_{\mu},z_{\mu+1}\partial_{\mu+1},z_{\mu+2}\partial_{\mu+2},...,z_{k}\partial_{k},\partial_{k+1},...\partial_{n}$
form a regular sequence in ${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}$,
$Q_{0}$ can be written as,

$Q_{0}=\sum^{\mu}_{a=0}\partial_{a}Q_{a}+\sum^{k}_{b=\mu+1}z_{b}\partial_{b}Q_{b}+\sum^{n}_{c=k+1}\partial_{c}Q_{c}.$

Substuiting in (7.31)

$\frac{1}{e_{0}}z_{0}\partial_{0}\left(P-\sum^{\mu}_{a=0}e_{a}\frac{t_{\alpha}}{z_{a}}Q_{a}-\sum^{k}_{b=\mu+1}e_{b}t_{\alpha}Q_{b}\right)\in(D_{1},D_{2},...,D_{n}){\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}.$

Now because $(z_{0}\partial_{0},D_{1},D_{2},...,D_{n})$ is a regular sequence
in ${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}$, $P$ is a linear
combination of ${t_{\alpha}}/{z_{a}}$ for $a\in\\{0,1,...,\mu\\}$ and
$D_{1},D_{2},...,D_{n}$ over ${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}$.
This concludes the case when $r=0$.

Assume the statement is true for the case when the exponent is less than $r$.
Because $[R_{\alpha}]$ sends the class of $P$ to $\ker[R_{\alpha}]^{r}$, by
induction hypothesis we have

(7.32)
$\frac{1}{e_{0}}z_{0}\partial_{0}P=\sum_{{\begin{subarray}{c}\\#J=\mu-r+1,\\\
J\subset
I_{\alpha}\end{subarray}}}z_{J}Q_{J}+\sum^{n}_{i=1}D_{i}Q_{i}\quad\text{
recalling that }z_{J}=\prod_{j\in J}z_{j}.$

Fixing a subset $J$, then $z_{J}Q_{J}$ is in the submodule generated by
$z_{a}$ for $a\in I_{\alpha}\setminus J$, $\partial_{b}$ for $b\in J$,
$z_{c}\partial_{c}$ for $c\in I\setminus I_{\alpha}$ and $\partial_{d}$ for
$d\notin I$ over ${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}$. Because the
elements $z_{a},\partial_{b},z_{c}\partial_{c},\partial_{d}$ for $a\in
I_{\alpha}\setminus J,b\in J,c\in I\setminus I_{\alpha},d\notin I$ together
with $z_{J}$ form a regular sequence in
${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}$, we deduce that

$Q_{J}=\sum_{a\in I_{\alpha}\setminus J}z_{a}Q_{a}+\sum_{b\in
J}\partial_{b}Q_{b}+\sum_{c\in I\setminus
I_{\alpha}}z_{c}\partial_{c}Q_{c}+\sum_{d\notin I}\partial_{d}Q_{d}.$

Substituting in (7.32), we deduce that

$\frac{1}{e_{0}}z_{0}\partial_{0}\left(P-\left(\sum_{b\in
J}e_{b}\frac{z_{J}}{z_{b}}Q_{b}+\sum_{c\in I\setminus
I_{\alpha}}e_{c}z_{J}Q_{c}\right)\right)-\sum_{a\in I_{\alpha}\setminus
J}z_{J}z_{a}Q_{a}$

is in the submodule generated by degree $\mu-r+1$ monomials dividing
$t_{\alpha}$ except $z_{J}$ and by $D_{1},D_{2},...,D_{n}$ over
${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{X}$. This means we can reduce
$z_{J}Q_{J}$ one by one for each $J$ on the right-hand side of the equation
(7.32) and at the last step we find that
$\frac{1}{e_{0}}z_{0}\partial_{0}(P-P^{\prime})$ is a linear combination of
degree $\mu-r+2$ monomials dividing $t_{\alpha}$ and $D_{1},D_{2},...,D_{n}$,
where $P^{\prime}$ is a linear combination of degree $\mu-r$ monomials
dividing $t_{\alpha}$.

Note that the left multiplication by $\frac{1}{e_{0}}z_{0}\partial_{0}$ has
the same effect as applying $[R_{\alpha}]$ on
${\mathrm{g}}{\mathrm{r}}^{F}{\mathcal{M}}_{\alpha}$. Therefore, the class
represented by $P-P^{\prime}$ is in $\ker[R_{\alpha}]^{r}$ since degree
$\mu-r+2$ monomials dividing $t_{\alpha}$ is in $\ker[R_{\alpha}]^{r-1}$. By
induction hypothesis the class represented $P-P^{\prime}$ is a linear
combination of degree $\mu-r+1$ monomials dividing $t_{\alpha}$. Therefore,
the class represented by $P$ in
${\mathrm{g}}{\mathrm{r}}^{F}{\mathcal{M}}_{\alpha}$ is a linear combination
of degree $\mu-r$ monomials dividing $t_{\alpha}$. This completes the proof. ∎

######  Corollary 7.7.

The $\ker R^{r+1}_{\alpha}$ is also generated by degree $\mu-r$ monomials
dividing $t_{\alpha}$ if one identifies ${\mathcal{M}}_{\alpha}$ locally with
${\mathscr{D}}_{X}/(t_{\alpha},D_{1},D_{2},...,D_{n}){\mathscr{D}}_{X}$.

The proof is the same as the one of Corollary 5.3

###  7.3. The weight filtration

Now the weight filtration of each generalized eigen-modules interacts well
with the good filtration because of the strictness. Recall that since
$R_{\alpha}$ is nilpotent on ${\mathcal{M}}_{\alpha}$, it induces a
${\mathbb{Z}}$-indexed filtration $W_{\bullet}{\mathcal{M}}_{\alpha}$. We
filtered the sub-module $W_{r}{\mathcal{M}}_{\alpha}$ by the induced
filtration
$F_{\bullet}W_{r}{\mathcal{M}}_{\alpha}=F_{\bullet}{\mathcal{M}}_{\alpha}\cap
W_{r}{\mathcal{M}}_{\alpha}$. Let

${\mathcal{P}}_{\alpha,r}=\frac{\ker R^{r+1}_{\alpha}}{\ker
R^{r}_{\alpha}+R_{\alpha}\ker R^{r+2}_{\alpha}}$

be the $r$-th primitive part of
${\mathrm{g}}{\mathrm{r}}^{W}{\mathcal{M}}_{\alpha}$ with the filtration
defined by

$F_{\ell}{\mathcal{P}}_{\alpha,r}=\frac{F_{\ell}\ker R^{r+1}_{\alpha}+\ker
R^{r}_{\alpha}+R_{\alpha}\ker R^{r+2}_{\alpha}}{\ker
R^{r}_{\alpha}+R_{\alpha}\ker R^{r+2}_{\alpha}}.$

As the formal proof in Theorem 5.6, we have

######  Corollary 7.8.

The induced operator
$R^{r}_{\alpha}:F_{\ell}{\mathrm{g}}{\mathrm{r}}^{W}_{r}{\mathcal{M}}_{\alpha}\rightarrow
F_{\ell+r}{\mathrm{g}}{\mathrm{r}}^{W}_{-r}{\mathcal{M}}_{\alpha}$ is an
isomorphism. Therefore, the Lefschetz decomposition of
${\mathrm{g}}{\mathrm{r}}^{W}{\mathcal{M}}_{\alpha}$ respects filtrations,
i.e.

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

###  7.4. Summands of the primitive part ${\mathcal{P}}_{\alpha,r}$

Recall that $Y^{J}=\bigcap_{j\in J}Y_{j}$ and $Y_{J}=\bigcup_{j\in J}Y_{j}$
for any subset $J$ of $I$ and $e_{j}$ is the multiplicity of $Y_{j}$ in $Y$.
Like the reduced case that ${\mathcal{P}}_{r}$ decomposes into the direct
images of $\omega_{Y^{J}}(-r)$ for all index subset s$J$ of cardinality $r+1$
(Theorem 5.7), the primitive part ${\mathcal{P}}_{\alpha,r}$ of the
generalized $\alpha$-eigemodule also decomposes into direct images of certain
filtered ${\mathscr{D}}_{Y^{J}}$-modules ${\mathcal{V}}_{\alpha,J}(-r)$ for
all $J$ of cardinality $r+1$ such that $e_{j}\alpha$ for every $j\in J$ is an
integer. The filtered ${\mathscr{D}}_{Y^{J}}$-modules
${\mathcal{V}}_{\alpha,J}$ comes from cyclic coverings so that
${\mathcal{P}}_{\alpha,r}$ carries the Hodge theory of the cyclic coverings.
In fact, by a well-know construction in [EV92, $\S$3] the direct image of the
de Rham complex of a cyclic covering decomposes into log de Rham complexes of
line bundles. A line bundle with an integrable log connection also can be
viewed as a log ${\mathscr{D}}$-module. This suggests that the
${\mathscr{D}}$-modules ${\mathcal{V}}_{\alpha,J}$ is generated by a certain
log ${\mathscr{D}}$-module ${\mathscr{V}}_{\alpha,J}$. If $Y$ is reduced and
$\alpha=0$, ${\mathcal{V}}_{\alpha,J}$ is just $\omega_{Y^{J}}$. We shall
construct auxiliary log ${\mathscr{D}}$-modules ${\mathscr{V}}_{\alpha,J}$
whose log de Rham complex will be used to construct the ${\mathscr{D}}$-module
${\mathcal{V}}_{\alpha,J}$, without using cyclic cover. The cyclic coverings
are involved only when we study the Hodge theory of those
${\mathscr{D}}$-modules. We fix a rational number $\alpha\in[0,1)$ to simplify
the notations and let $I_{\alpha}$ be a subset of indices consisting of $i$
such that $\alpha e_{i}$ is an integer.

