# Vanishing Theorems for the de Rham Complex of unitary local system

Hongshan Li  Department of Mathematics
Purdue University
West Lafayette, IN 47906
U.S.A

###### Abstract.

We will prove a Kodaira-Nakano type of vanishing theorem for the logarithmic
de Rham complex of unitary local system. We will then study the weight
filtration on the logarithmic de Rham complex, and prove a stronger statement
for the associated graded complex.

## Introduction

Let $X$ be a smooth projective variety of dimension $n$ over $\mathbb{C}$ and
$D\subset X$ a simple normal crossing divisor. In [2], Deligne constructed the
canonical extension $E$ for any local system $\mathcal{L}$ defined on $U$
(over complex topology). $E$ is equipped with a flat connection,

$\nabla:E\rightarrow E\otimes\Omega_{X}(\log D)$

and it is characterized by the following two properties

  1. (1)

The flat sections of $\nabla$ coincide with $\mathcal{L}$ on $U$.

  2. (2)

The eigenvalues of the residue of $\nabla$ lie in the strip

$\\{z\in\mathbb{C}|0\leq\text{Re}(z)<1\\}$

Let $V$ be a unitary local system on
$U{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}:=}X-D$
(over complex topology). Let $(E,\nabla)$ be the canonical extension of $V$.
Write $\text{DR}_{X}(D,E)$ for the de Rham complex

$E\xrightarrow{\nabla}E\otimes\Omega_{X}(\log
D)\rightarrow\cdots\xrightarrow{\nabla}E\otimes\Omega^{n}_{X}(\log D)$

First, we will prove a Kodaira-Nakano type of vanishing theorem

######  Theorem 1.

Let $L$ be an ample line bundle on $X$, then

$H^{q}(X,E\otimes\Omega^{p}_{X}(\log D)\otimes L)=0$

for $p+q>\dim X$.

The de Rham complex $\text{DR}_{X}(D,E)$ comes with an decreasing filtration
$F^{.}$ (Hodge filtration or ”naive” filtration) and a increasing filtration