Denote by ${\mathcal{L}}$ the line bundle ${\mathscr{O}}_{X}\left(-\sum_{i\in
I_{\alpha}}{\frac{e_{i}}{N}}Y_{i}\right)$, where $N$ is the greatest common
divisor of $e_{i}$ for $i\in I_{\alpha}$. In this notation,
${\mathscr{O}}_{X}\left(-\left\lceil\alpha
Y\right\rceil\right)={\mathcal{L}}^{\alpha N}\left(-\sum_{i\in I\setminus
I_{\alpha}}\left\lceil\alpha e_{i}Y_{i}\right\rceil\right)$. Because the line
bundle ${\mathscr{O}}_{X}(Y)$ can be trivialized by a global section, we get
an isomorphism of ${\mathscr{O}}_{X}$-modules:

(7.33) ${\mathcal{L}}^{N}={\mathscr{O}}_{X}\left(-\sum_{i\in
I_{\alpha}}e_{i}Y_{i}\right)\rightarrow{\mathscr{O}}_{X}\left(\sum_{i\in
I\setminus I_{\alpha}}e_{i}Y_{i}\right).$

Choose a local section $l$ of ${\mathcal{L}}$ such that
$l^{N}\mapsto\prod_{i\in I\setminus I_{\alpha}}z^{-e_{i}}_{i}$ under (7.33).
Now we shall put a log connection $\nabla$ on

${\mathscr{O}}_{X}(-\lceil\alpha
Y\rceil)={\mathcal{L}}^{\alpha{N}}\left(-\sum_{i\in I\setminus
I_{\alpha}}\left\lceil\alpha e_{i}Y_{i}\right\rceil\right).$

First we define, using the product rule

(7.34) $\frac{\nabla l^{N}}{l^{N}}=N\frac{\nabla l}{l}=\sum_{i\in I\setminus
I_{\alpha}}-e_{i}\frac{dz_{i}}{z_{i}}$

due to (7.33). Then, let $s=l^{\alpha N}\prod_{i\in I\setminus
I_{\alpha}}z^{\left\lceil\alpha e_{i}\right\rceil}_{i}$ be the local frame of
${\mathscr{O}}_{X}(-\lceil\alpha Y\rceil)$. Noting that $\alpha N$ is a non-
negative integer, the induced log connection works as

(7.35) $\displaystyle\frac{\nabla s}{s}=\frac{\nabla(l^{\alpha N}\prod_{i\in
I\setminus I_{\alpha}}z^{\left\lceil\alpha e_{i}\right\rceil}_{i})}{l^{\alpha
N}\prod_{i\in I\setminus I_{\alpha}}z^{\left\lceil\alpha
e_{i}\right\rceil}_{i}}$ $\displaystyle=\alpha N\frac{\nabla l}{l}+\sum_{i\in
I\setminus I_{\alpha}}{\left\lceil\alpha
e_{i}\right\rceil}\frac{dz_{i}}{z_{i}}$ $\displaystyle=\sum_{i\in I\setminus
I_{\alpha}}\left({\left\lceil\alpha e_{i}\right\rceil}-\alpha
e_{i}\right)\frac{dz_{i}}{z_{i}}=\sum_{i\in I\setminus I_{\alpha}}\\{-\alpha
e_{i}\\}\frac{dz_{i}}{z_{i}},$

where $\\{-\\}$ denotes the function of taking fractional part. Putting in
more standard form,

$\nabla s=\sum_{i\in I\setminus I_{\alpha}}\\{-\alpha
e_{i}\\}\frac{dz_{i}}{z_{i}}\otimes s.$

This log connection is integrable and has poles along $Y_{i}$ for $i\in
I\setminus I_{\alpha}$ with eigenvalues $\\{-\alpha e_{i}\\}$. We endow the
line bundle ${\mathscr{O}}_{X}(-\lceil\alpha Y\rceil)$ with this integrable
log connection $\nabla$.

Fix a subset $J$ of $I_{\alpha}$ with $\\#J=r+1$ so that $\dim Y^{J}=n-r$. The
pullback of $({\mathscr{O}}_{X}(\lceil-\alpha Y\rceil),\nabla)$ by the
inclusion $\tau^{J}:Y^{J}\rightarrow X$ gives an integrable log connection
$(\mathscr{V},\nabla)=({\mathscr{V}}_{\alpha,J},\nabla)$ on $Y^{J}$ with poles
along $E=E^{\alpha,J}$ the pullback of $Y_{I\setminus I_{\alpha}}$. Moreover,
the log de Rham complex of $({\mathscr{V}},\nabla)$

$\\{{\mathscr{V}}\rightarrow\Omega_{Y^{J}}(\log
E)\otimes{\mathscr{V}}\rightarrow\cdots\rightarrow\Omega^{{n-r}}_{Y^{J}}(\log
E)\otimes{\mathscr{V}}\\}[{n-r}],$

induces a complex of ${\mathscr{D}}_{Y^{J}}$-modules

(7.36)
$\\{{\mathscr{V}}\otimes{\mathscr{D}}_{Y^{J}}\rightarrow\Omega_{Y^{J}}(\log
E)\otimes{\mathscr{V}}\otimes{\mathscr{D}}_{Y^{J}}\rightarrow\cdots\rightarrow\Omega^{{n-r}}_{Y^{J}}(\log
E)\otimes{\mathscr{V}}\otimes{\mathscr{D}}_{Y^{J}}\\}[{n-r}],$

which is nothing but the log de Rham complex of
${\mathscr{V}}\otimes{\mathscr{D}}_{Y^{J}}$. It follows from Lemma 2.3 that
the complex is a resolution of

${\mathcal{V}}={\mathcal{V}}_{\alpha,J}=_{\mathrm{def}}\omega_{Y^{J}}(\log
E)\otimes{\mathscr{V}}\mathbin{\mathop{\otimes}\displaylimits_{{\mathscr{D}}_{\left(Y^{J},E\right)}}}{\mathscr{D}}_{Y^{J}}.$

We endow ${\mathcal{V}}$ with the filtration
$F_{\ell}{\mathcal{V}}=F_{\ell}{\mathcal{V}}_{\alpha,J}$ induced the
subcomplex

$\\{{\mathscr{V}}\otimes
F_{\ell}{\mathscr{D}}_{Y^{J}}\rightarrow\Omega_{Y^{J}}(\log
E)\otimes{\mathscr{V}}\otimes
F_{\ell+1}{\mathscr{D}}_{Y^{J}}\rightarrow\cdots\rightarrow\Omega^{{n-r}}_{Y^{J}}(\log
E)\otimes{\mathscr{V}}\otimes F_{\ell+{n-r}}{\mathscr{D}}_{Y^{J}}\\}[{n-r}].$

It is clear that $F_{\bullet}{\mathcal{V}}$ is a good filtration. For example,
if $\alpha=0$, then $E$ is empty and ${\mathscr{V}}$ is just
${\mathscr{O}}_{Y^{J}}$ so that ${\mathcal{V}}=\omega_{Y^{J}}$ as
${\mathscr{D}}_{Y^{J}}$-modules. Since the eigenvalues of the log connection
are in $(0,1)$ if poles exist, the log de Rham complex of
$({\mathscr{V}},\nabla)$ is the minimal extension $R_{!*}\mathbb{V}$ of the
local system $\mathbb{V}$ consisting of the flat sections of $\nabla$ on
${\mathscr{V}}$ over $Y^{J}\setminus Y_{I\setminus J}$ (see [EV92, 1.6]).
Later we will put a sesquilinear pairing on ${\mathcal{V}}$ and all the data
will yield a pure Hodge structure of the log de Rham complex of
${\mathscr{V}}$.

######  Lemma 7.9.

The de Rham complex ${\mathrm{D}}{\mathrm{R}}_{Y^{J}}{\mathcal{V}}$ together
with the filtration $F_{\bullet}{\mathrm{D}}{\mathrm{R}}_{Y^{J}}{\mathcal{V}}$
is isomorphic to the log de Rham complex $\Omega^{n-r+\bullet}_{Y^{J}}(\log
E)\otimes{\mathscr{V}}$ with the stupid filtration in the derived category of
filtered complexes of ${\mathbb{C}}$-vector spaces. In addition,
${\mathcal{V}}$ is holonomic and the characteristic cycle of ${\mathcal{V}}$
is

$cc({\mathcal{V}})=\sum_{K\subset I\setminus
I_{\alpha}}\left[T^{*}_{Y^{K\cup{J}}}Y^{J}\right].$

###### Proof.

We can choose the local frame $s$ of ${\mathscr{V}}$ such that

$\nabla s=\sum_{i\in I\setminus
I_{\alpha}}\frac{dz_{i}}{z_{i}}\otimes\\{-\alpha e_{i}\\}s$

where $z_{i}$ is the defining equation of $E_{i}$ for each $i$. Therefore, the
complex (7.36) locally is the Koszul complex over ${\mathscr{D}}_{Y^{J}}$
associated to the sequence

$x_{1}\partial_{1}+\\{-\alpha e_{1}\\},x_{2}\partial_{2}+\\{-\alpha
e_{2}\\},...,x_{p}\partial_{p}+\\{-\alpha
e_{p}\\},\partial_{p+1},\partial_{p+2},...,\partial_{n-r},$

for some rearrangement of coordinates and under the trivialization of
${\mathscr{V}}$ given by $s$. It follows that the associated graded of (7.36)
is the Koszul complex associated to the regular sequence

$x_{1}\partial_{1},x_{2}\partial_{2},...,x_{p}\partial_{\nu},\partial_{p+1},\partial_{p+2},...,\partial_{n-r}$

over ${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{Y^{J}}$. Thus, the complex
(7.36) is filtered acyclic. By the similar argument in Theorem 4.5, the
${\mathscr{D}}_{Y^{J}}$-module ${\mathcal{V}}$ is holonomic and the
charateristic cycle $cc({\mathcal{V}})=\sum_{K\subset I\setminus
I_{\alpha}}\left[T^{*}_{Y^{K\cup{J}}}Y^{J}\right]$.

Moreover, we have isomorphisms in the derived category of complexes of
${\mathbb{C}}$-vector spaces:

$\displaystyle
F_{\ell}{\mathrm{D}}{\mathrm{R}}{\mathcal{V}}=F_{\ell+*}{\mathcal{V}}\otimes\bigwedge^{-\star}{\mathscr{T}}_{Y^{J}}$
$\displaystyle\simeq\Omega^{n-r+\bullet}_{Y^{J}}(\log
E)\otimes{\mathscr{V}}\otimes
F_{\ell+n-r+\bullet+*}{\mathscr{D}}_{Y^{J}}\otimes\bigwedge^{-\star}{\mathscr{T}}_{Y^{J}}$
$\displaystyle\simeq\Omega^{n-r+\bullet}_{Y^{J}}(\log
E)\otimes{\mathscr{V}}\otimes F_{\ell+n-r}{\mathscr{O}}_{Y^{J}}.$

Since $F_{\ell}{\mathscr{O}}_{Y^{J}}$ is ${\mathscr{O}}_{Y^{J}}$ or vanishes
if $\ell<0$, the complex $\Omega^{n-r+\bullet}_{Y^{J}}(\log
E)\otimes{\mathscr{V}}\otimes F_{\ell+n-r}{\mathscr{O}}_{Y^{J}}$ is the stupid
filtration on the log de Rham complex on ${\mathscr{V}}$. We conclude the
proof. ∎

We also need an auxiliary ${\mathscr{D}}_{Y^{J}}$-module
${\mathcal{V}}^{*}_{\alpha,J}$ to identify the primitive part
${\mathcal{P}}_{\alpha,r}$ which plays the role as $\omega_{Y^{J}}(*D^{J})$ in
the counterpart for the reduced case (Theorem 5.7). The log de Rham complex of
$({\mathscr{V}},\nabla)$ can be enlarged into

$\\{{\mathscr{V}}\rightarrow\Omega_{Y^{J}}(\log
D)\otimes{\mathscr{V}}\rightarrow\cdots\rightarrow\Omega^{{n-r}}_{Y^{J}}(\log
D)\otimes{\mathscr{V}}\\}[{n-r}],\quad\text{for }D=D^{J}\text{ the pullback of
the divisor }Y_{I\setminus J}.$

It is quasi-isomorphic to $Rj_{*}\mathbb{V}$ for $j:Y^{J}\setminus
Y_{I_{\alpha}}\rightarrow Y^{J}$ is the open immersion. By the similar process
of the above, it induces a filtered acyclic complex of
${\mathscr{D}}_{Y^{J}}$-modules

(7.37)
$\\{{\mathscr{V}}\otimes{\mathscr{D}}_{Y^{J}}\rightarrow\Omega_{Y^{J}}(\log
D)\otimes{\mathscr{V}}\otimes{\mathscr{D}}_{Y^{J}}\rightarrow\cdots\rightarrow\Omega^{{n-r}}_{Y^{J}}(\log
D)\otimes{\mathscr{V}}\otimes{\mathscr{D}}_{Y^{J}}\\}[{n-r}].$

Let ${\mathcal{V}}^{*}={\mathcal{V}}^{*}_{\alpha,J}$ be the $0$-th cohomology
of the above complex and endow it with the filtration such that
$F_{\ell}{\mathcal{V}}^{*}=F_{\ell}{\mathcal{V}}^{*}_{\alpha,J}$ is induced by
the subcomplex

$\\{{\mathscr{V}}\otimes
F_{\ell}{\mathscr{D}}_{Y^{J}}\rightarrow\Omega_{Y^{J}}(\log
D)\otimes{\mathscr{V}}\otimes
F_{\ell+1}{\mathscr{D}}_{Y^{J}}\rightarrow\cdots\rightarrow\Omega^{{n-r}}_{Y^{J}}(\log
D)\otimes{\mathscr{V}}\otimes F_{\ell+{n-r}}{\mathscr{D}}_{Y^{J}}\\}[{n-r}].$

We naturally get an induced morphism
$({\mathcal{V}},F_{\bullet}{\mathcal{V}})\rightarrow({\mathcal{V}}^{*},F_{\bullet}{\mathcal{V}}^{*})$
from the inclusion of the log de Rham complexes.

######  Lemma 7.10.

The canonical morphism
$({\mathcal{V}},F_{\bullet}{\mathcal{V}})\rightarrow({\mathcal{V}}^{*},F_{\bullet}{\mathcal{V}}^{*})$
is injective, whose image is generated by the monomials defining $D-E$.

###### Proof.

Suppose $x_{1}x_{2}\cdots x_{p}$ is the local defining equation of $E$ and
$x_{1}x_{2}\cdots x_{q}$ is the local defining equation of $D$ for $q\geq
p+1$. Since ${\mathcal{V}}$ is locally generated by the class of

$\bigwedge^{p}_{i=1}\frac{dx_{i}}{x_{i}}\wedge dx_{p+1}\wedge\cdots\wedge
dx_{n-r}\otimes s\otimes 1$

and ${\mathcal{V}}^{*}$ is locally generated by the class of

$\bigwedge^{q}_{i=1}\frac{dx_{i}}{x_{i}}\wedge dx_{q+1}\wedge\cdots\wedge
dx_{n-r}\otimes s\otimes 1,$

the image is generated by the class of
$\bigwedge^{q}_{i=1}\frac{dx_{i}}{x_{i}}\wedge dx_{q+1}\wedge\cdots\wedge
dx_{n-r}\otimes s\otimes x_{p+1}x_{p+2}\cdots x_{q}$. The morphism locally is

$\displaystyle{\mathscr{D}}_{Y^{J}}/(x_{1}\partial_{1}+r_{1},...,x_{p}\partial_{p}+r_{p},\partial_{p+1},...,\partial_{n-r}){\mathscr{D}}_{Y^{J}}$
$\displaystyle\rightarrow{\mathscr{D}}_{Y^{J}}/(x_{1}\partial_{1}+r_{1},...,x_{q}\partial_{q}+r_{q},\partial_{q+1}...,\partial_{n-r}){\mathscr{D}}_{Y^{J}},$

with $[P]\mapsto[x_{p+1}x_{p+2}\cdots x_{q}P]$ where $r_{1},r_{2},...,r_{p}$
are the eigenvalues of $\nabla$ on ${\mathscr{V}}$ and
$r_{p+1}=r_{p+2}=\cdots=r_{q}=0$. Since

$\Omega^{n-r}_{Y^{J}}(\log
E)\otimes{\mathscr{V}}=F_{-(n-r)}{\mathcal{V}}\rightarrow
F_{-(n-r)}{\mathcal{V}}^{*}=\Omega^{n-r}_{Y^{J}}(\log D)\otimes{\mathscr{V}}$

is injective, by induction, it suffices to show that
${\mathrm{g}}{\mathrm{r}}^{F}{\mathcal{V}}\rightarrow{\mathrm{g}}{\mathrm{r}}^{F}{\mathcal{V}}^{*}$
is injective. Due to the complexes (7.36) and (7.37) is filtered acyclic, the
morphism on the associated graded modules works as, in the local
representation,

${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{Y^{J}}/(x_{1}\partial_{1},...,x_{p}\partial_{p},\partial_{p+1},...,\partial_{n-r}){\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{Y^{J}}\rightarrow{\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{Y^{J}}/(x_{1}\partial_{1},...,x_{q}\partial_{q},\partial_{q+1}...,\partial_{n-r}){\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{Y^{J}},$

with $[P]\mapsto[x_{p+1}x_{p+2}\cdots x_{q}P]$. By induction on the number of
components of $D-E$, we can assume $q=p+1$. Let
$P\in{\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{Y^{J}}$ represent a class in
the kernel. Then

$x_{q}P=\sum^{q}_{i=1}x_{i}\partial_{i}P_{i}+\sum^{n-r}_{j=q+1}\partial_{j}P_{j}\in{\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{Y^{J}}.$

Subtracting $x_{q}\partial_{q}P_{q}$ on the both sides yeilds

$x_{q}(P-\partial_{q}P_{q})=\sum^{q-1}_{i=1}x_{i}\partial_{i}P_{i}+\sum^{n-r}_{j=q+1}\partial_{j}P_{j}\in{\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{Y^{J}}.$

Since
$x_{q},x_{1}\partial_{1},...,x_{q-1}\partial_{q-1},\partial_{q+1},...,\partial_{n-r}$
is a regular sequence over
${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{Y^{J}}$,

$(P-\partial_{q}P_{q})=\sum^{q-1}_{i=1}x_{i}\partial_{i}P^{\prime}_{i}+\sum^{n-r}_{j=q+1}\partial_{j}P^{\prime}_{j}\in{\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{Y^{J}}.$

We find that $P$ is a linear combination of
$x_{1}\partial_{1},x_{2}\partial_{2},...,x_{p}\partial_{p},\partial_{p+1},...,\partial_{n-r}$
over ${\mathrm{g}}{\mathrm{r}}^{F}{\mathscr{D}}_{Y^{J}}$. We conclude the
proof. ∎

######  Remark 7.11.

One can use Riemann-Hilbert correspondence to conclude that ${\mathcal{V}}$ is
the minimal extension of ${\mathscr{V}}|_{Y^{J}\setminus D}$ and
${\mathcal{V}}^{*}$ is the $*$-extension of ${\mathscr{V}}|_{Y^{J}\setminus
D}$, which is overkill in our situation. The above argument also showed the
strictness, i.e.,
$F_{\ell}{\mathcal{V}}=F_{\ell}{\mathcal{V}}^{*}\cap{\mathcal{V}}$.

Putting in more general notations and summarizing what we have proved in the
above two lemmas:

######  Theorem 7.12.

The filtered ${\mathscr{D}}_{Y^{J}}$-module
$({\mathcal{V}}_{\alpha,J},F_{\bullet})$ is holonomic whose de Rham complex
${\mathrm{D}}{\mathrm{R}}_{Y^{J}}{\mathcal{V}}_{\alpha,J}$ together with the
induced filtration is isomorphic to the log de Rham complex
$\Omega^{n-r+\bullet}_{Y^{J}}(\log
E^{\alpha,J})\otimes{\mathscr{V}}_{\alpha,J}$ with the stupid filtration in
the derived category of filtered complexes of ${\mathbb{C}}$-vector spaces and
whose characteristic cycle is

$cc({\mathcal{V}}_{\alpha,J})=\sum_{K\subset I\setminus
I_{\alpha}}\left[T^{*}_{Y^{K\cup J}}Y^{J}\right].$

The canonical filtered morphism
$({\mathcal{V}}_{\alpha,J},F_{\bullet}{\mathcal{V}}_{\alpha,J})\rightarrow({\mathcal{V}}^{*}_{\alpha,J},F_{\bullet}{\mathcal{V}}^{*}_{\alpha,J})$
is injective and the image is generated by the monomial defining the divisor
$D^{J}-E^{\alpha,J}$.

###  7.5. Identifying the primitive part ${\mathcal{P}}_{\alpha,r}$

Now we are going to identify the $r$-th primitive part
$({\mathcal{P}}_{\alpha,r},F_{\bullet}{\mathcal{P}}_{\alpha,r})$ with a direct
sum of ${\mathcal{V}}_{\alpha,J}(-r)$ for $J$ ranging over subsets
$I_{\alpha}$ of cardinality $r+1$. The argument is parallel to the one of the
reduced case (Theorem 5.7), replacing ${\mathcal{M}}$ by
${\mathcal{M}}_{\alpha}$, $R$ by $R_{\alpha}$, $\omega_{Y^{J}}$ by
${\mathcal{V}}_{\alpha,J}$, $\omega_{Y^{J}}(*D^{J})$ by
${\mathcal{V}}^{*}_{\alpha,J}$, the complex
$\Omega^{n+\bullet}_{X/\Delta}(\log Y)|_{Y}$ by
$C^{\bullet}_{\alpha}=\Omega^{n+\bullet}_{X/\Delta}(\log Y)(-\lceil\alpha
Y\rceil)|_{Y_{I_{\alpha}}}$ and the log de Rham complex
$\Omega^{n-r+\bullet}_{Y^{J}}(\log D^{J})$ by
$\Omega^{n-r+\bullet}_{Y^{J}}(\log D^{J})\otimes{\mathscr{V}}_{\alpha,J}$.

######  Theorem 7.13.

Let
${\mathcal{V}}_{\alpha,r}=\bigoplus_{J}\tau^{J}_{+}{\mathcal{V}}_{\alpha,J}$
for $J$ running over the subsets of $I_{\alpha}$ of cardinality $r+1$, where
$\tau^{J}:Y^{J}\hookrightarrow X$ is the closed embedding. Then there exists
an isomorphism
$\phi_{\alpha,r}:({\mathcal{P}}_{\alpha,r},F_{\bullet}{\mathcal{P}}_{\alpha,r})\rightarrow{\mathcal{V}}_{\alpha,r}(-r)$
in the category of filtered ${\mathscr{D}}_{X}$-modules.

###### Proof.

Because the log connection (7.33) we constructed on
${\mathscr{O}}_{X}(-\lceil\alpha Y\rceil)$ has zero residue on $Y_{i}$ for
$i\in I_{\alpha}$, we have the residue morphism between log de Rham complexes.

$\displaystyle{\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{Y^{J}}:\Omega^{\bullet+n+1}_{X}(\log
Y)\otimes{\mathscr{O}}_{X}(-\lceil\alpha Y\rceil)|_{Y_{I_{\alpha}}}$
$\displaystyle\rightarrow\Omega^{\bullet+n-r}_{Y^{J}}(\log
D^{J})\otimes{\mathscr{V}}_{\alpha,J},\text{ where }D^{J}\text{ is the pull
back of }Y_{I\setminus J}$

for $J\subset I_{\alpha}$ of cardinality $r+1$, up to a sign depending on the
order of the indices. Denote by $B^{\bullet}_{\alpha}$ the log de Rham complex
$\Omega^{\bullet+n+1}_{X}(\log Y)\otimes{\mathscr{O}}_{X}(-\lceil\alpha
Y\rceil)$ of ${\mathscr{O}}_{X}(-\lceil\alpha Y\rceil)$. The residue morphism
${\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{Y^{J}}$ extends to a morphism of the
complexes of induced ${\mathscr{D}}_{X}$-modules

${\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{Y^{J}}:B^{\bullet}_{\alpha}|_{Y_{I_{\alpha}}}\otimes{\mathscr{D}}_{X}\rightarrow\Omega^{\bullet+n-r}_{Y^{J}}(\log
D^{J})\otimes{\mathscr{V}}_{\alpha,J}\otimes{\mathscr{D}}_{X}.$

Let ${\mathcal{H}}^{\ell}_{\alpha}$ be the $\ell$-th cohomology of
$B^{\bullet}_{\alpha}|_{Y_{I_{\alpha}}}\otimes{\mathscr{D}}_{X}$. Then we have
an induced morphism
${\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{Y^{J}}:{\mathcal{H}}^{0}_{\alpha}\rightarrow{\mathcal{V}}^{*}_{\alpha,J}$
by taking cohomology. Let
${\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{\alpha,r}=\bigoplus{\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{Y^{J}}:{\mathcal{H}}^{0}_{\alpha}\rightarrow{\mathcal{V}}^{*}_{\alpha,r}(-r)$
where ${\mathcal{V}}^{*}_{\alpha,r}=\bigoplus_{J}{\mathcal{V}}^{*}_{\alpha,J}$
for $J$ running over cardinality $r+1$ subsets of $I_{\alpha}$. Because
$\frac{dt}{t}\wedge:\Omega^{\bullet+n}_{X/\Delta}(\log Y)(-\lceil\alpha
Y\rceil)\rightarrow\Omega^{\bullet+n+1}_{X}(\log Y)(-\lceil\alpha Y\rceil)$
also extends to the complexes of the induced ${\mathscr{D}}_{X}$-modules, we
obtain a short exact sequence

${0}$${C^{\bullet-1}_{\alpha}\otimes{\mathscr{D}}_{X}}$${B^{\bullet-1}|_{Y_{I_{\alpha}}}\otimes{\mathscr{D}}_{X}}$${C^{\bullet}_{\alpha}\otimes{\mathscr{D}}_{X}}$${0.}$$\scriptstyle{\frac{dt}{t}\wedge}$

The associated long exact sequence gives

(7.38)
${0}$${{\mathcal{H}}^{-1}_{\alpha}}$${{\mathcal{M}}_{\alpha}}$${{\mathcal{M}}_{\alpha}}$${{\mathcal{H}}^{0}_{\alpha}}$${0.}$$\scriptstyle{R_{\alpha}}$$\scriptstyle{\frac{dt}{t}\wedge}$

By pre-composing $\frac{dt}{t}\wedge$, we get a morphism

${\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{\alpha,J}\circ\frac{dt}{t}\wedge:{\mathcal{M}}_{\alpha}\rightarrow{\mathcal{V}}^{*}_{\alpha,r}(-r),\quad[\zeta_{\alpha}\otimes
P]\rightarrow[{\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{\alpha,J}\frac{dt}{t}\wedge\zeta_{\alpha}\otimes
P].$

Recall that every element in ${\mathcal{M}}_{\alpha}$ is locally represented
by $\zeta_{\alpha}\otimes P$ for
$\zeta_{\alpha}=z^{\lceil\alpha\mathbf{e}\rceil}_{I}\frac{dz_{1}}{z_{1}}\wedge\frac{dz_{2}}{z_{2}}\wedge\cdots\wedge\frac{dz_{k}}{z_{k}}\wedge\cdots\wedge
dz_{n}$ given that locally $I=\\{0,1,\dots,k\\}$, and $P\in{\mathscr{D}}_{X}$.
By Corollary 7.7, every class in $\ker R^{r+1}_{\alpha}$ is represented by
$\zeta_{\alpha}\otimes z_{\overline{J}}P$ for some ordered index subset $J$ of
$I_{\alpha}$ of cardinality $r+1$ and $\overline{J}$ is the complement of $J$
in $I_{\alpha}$ and $z_{\overline{J}}=\prod_{j\in\overline{J}}z_{j}$. Thus,
its image under the above morphism only contained in the component
${\mathcal{V}}^{*}_{\alpha,J}(-r)$ because $z_{\overline{J}}$ vanishes on
other components. The image is the class represented by

(7.39)
${\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{\alpha,J}\frac{dz_{0}}{z_{0}}\wedge\frac{dz_{1}}{z_{1}}\wedge\cdots\wedge\frac{dz_{k}}{z_{k}}\wedge\cdots\wedge
dz_{n}z^{\lceil\alpha\mathbf{e}\rceil}_{I}\otimes
z_{\overline{J}}P=\pm\frac{dz_{I\setminus J}}{z_{I\setminus J}}\wedge
dz_{k+1}\wedge\cdots\wedge dz_{n}\otimes s_{\alpha,J}\otimes
z_{\overline{J}}P\in\Omega^{n-r}_{Y^{J}}\otimes{\mathscr{V}}_{\alpha,J}\otimes{\mathscr{D}}_{X},$

where $s_{\alpha,J}$ is the local frame of ${\mathscr{V}}_{\alpha,J}$ by
restricting $z^{\lceil\alpha\mathbf{e}\rceil}_{I}$ and the sign is depending
on the order of $J$. It also follows from the calculation that the image does
not have pole along the pull-back of $Y_{\overline{J}}$. So it is contained in
the subsheaf consisting of classes represented by $\Omega^{n-r}_{Y^{J}}(\log
E^{\alpha,J})\otimes{\mathscr{V}}_{\alpha,J}\otimes{\mathscr{D}}_{X}$, where
$E^{\alpha,J}$ is the pull-back of $Y_{I\setminus I_{\alpha}}$ so that
$D^{J}-E^{\alpha,J}$ is the pull-back of $Y_{\overline{J}}$. This means that
the image of the class represented by (7.39) is also in the image of the
canonical inclusion:

$\displaystyle\tau^{J}_{+}{\mathcal{V}}_{\alpha,J}(-r)$
$\displaystyle\hookrightarrow\tau^{J}_{+}{\mathcal{V}}^{*}_{\alpha,J}(-r),$
$\displaystyle[dz_{\overline{J}}\wedge\frac{dz_{I\setminus
I_{\alpha}}}{z_{I\setminus I_{\alpha}}}\wedge dz_{k+1}\wedge\cdots\wedge
dz_{n}\otimes s_{\alpha,J}\otimes P]$
$\displaystyle\mapsto[\frac{dz_{\overline{J}}}{z_{\overline{J}}}\wedge\frac{dz_{I\setminus
I_{\alpha}}}{z_{I\setminus I_{\alpha}}}\wedge dz_{k+1}\wedge\cdots\wedge
dz_{n}\otimes s_{\alpha,J}\otimes z_{\overline{J}}P].$

See Theorem 7.12. Therefore, the morphism $\ker
R_{\alpha}^{r+1}\rightarrow{\mathcal{V}}^{*}_{\alpha,r}(-r)$ constructed above
factors through ${\mathcal{V}}_{\alpha,r}(-r)$. Summarizing, we have the
following diagram.

${\ker
R_{\alpha}^{r+1}}$${{\mathcal{V}}_{\alpha,r}(-r)}$${{\mathcal{M}}_{\alpha}}$${{\mathcal{H}}^{0}_{\alpha}}$${{\mathcal{V}}^{*}_{\alpha,r}(-r)}$$\scriptstyle{\rho_{\alpha,r}}$$\scriptstyle{{\frac{dt}{t}\wedge}}$$\scriptstyle{{\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{\alpha,r}}$

In fact, the kernel of $\rho_{r}$ contains $\ker R_{\alpha}^{r}$: for an
element in $\ker R_{\alpha}^{r}$ locally represented by $\zeta_{\alpha}\otimes
z_{K}P$ for $K$ a subset of $I_{\alpha}$ such that the cardinality of
$I_{\alpha}\setminus K$ is $r$, its image under $\rho_{\alpha,r}$ is zero
because $z_{K}$ annihilates all $\Omega^{n-r}_{Y^{J}}(\log
D^{J})\otimes{\mathscr{V}}_{\alpha,J}$ for any $J\subset I_{\alpha}$ of
cardinality $r+1$. The morphism $\rho_{\alpha,r}$ also kills $R_{\alpha}\ker
R^{r+2}_{\alpha}$ because $\frac{dt}{t}\wedge$ vanishes on the image of
$R_{\alpha}$ by (7.38). It follows that $\rho_{\alpha,r}$ factors through a
filtered morphism

$\phi_{\alpha,r}:{\mathcal{P}}_{\alpha,r}=\frac{\ker R^{r+1}_{\alpha}}{\ker
R^{r}_{\alpha}+R_{\alpha}\ker
R^{r+2}_{\alpha}}\rightarrow{\mathcal{V}}_{\alpha,r}(-r).$

For $dz_{\overline{J}}\wedge\frac{dz_{I\setminus I_{\alpha}}}{z_{I\setminus
I_{\alpha}}}\wedge dz_{k+1}\wedge\cdots\wedge dz_{n}\otimes
s_{\alpha,J}\otimes P\in\Omega^{n-r}_{Y^{J}}(\log
E^{\alpha,J})\otimes{\mathscr{V}}_{\alpha,J}\otimes F_{\ell}{\mathscr{D}}_{X}$
representing a class in $F_{\ell}\tau^{J}_{+}{\mathcal{V}}_{\alpha,J}(-r)$
where $J\subset I_{\alpha}$ of cardinality $r+1$, we can find a lifting
represented by $\zeta_{\alpha}\otimes z_{\overline{J}}P$ in $F_{\ell}\ker
R^{r+1}_{\alpha}$, which means

$F_{\ell}\ker R_{\alpha}^{r+1}\rightarrow F_{\ell+r}{\mathcal{V}}_{\alpha,r}$

is surjective, i.e. the morphism $\phi_{\alpha,r}$ is filtered surjective. It
remains to prove that $\phi_{\alpha,r}$ is injective. We prove that
$\phi_{\alpha,r}$ is an isomorphism by counting the characteristic cycles as
in Theorem 5.7. Because $\phi_{\alpha,r}$ is surjective, one gets

$cc({\mathcal{P}}_{\alpha,r})\geq cc({\mathcal{V}}_{\alpha,r}).$

It follows from Corollary 7.12 that

$cc({\mathcal{V}}_{\alpha,r})=\sum_{\begin{subarray}{c}J\subset I_{\alpha},\\\
\\#J=r+1\end{subarray}}cc(\tau^{J}_{+}{\mathcal{V}}_{\alpha,J})=\sum_{\begin{subarray}{c}J\subset
I_{\alpha},\\\ \\#J=r+1\end{subarray}}\sum_{K\subset I\setminus
I_{\alpha}}\left[T^{*}_{Y^{J\cup K}}X\right]=\sum_{\begin{subarray}{c}J\subset
I,\\\ \\#J\cap I_{\alpha}=r+1\end{subarray}}\left[T^{*}_{Y^{J}}X\right].$

One the other hand, by the Lefschetz decomposition and Theorem 7.2,

$\displaystyle\sum_{J\subset I}\\#(J\cap
I_{\alpha})\left[T^{*}_{Y^{J}}X\right]=cc({\mathcal{M}}_{\alpha})$
$\displaystyle=cc({\mathrm{g}}{\mathrm{r}}^{W}{\mathcal{M}}_{\alpha})=\sum_{r\geq
0}(r+1)cc({\mathcal{P}}_{\alpha,r})\geq\sum_{r\geq
0}(r+1)cc({\mathcal{V}}_{\alpha,r})$ $\displaystyle=\sum_{r\geq
0}\sum_{\begin{subarray}{c}J\subset I,\\\ \\#J\cap
I_{\alpha}=r+1\end{subarray}}(r+1)\left[T^{*}_{Y^{J}}X\right]=\sum_{J\subset
I}\\#(J\cap I_{\alpha})\left[T^{*}_{Y^{J}}X\right].$

It follows that all inequalities above are equalities and in particular,

$cc({\mathcal{P}}_{\alpha,r})=cc({\mathcal{V}}_{\alpha,r})$

from which we conclude that $\phi_{\alpha,r}$ is an isomorphism between the
underlying ${\mathscr{D}}_{X}$-modules. Plus it is filtered surjective, we
conclude that $\phi_{\alpha,r}$ is filtered isomorphism. ∎

##  8\. Non-reduced case: Sesquilinear pairing and limiting mixed Hodge
structure

###  8.1. Kähler package of cyclic covering

To accomplish our goal, we need to show that the sum of all hypercohomologies
of the complex

$\Omega^{\bullet}_{Y^{J}}(\log
E^{\alpha,J})\otimes{\mathscr{V}}_{\alpha,J}[n-r]$

has a polarized Hodge-Lefschetz structure and hard Lefschetz so that the
hypercohomology of the de Rham complex of the primitive part
${\mathcal{P}}_{\alpha,r}$ will inherit the properties by Theorem 7.12 and
Theorem 7.13. For this, we need to use the geometry of cyclic coverings.

We first give another description of the integrable log connection (7.33)
using cyclic coverings. Fix a rational number $\alpha$ in $[0,1)$, Because the
isomorphism,

${\mathcal{L}}^{N}={\mathscr{O}}_{X}\left(-\sum_{i\in
I_{\alpha}}e_{i}Y_{i}\right)\rightarrow{\mathscr{O}}_{X}\left(\sum_{i\in
I\setminus I_{\alpha}}e_{i}Y_{i}\right),$

we obtain a cyclic covering $\pi_{\alpha}:X_{\alpha}\rightarrow X$ by taking
the $N$-th roots out of $\sum_{i\in I\setminus I_{\alpha}}e_{i}Y_{i}$ and
normalizing it. The direct image
${\pi_{\alpha}}_{*}{\mathscr{O}}_{X_{\alpha}}$ decomposes into eigenspaces
with respect the Galois action as well as the direct image of exterior
differential
${\pi_{\alpha}}_{*}{\mathscr{O}}_{X_{\alpha}}\rightarrow{\pi_{\alpha}}_{*}\Omega_{X_{\alpha}}$
[EV92, Theorem 3.2]. The line bundle

${\mathcal{L}}^{\alpha{N}}\left(-\sum_{i\in I\setminus
I_{\alpha}}\left\lceil\alpha e_{i}Y_{i}\right\rceil\right),$

is the $\alpha$-eigenspaces of ${\pi_{\alpha}}_{*}{\mathscr{O}}_{X_{\alpha}}$
for some suitable choice of a generator of the Galois group. Because the
decomposition respects the exterior differential, we obtained an integrable
log connection with eigenvalues $\\{\alpha e_{i}\\}$ along $Y_{i}$ for each
$i\in I_{\alpha}$. Note that $X_{\alpha}$ might not be smooth.

Let $J\subset I_{\alpha}$ of cardinality $r+1$. Since $Y^{J}$ is not contained
in $Y_{I\setminus I_{\alpha}}$, the fiber product
$Y^{J}_{\alpha}=X_{\alpha}\times_{X}Y^{J}$ is again a cyclic covering of
$Y^{J}$ by taking the $N$-th roots out of $\sum_{i\in I\setminus
I_{\alpha}}e_{i}Y_{i}\cap Y^{J}$. Let
$\pi^{J}_{\alpha}:Y^{J}_{\alpha}\rightarrow Y^{J}$ be the second projection.

(8.40)
${Y^{J}_{\alpha}}$${X_{\alpha}}$${Y^{J}}$${X}$$\scriptstyle{\pi^{J}_{\alpha}}$$\scriptstyle{\pi_{\alpha}}$$\scriptstyle{\tau^{J}}$

We conclude that $({\mathscr{V}}_{\alpha,J},\nabla)$ is the
$\alpha$-eigenspace of
${\pi^{J}_{\alpha}}_{*}({\mathscr{O}}_{Y^{J}_{\alpha}},d)$. The log de Rham
complex of $({\mathscr{V}}_{\alpha,J},\nabla)$ is a summand of the direct
image of the de Rham compolex
${\pi^{J}_{\alpha}}_{*}\Omega^{\bullet+n-r}_{Y^{J}_{\alpha}}$ of
$Y^{J}_{\alpha}$.

We shall work in the general setting and adopt the convention in [EV86] and
[EV92]. Let ${\mathcal{L}}$ be a line bundle on a Kähler manifold $Z$ with a
Kähler form $\omega$ and $D=\sum_{i}\nu_{i}D_{i}$ be a simple normal crossings
divisor such that for some $N>1$ one has
${\mathcal{L}}^{N}={\mathscr{O}}_{Z}(D)$. Define
${\mathcal{L}}^{(j)}={\mathcal{L}}^{j}(-\lfloor\frac{jD}{N}\rfloor)$ for
$1\leq j\leq N-1$. One puts an integrable logarithmic connection on
${\mathcal{L}}^{(j)}$ with poles along $D^{(j)}$, where

$D^{(j)}=\sum_{\frac{j\nu_{i}}{N}\notin{\mathbb{Z}}}D_{i}.$

Let $\iota:U\hookrightarrow Z$ be the complement of $D$ and $\mathbb{V}$ is
the underlying local system of ${\mathcal{L}}|_{U}$. Let
$\tau:Z^{\prime}\rightarrow Z$ be the cyclic covering obtained by first taking
$N$-th root out of $D$ then taking the normalization and
$\pi:\tilde{Z}\rightarrow Z^{\prime}$ be a log resolution of singularity
equivariant with respect to the Galois group
${\mathrm{G}}{\mathrm{a}}{\mathrm{l}\,}(Z^{\prime}/Z)=\langle\sigma\rangle$
and let $E$ be the simple normal crossing exceptional divisor.

${\tilde{Z}}$${Z^{\prime}}$${Z}$$\scriptstyle{\pi}$$\scriptstyle{\eta}$$\scriptstyle{\tau}$

Note that $\tilde{Z}$ is Kähler because it is a resolution of subvariety of
the geometric line bundle of ${\mathcal{L}}$, which is Kähler, although the
induced Kähler class does not relate well with $\omega$ on $X$. The pullback
$\eta^{*}\omega$ is only positive over $\tilde{U}=\eta^{-1}(U)$, but one can
still cook up a Kähler class by adding a small multiple of the first Chern
class $\Theta\in H^{2}(\tilde{Z},{\mathbb{Z}}(1))$ of the relative ample line
bundle of the projective morphism $\pi:\tilde{Z}\rightarrow Z^{\prime}$. We
can assume $\Theta$ is invariant under $\sigma$ by averaging it.

######  Lemma 8.1.

Notations as above, the cohomology class
$[\eta^{*}\omega]+\lambda(2\pi\sqrt{-1})^{-1}\Theta\in H^{1,1}(Z)\cap
H^{2}(Z,{\mathbb{R}})$ is an invariant Kähler class for $\lambda$ is a
sufficient small positive number.

###### Proof.

Let $\tilde{D}_{i}$ be the strict transformation of $\tau^{-1}(D_{i})$ and
$s_{i}\in H^{0}(\tilde{Z},{\mathscr{O}}_{\tilde{Z}}(\tilde{D}_{i}))$ whose
zero locus is $\tilde{D}_{i}$. Let $h_{i}$ be a Hermitian metric on each line
bundle ${\mathscr{O}}_{\tilde{Z}}(\tilde{D}_{i})$ and $\rho_{i}$ be sufficient
small positive bump function supported in a small neighborhood of
$\tilde{D}_{i}$ for each $i$. Then the $(1,1)$-form

$\eta^{*}\omega+\sum_{i}\frac{\sqrt{-1}}{2\pi}\partial\bar{\partial}\rho_{i}h_{i}(s_{i},s_{i})$

is positive on $\tilde{Z}-E$ but only semi-positive over $E$. However, the
class $(2\pi\sqrt{-1})^{-1}\Theta$ is positive over $E$. Therefore, for
$\lambda$ sufficient small positive, the class of

$\eta^{*}\omega+\sum_{i}\frac{\sqrt{-1}}{2\pi}\partial\bar{\partial}\rho_{i}h_{i}(s_{i},s_{i})+\lambda(2\pi\sqrt{-1})^{-1}\Theta$

is a Kähler class. But $\partial\bar{\partial}\rho_{i}h_{i}(s_{i},s_{i})$ is
exact. The cohomology class of above just equals
$[\eta^{*}\omega]+\lambda(2\pi\sqrt{-1})^{-1}\Theta$ in
$H^{1,1}(\tilde{Z})\cap H^{2}(Z,{\mathbb{R}})$. It is invariant because both
$[\eta^{*}\omega]$ and $\Theta$ are invariant. ∎

######  Lemma 8.2.

The hypercohomology $H^{k}\left(Z,\Omega^{\bullet}_{Z}(\log
D^{(j)})\otimes{{\mathcal{L}}^{(j)}}^{-1}\right)$ is a summand of
$\xi^{-j}$-eigenspace of $H^{k}(\tilde{Z})$, and thus it is a sub-Hodge
structure of weight $k$.

###### Proof.

It follows from $(1.6)$ in [EV86] that $R\iota_{!}\mathbb{V}^{-j}$,
$R\iota_{*}\mathbb{V}^{-j}$ and $\Omega^{\bullet}_{Z}(\log
D^{(j)})\otimes{{\mathcal{L}}^{(j)}}^{-1}$ are all quasi-isomorphic. Taking
hypercohomology gives canonical isomorphisms

$H^{k}(Z,\Omega^{\bullet}_{Z}(\log
D^{(j)})\otimes{{\mathcal{L}}^{(j)}}^{-1})\simeq
H^{k}_{c}(U,\mathbb{V}^{-j})\simeq H^{k}(U,\mathbb{V}^{-j}).$

Because $\eta$ is étale over $U$, $H^{k}(U,\mathbb{V}^{j})$ (resp.
$H^{k}_{c}(U,\mathbb{V}^{j})$) is a $\xi^{j}$-eigenspace of
$H^{k}(\tilde{U},{\mathbb{C}})$ (resp. $H^{k}_{c}(\tilde{U},{\mathbb{C}})$)
for the cyclic action $\sigma$, where $\xi$ is a $N$-th root of unity. Then
the canonical morphisms of mixed Hodge structures

(8.41) $H^{k}_{c}(\tilde{U})\rightarrow H^{k}(\tilde{Z})\rightarrow
H^{k}(\tilde{U})$

respect the eigenspaces decomposition because we make $\tilde{Z}$ equivariant.
We complete the proof. ∎

######  Lemma 8.3.

Let ${\mathsf{X}}={2\pi\sqrt{-1}}L$ where $L=[\omega]\wedge$ is the Lefschetz
operator on $Z$. The following two statements hold:

  1. (1)

Hard Lefschetz is valid on the hypercohomolgy, i.e.

${\mathsf{X}}^{k}:H^{\dim Z-k}\left(Z,\Omega^{\bullet}_{Z}(\log
D^{(j)})\otimes{{\mathcal{L}}^{(j)}}^{-1}\right)\rightarrow H^{\dim
Z+k}\left(Z,\Omega^{\bullet}_{Z}(\log
D^{(j)})\otimes{{\mathcal{L}}^{(j)}}^{-1}\right)(k)$

is an isomorphism of Hodge structures.

  2. (2)

The pairing

(8.42) $(m^{\prime},m^{\prime\prime})\mapsto\frac{\varepsilon(\dim
Z+k+1)}{(2\pi\sqrt{-1})^{\dim
Z}}\int_{\tilde{Z}}\eta^{*}\left({\mathsf{X}}^{\dim
Z-k}m^{\prime}\wedge\overline{m^{\prime\prime}}\right)$

is a polarization on the primitive part of
$H^{k}\left(Z,\Omega^{\bullet}_{Z}(\log
D^{(j)})\otimes{{\mathcal{L}}^{(j)}}^{-1}\right)$, where
$\eta^{*}\left({\mathsf{X}}^{\dim Z-k}\alpha\wedge\overline{\beta}\right)$ is
the top form determined by the inclusion $\eta^{*}\Omega^{\dim Z}_{Z}(\log
D^{(j)})\otimes{{\mathcal{L}}^{(j)}}^{-1}\subset\omega_{\tilde{Z}}$.

###### Proof.

Let $\tilde{L}=[\eta^{*}\omega+\lambda\Theta]\wedge$ be the Lefschetz operator
on $\tilde{Z}$. Then the Hard Lefschetz on $\tilde{Z}$ says

$\tilde{\mathsf{X}}^{k}:H^{\dim Z-k}(\tilde{Z})\rightarrow H^{\dim
Z+k}(\tilde{Z})(k)$

is an isomorphism, where
$\tilde{\mathsf{X}}=_{\mathrm{def}}2\pi\sqrt{-1}\tilde{L}$. Because
$\tilde{L}$ is invariant and respects the morphisms in (8.41), the above
isomorphism is compatible with eigenspaces decomposition, it follows that

(8.43) ${\tilde{\mathsf{X}}}^{k}:H^{\dim Z-k}\left(Z,\Omega^{\bullet}_{Z}(\log
D^{(j)})\otimes{{\mathcal{L}}^{(j)}}^{-1}\right)\rightarrow H^{\dim
Z+k}\left(Z,\Omega^{\bullet}_{Z}(\log
D^{(j)})\otimes{{\mathcal{L}}^{(j)}}^{-1}\right)\left(k\right)$

is injective by Lemma 8.2. In fact, the $\xi^{i}$-eigenspace of
$H^{k}_{c}(\tilde{U})$ is orthogonal to the $\xi^{j}$-eigenspace of $H^{2\dim
Z-k}(\tilde{U})$ with respect to Poincaré pairing unless $i+j\equiv
0\,(\mathrm{mod}\,N)$: for $a$ in the $\xi^{i}$-eigenspace of
$H^{k}_{c}(\tilde{U})$ and $b$ in the $\xi^{j}$-eigenspace of $H^{2\dim
Z-k}(\tilde{U})$ then

$\xi^{i}\int_{\tilde{U}}a\wedge b=\int_{\tilde{U}}\sigma^{*}a\wedge
b=\int_{\tilde{U}}a\wedge(\sigma^{-1})^{*}b=\xi^{-j}\int_{\tilde{U}}a\wedge
b,$

which means $\int_{\tilde{U}}a\wedge b$ is zero unless $i+j\equiv
0\,(\mathrm{mod}\,N)$. It follows from Poincaré duality on
$H^{k}_{c}(\tilde{U})\times H^{2\dim Z-k}(\tilde{U})$ that the
$\xi^{i}$-eigenspace of $H^{k}_{c}(\tilde{U})$ is Poincaré dual to the
$\xi^{-i}$-eigenspace of $H^{2\dim Z-k}(\tilde{U})$. On the other hand, since
the $\xi^{i}$-eigenspace is complex conjugate to the $\xi^{-i}$-eigenspace,
the $\xi^{i}$-eigenspace of $H^{k}_{c}(\tilde{U})$ and the
$\xi^{i}$-eigenspace of $H^{2\dim Z-k}(\tilde{U})$ have the same dimension. It
follows that the morphism (8.43) is an isomorphism.

The operator $\tilde{L}$ has the same effect as $\eta^{*}L$ over
$H^{\bullet}_{c}(\tilde{U})$, because $\Theta$ is supported on $E$. Therefore,

${\mathsf{X}}^{k}:H^{\dim Z-k}\left(Z,\Omega^{\bullet}_{Z}(\log
D^{(j)})\otimes{{\mathcal{L}}^{(j)}}^{-1}\right)\rightarrow H^{\dim
Z+k}\left(Z,\Omega^{\bullet}_{Z}(\log
D^{(j)})\otimes{{\mathcal{L}}^{(j)}}^{-1}\right)(k)$

is an isomorphism. We conclude $(1)$. It also follows that $\eta^{*}$
identifies the primitive part of ${\mathsf{X}}$

$H^{\dim
Z-k}_{{\mathrm{p}}{\mathrm{r}}{\mathrm{i}}{\mathrm{m}\,}}\left(Z,\Omega^{\bullet}_{Z}(\log
D^{(j)})\otimes{{\mathcal{L}}^{(j)}}^{-1}\right)$

with the primitive part of $\tilde{\mathsf{X}}$

$\ker\left({\tilde{\mathsf{X}}}^{k+1}:H^{\dim
Z-k}\left(Z,\Omega^{\bullet}_{Z}(\log
D^{(j)})\otimes{{\mathcal{L}}^{(j)}}^{-1}\right)\rightarrow H^{\dim
Z+k+2}\left(Z,\Omega^{\bullet}_{Z}(\log
D^{(j)})\otimes{{\mathcal{L}}^{(j)}}^{-1}\right)\right).$

Thus, $H^{\dim
Z-k}_{{\mathrm{p}}{\mathrm{r}}{\mathrm{i}}{\mathrm{m}\,}}\left(Z,\Omega^{\bullet}_{Z}(\log
D^{(j)})\otimes{{\mathcal{L}}^{(j)}}^{-1}\right)$ is a sub-Hodge structure of
$H^{\dim
Z-k}_{{\mathrm{p}}{\mathrm{r}}{\mathrm{i}}{\mathrm{m}\,}}(\tilde{Z})$. And the
restriction of the polarization is again a polarization. This proves $(2)$. ∎

The above two lemmas indicate that the sum of hypercohomologies

$\bigoplus_{k\in{\mathbb{Z}}}H^{k}\left(Z,\Omega^{\bullet}_{Z}(\log
D^{(j)})\otimes{{\mathcal{L}}^{(j)}}^{-1}\right)$

is a polarized sub-Hodge-Lefschetz structure of
$\bigoplus_{k\in{\mathbb{Z}}}H^{k}(\tilde{Z},{\mathbb{C}})$. In practice, it
is more convenient to make the polarization independent of the resolution of
singularities and intrinsic on $Z$. Heuristically, the local system
$\mathbb{V}^{-j}$ over $U$ inherits a pairing from ${\mathbb{C}}_{\tilde{U}}$
and it has a Hodge theoretic extension on its canonical extension. First, we
can resolve $\Omega^{\bullet}_{Z}(\log D^{(j)})$ using
${\mathcal{A}}^{\bullet}_{Z}(\log D^{(j)})$, the complex of
$\mathscr{C}^{\infty}$-forms with log poles along $D^{(j)}$. Note that we have
the inclusion of sheaves

${\mathcal{A}}^{\dim Z+k}_{Z}(\log
D^{(j)})\otimes{{\mathcal{L}}^{(j)}}^{-1}\wedge\overline{{\mathcal{A}}^{\dim
Z-k}_{Z}(\log
D^{(j)})\otimes{{\mathcal{L}}^{(j)}}^{-1}}\subset{\mathcal{A}}^{2\dim
Z}_{Z}\otimes{{\mathcal{L}}^{(j)}}^{-1}(D^{(j)})\mathbin{\mathop{\otimes}\displaylimits_{{\mathbb{C}}}}\overline{{{\mathcal{L}}^{(j)}}^{-1}(D^{(j)})}.$

Since ${\mathcal{L}}^{N}\simeq{\mathscr{O}}_{Z}(D)$, picking local section of
$l$ such that $l^{N}=\prod_{i}x^{-\nu_{i}}_{i}$ we can put a canonical
singular Hermitian metric on ${\mathcal{L}}$ by setting the weight function as

$|l|_{h}=\prod_{i}|x_{i}|^{{-\nu_{i}}/{N}},\quad\text{where }x_{i}\text{ is
the local defining equation of }D_{i}.$

Then the induced singular Hermitian metric on
${{\mathcal{L}}^{(j)}}^{-1}(D^{(j)})={\mathcal{L}}^{-j}(\lfloor\frac{jD}{N}\rfloor+D^{(j)})$
locally is

$\left|l^{-j}\prod_{i}x^{-\lfloor
j\mu_{i}/N\rfloor}_{i}\prod_{j\nu_{i}/N\notin{\mathbb{Z}}}x^{-1}_{i}\right|_{h}=\prod_{i}|x_{i}|^{{j\nu_{i}}/{N}-\lfloor{j\nu_{i}}/{N}\rfloor}\prod_{j\nu_{i}/N\notin{\mathbb{Z}}}|x_{i}|^{-1}=\prod_{i}|x_{i}|^{-\\{-j\nu_{i}/N\\}}.$

For any smooth top form $\Upsilon$ with values in
${{\mathcal{L}}^{(j)}}^{-1}(D^{(j)})\otimes_{\mathbb{C}}\otimes\overline{{{\mathcal{L}}^{(j)}}^{-1}(D^{(j)})}$
we can associate an integrable top form
$(\Upsilon)_{h}=f\overline{g}|s|^{2}_{h}\,\mathrm{vol}(Z)$ by fixing a volume
form $\mathrm{vol}(Z)$ on $Z$ and writing locally
$\Upsilon=fs\otimes\overline{gs}\,\mathrm{vol}(Z)$ for $s$ a local fram of
${{\mathcal{L}}^{(j)}}^{-1}(D^{(j)})$. Therefore, we obtain a well-defined
pairing,

(8.44) ${\mathcal{A}}^{k}_{Z}(\log
D^{(j)})\otimes{{\mathcal{L}}^{(j)}}^{-1}\wedge\overline{{\mathcal{A}}^{k}_{Z}(\log
D^{(j)})\otimes{{\mathcal{L}}^{(j)}}^{-1}}\rightarrow{\mathbb{C}},\quad(m^{\prime},m^{\prime\prime})\mapsto\frac{\varepsilon(\dim
Z+k+1)}{(2\pi\sqrt{-1})^{\dim Z}}\int_{Z}\left({\mathsf{X}}^{\dim
Z-k}m^{\prime}\wedge\overline{m^{\prime\prime}}\right)_{h}.$

Since $\eta:\tilde{Z}\rightarrow Z$ is generic finite, it follows from

$\int_{\tilde{Z}}\eta^{*}\left({\mathsf{X}}^{\dim
Z-k}m^{\prime}\wedge\overline{m^{\prime\prime}}\right)=N\int_{Z}\left({\mathsf{X}}^{\dim
Z-k}m^{\prime}\wedge\overline{m^{\prime\prime}}\right)_{h}$

that (8.44) induces the same polarization in the statement $(2)$ of the above
lemma except for the constant $N$.

Applying to our situation yields that ${\mathscr{V}}_{\alpha,J}(E^{\alpha,J})$
carries a canonical singular Hermitian metric $|-|_{h}$ with local weight
functions $\prod_{j\in I\setminus I_{\alpha}}|z_{j}|^{-\\{\alpha e_{j}\\}}$
restricted on $Y^{J}$, where $z_{i}$ is the defining equation of $Y_{i}$.
Provided the above two lemmas, the sum of hypercohomologies

$\bigoplus_{k\in{\mathbb{Z}}}H^{k}\left(Y^{J},\Omega_{Y^{J}}^{\bullet+\dim
Y^{J}}(\log E^{\alpha,J})\otimes{\mathscr{V}}_{\alpha,J}\right)$

is a polarized Hodge-Lefschetz structure of central weight $\dim Y^{J}$ for
any non-empty subset $J$ of $I_{\alpha}$. Similarly to Example 2.9 this is
also determined by the filtered ${\mathscr{D}}_{Y^{J}}$-module
$({\mathcal{V}}_{\alpha,J},F_{\bullet}{\mathcal{V}}_{\alpha,J})$ with the
sesquilinear pairing
$S_{\alpha,J}:{\mathcal{V}}_{\alpha,J}\otimes_{{\mathbb{C}}}\overline{{\mathcal{V}}_{\alpha,J}}\rightarrow\mathfrak{C}_{Y^{J}}$
is given by

(8.45) $\left(\left[s_{1}\otimes P_{1}\right],\left[s_{2}\otimes
P_{2}\right]\right)\mapsto\frac{\varepsilon(\dim
Y^{J}+1)}{(2\pi\sqrt{-1})^{\dim
Y^{J}}}\int_{Y^{J}}(P_{1}\overline{P_{2}}-)\left(s_{1}\wedge\overline{s_{2}}\right)_{h}\
$

for local sections of ${\mathcal{V}}_{\alpha,J}$ (see (8.40)) represented by
$s_{i}\otimes P_{i}$ such that $s_{i}$ local sections of

$\omega_{Y^{J}}(\log
E^{\alpha,J})\otimes{\mathscr{V}}_{\alpha,J}=\omega_{Y^{J}}\otimes{\mathscr{V}}_{\alpha,J}(E^{\alpha,J})$

and $P_{i}$ is a differential operator $i=1,2$. Here,
$(s_{1}\wedge\overline{s_{2}})_{h}$ is the top form induced by the singular
Hermitian metric on ${\mathscr{V}}_{\alpha,J}(E^{\alpha,J})$. Summarizing the
results we proved in this subsection:

######  Corollary 8.4.

With notations as above, the direct sum of all hypercohomologies of the de
Rham complex of
$({\mathcal{V}}_{\alpha,J},F_{\bullet}{\mathcal{V}}_{\alpha,J})$ underlies a
polarized Hodge-Lefschetz structure of central weight $\dim Y^{J}$ with the
Hodge filtration induced by $F_{\bullet}{\mathcal{V}}_{\alpha,J}$ and with the
polarization, on degree $k$, given by the following induced pairing scaled by
$\varepsilon(k)$,

${H^{k}(Y^{J},{\mathrm{D}}{\mathrm{R}}_{Y^{J}}{\mathcal{V}}_{\alpha,J})\otimes
H^{-k}(Y^{J},{\mathrm{D}}{\mathrm{R}}_{Y^{J}}{\mathcal{V}}_{\alpha,J})}$${H^{0}(Y^{J},{\mathrm{D}}{\mathrm{R}}_{Y^{J},\overline{Y^{J}}}{\mathcal{V}}_{\alpha,J}\otimes_{\mathbb{C}}\overline{{\mathcal{V}}_{\alpha,J}})}$${H^{0}(Y^{J},{\mathrm{D}}{\mathrm{R}}_{Y^{J},\overline{Y^{J}}}\mathfrak{C}_{Y^{J}})\simeq{\mathbb{C}}.}$$\scriptstyle{S_{\alpha,J}}$

######  Remark 8.5.

We cannot make the Hodge structure in the above corollary over ${\mathbb{Q}}$
because there is no eigenvalue decomposition of ${\mathbb{Q}}$-structure.

###  8.2. Sesquilinear pairing

As in the reduced case, we need a sesquilinear pairing to construct the
limiting mixed Hodge structure. In fact, the construction for the reduced case
still works with a little modification. Note that for any test function $\eta$
over a local chart $U$ and two local sections $\zeta_{1}\otimes
P_{1},\zeta_{2}\otimes P_{2}$ of $H^{0}\left(U,\Omega^{n}_{X/\Delta}(\log
Y)(-\lceil\alpha Y\rceil)\otimes{\mathscr{D}}_{X}\right)$, the function

$t\mapsto\frac{\varepsilon(n+1)}{(2\pi\sqrt{-1})^{n}}\int_{X_{t}}P_{1}\overline{P_{2}}(\eta)\zeta_{1}\wedge\overline{\zeta_{2}}.$

may have order approximately at most
${|t|^{2\alpha}\left(-\log|t^{2}|\right)^{k}}$ near $t=0$ where $k+1$ is the
number of components of $Y_{I_{\alpha}}$ that intersect in $U$. This suggests
that we can define the pairing $S_{\alpha}$ on ${\mathcal{M}}_{\alpha}$ by

$\displaystyle\langle S_{\alpha}([\zeta_{1}\otimes P_{1}],[\zeta_{2}\otimes
P_{2}]),\eta\rangle$
$\displaystyle=_{\mathrm{def}}{\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{s=-\alpha}\frac{\varepsilon(n+2)}{(2\pi\sqrt{-1})^{n+1}}\int_{X}|t|^{2s}P_{1}\overline{P_{2}}(\eta)\frac{dt}{t}\wedge\zeta_{1}\wedge\overline{\frac{dt}{t}\wedge\zeta_{2}}$
$\displaystyle={\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{s=-\alpha}\frac{\varepsilon(2)}{2\pi\sqrt{-1}}\int_{\Delta}|t|^{2s}\frac{dt}{t}\wedge\overline{\frac{dt}{t}}\left(\frac{\varepsilon(n+1)}{(2\pi\sqrt{-1})^{n}}\int_{X_{t}}P_{1}\overline{P_{2}}(\eta)\zeta_{1}\wedge\overline{\zeta_{2}}\right).$

Again, we have not check that $S_{\alpha}$ is well-defined but let us do some
local calculations to see what is going on.

######  Example 8.6.

Suppose $Y=2Y_{0}$ for $Y_{0}$ is a smooth manifold and $t$ is equal to
$z^{2}_{0}$ on $X$. Then $R$ satisfies the equation $R(R-\frac{1}{2})=0$. We
deduce that ${\mathcal{M}}$ has two eigenspaces ${\mathcal{M}}_{0}$ and
${\mathcal{M}}_{\frac{1}{2}}$ by (4.19). Then for any local sections
$\zeta_{i}\otimes P_{i}=dz_{1}\wedge dz_{2}\wedge\cdots\wedge dz_{n}\otimes
P_{i}$ of $\Omega^{n}_{X/\Delta}(\log Y)\otimes{\mathscr{D}}_{X}$, $i=1,2$
representing classes of ${\mathcal{M}}_{0}$, the calculation of the pairing
$S_{0}([\zeta_{1}\otimes P_{1}],[\zeta_{2}\otimes P_{2}])$ is exactly as in
the reduced case and as it turned out

$S_{0}([\zeta_{1}\otimes P_{1}],[\zeta_{2}\otimes
P_{2}])={i_{Y_{0}}}_{+}S_{Y_{0}}\left([\zeta_{1}\otimes
P_{1}],[\zeta_{2}\otimes P_{2}]\right).$

By Theorem 7.3 ${\mathcal{M}}_{\frac{1}{2}}$ is locally generated by the class
represented by $dz_{1}\wedge dz_{2}\wedge\cdots\wedge dz_{n}\otimes z_{0}$.
Now for any local sections $\zeta\otimes z_{0}P_{i}=dz_{1}\wedge
dz_{2}\wedge\cdots\wedge dz_{n}\otimes z_{0}P_{i}$ representing classes of
${\mathcal{M}}_{\frac{1}{2}}$, we have

$\displaystyle\langle S_{\frac{1}{2}}([\zeta\otimes z_{0}P_{1}],[\zeta\otimes
z_{0}P_{2}]),\eta\rangle$
$\displaystyle={\mathrm{R}}{\mathrm{e}}{\mathrm{s}}_{s=-\frac{1}{2}}\int_{X}|z_{0}|^{4s}P_{1}\overline{P_{2}}(\eta)\bigwedge^{n}_{i=0}\left(\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge
d\overline{z_{i}}\right)$
$\displaystyle=\int_{X}\frac{1}{2}\log|z_{0}|^{2}\partial_{0}\overline{\partial_{0}}P_{1}\overline{P_{2}}(\eta)\bigwedge^{n}_{i=0}\left(\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge
d\overline{z_{i}}\right)$ by Poincaré-Lelong equation [GH14, Page 388]
$\displaystyle=\int_{Y_{0}}\frac{1}{2}P_{1}\overline{P_{2}}(\eta)\bigwedge^{n}_{i=1}\left(\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge
d\overline{z_{i}}\right)$
$\displaystyle=\frac{1}{2}\langle{i_{Y_{0}}}_{+}S_{Y_{0}}(\left[\zeta_{1}\otimes
P_{1}\right],\left[\zeta_{2}\otimes P_{2}\right]),\eta\rangle$
$\displaystyle=\frac{1}{2}\langle{i_{Y_{0}}}_{+}S_{\frac{1}{2},\\{0\\}}(\left[\zeta_{1}\otimes
z_{0}P_{1}\right],\left[\zeta_{2}\otimes z_{0}P_{2}\right]),\eta\rangle,$

Recall $S_{\frac{1}{2},\\{0\\}}$ defined in (8.45): since we have the
isomorphism
${\mathscr{O}}_{Y_{0}}(2Y_{0})={\mathscr{O}}_{Y_{0}}(Y)\simeq{\mathscr{O}}_{Y_{0}}$
there exists a canonical singular Hermitian metric (this case is smooth)
$|-|_{h}$ on ${\mathscr{O}}_{Y_{0}}(-Y_{0})$ by setting the local frame
${z_{0}}$ has norm $1$ so that

${i_{Y_{0}}}_{+}S_{\frac{1}{2},\\{0\\}}(\left[\zeta_{1}\otimes
z_{0}P_{1}\right],\left[\zeta_{2}\otimes
z_{0}P_{2}\right]),\eta\rangle=\int_{X}|z_{0}|^{2}_{h}P_{1}\overline{P_{2}}(\eta)\bigwedge^{n}_{i=0}\left(\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge
d\overline{z_{i}}\right)={i_{Y_{0}}}_{+}S_{Y_{0}}(\left[\zeta_{1}\otimes
P_{1}\right],\left[\zeta_{2}\otimes P_{2}\right]),\eta\rangle.$

The above equality can also be explained as follows: the cyclic covering
constructed by taking out of the second root of the constant section of
${\mathscr{O}}_{Y_{0}}(2Y_{0})\simeq{\mathscr{O}}_{Y_{0}}$ has two connected
components and each component is isomorphic to $Y_{0}$.

Let $\eta$ be a test function over an open subset $U$. For any two sections
$m_{1},m_{2}\in H^{0}(U,\Omega^{n}_{X/\Delta}(\log Y)(-\lceil\alpha
Y\rceil)\otimes{\mathscr{D}}_{X})$, the $(2n+2)$-form $\frac{dt}{t}\wedge
m_{1}\wedge\overline{\frac{dt}{t}\wedge m_{2}}$ is smooth of outside $Y$ and
has pole along $Y$. Locally, the $(2n+2)$-form just is
$|z_{I}|^{2\lceil\alpha\mathbf{e}\rceil}P_{1}\overline{P_{2}}(\eta)\frac{dt}{t}\wedge\zeta\wedge\overline{\frac{dt}{t}\wedge\zeta}$,
where $m_{j}=\zeta\otimes z^{\lceil\alpha\mathbf{e}\rceil}_{I}P_{j}$ for
$\zeta=\frac{dz_{1}}{z_{1}}\wedge\frac{dz_{2}}{z_{2}}\wedge\cdots\wedge\frac{dz_{k}}{z_{k}}\wedge\cdots\wedge
dz_{n}$ and $j=1,2$. Let $F(s)=F(s,m_{1},m_{2},\eta)$ be the meromorphic
extension of

$\frac{\varepsilon(n+2)}{(2\pi\sqrt{-1})^{n+1}}\int_{X}|t|^{2s}\frac{dt}{t}\wedge
m_{1}\wedge\overline{\frac{dt}{t}\wedge m_{2}}(\eta)$

via integration by parts. The function $F(s)$ is well defined when
$\mathrm{Re\,}s>-\alpha$ and has a pole at $s=-\alpha$. We only care about the
polar part of $F(s)$ at $s=-\alpha$.

######  Theorem 8.7.

The polar part of $F(s)$ at $s=-\alpha$ is only depends on the classes of
$m_{1}$ and $m_{2}$ in ${\mathcal{M}}_{\alpha}$.

###### Proof.

Let $\\{\rho_{\lambda}\\}$ be a partition of unity of the open covering
$\\{U_{\lambda}\\}$ by local charts. Then

$F(s)=\sum_{\lambda}\frac{\varepsilon(n+2)}{\left(2\pi\sqrt{-1}\right)^{n+1}}\int_{U_{\lambda}}|t|^{2s}\frac{dt}{t}\wedge
m_{1}\wedge\overline{\frac{dt}{t}\wedge m_{2}}(\rho_{\lambda}\eta).$

Since $\rho_{\lambda}\eta$ is a test function over local chart $U_{\lambda}$,
we can assume $U$ itself is a local chart. We assume $k+1$ components of $Y$
intersect in $U$.

######  Lemma 8.8.

Under the assumption that $m_{i}=\zeta_{\alpha}\otimes P_{i}$ for
$\zeta_{\alpha}=z^{\lceil\alpha\mathbf{e}\rceil}_{I}\frac{dz_{1}}{z_{1}}\wedge\frac{dz_{2}}{z_{2}}\wedge\cdots\wedge\frac{dz_{k}}{z_{k}}\wedge
dz_{k+1}\wedge\cdots\wedge dz_{n}$ and for $i=1,2$, the followings are valid.

  1. (1)

the order of the pole of $F(s)$ at $s=-\alpha$ is at most $k+1$;

  2. (2)

if $P_{i}=t_{\alpha}P^{\prime}_{i}$ for one of $i=1,2$, then $F(s)$ is
holomorphic at $s=-\alpha$;

  3. (3)

for $0\leq j\leq k$ we have,

$\displaystyle F\left(s,\zeta_{\alpha}\otimes
P_{1},\zeta_{\alpha}\otimes\frac{1}{e_{j}}z_{j}\partial_{j}P_{2},\eta\right)=$
$\displaystyle
F\left(s,\zeta_{\alpha}\otimes\frac{1}{e_{j}}z_{j}\partial_{j}P_{1},\zeta_{\alpha}\otimes
P_{2},\eta\right)=$ $\displaystyle-\left(s+\frac{\lceil\alpha
e_{j}\rceil}{e_{j}}\right)F(s,\zeta_{1}\otimes P_{1},\zeta_{2}\otimes
P_{2},\eta).$

###### Proof of the lemma.

We work out Laurent series of $F(s)$ at $s=-\alpha$:

$\displaystyle F(s)$
$\displaystyle=\int_{X}|z_{I}|^{2s\mathbf{e}+2\lceil\alpha\mathbf{e}\rceil-2\cdot\mathbf{1}}P_{1}\overline{P_{2}}(\eta)\bigwedge^{n}_{i=0}(\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge
d\overline{z_{i}})$
$\displaystyle=\int_{X}|z_{I}|^{2(s+\alpha)\mathbf{e}-2\cdot\mathbf{1}}|z_{I}|^{2\\{-\alpha\mathbf{e}\\}}P_{1}\overline{P_{2}}(\eta)\bigwedge^{n}_{i=0}(\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge
d\overline{z_{i}})$
$\displaystyle=\int_{X}(s+\alpha)^{-2(k+1)}|z_{I}|^{2(s+\alpha)\mathbf{e}}\eta^{\prime}\bigwedge^{n}_{i=0}(\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge
d\overline{z_{i}})\quad\text{where
}\eta^{\prime}=\partial_{I}\overline{\partial_{I}}\left(|z_{I}|^{2\\{-\alpha\mathbf{e}\\}}P_{1}\overline{P_{2}}\eta\right)$
$\displaystyle=\sum^{\infty}_{\ell=0}\frac{1}{\ell!}(s+\alpha)^{\ell-2(k+1)}\int_{X}(\log|z_{I}|^{2\mathbf{e}})^{\ell}\eta^{\prime}\bigwedge^{n}_{i=0}(\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge
d\overline{z_{i}}).$

When $\ell<k+1$, then the form

$(\log|z_{I}|^{2\mathbf{e}})^{\ell}\eta^{\prime}\bigwedge^{n}_{i=0}(\frac{\sqrt{-1}}{2\pi}dz_{i}\wedge
d\overline{z_{i}}).$

is actually exact because one of the $a_{i}$ must be zero in the expansion of
$(\log|z_{I}|^{2\mathbf{e}})^{\ell}$ into the linear combination of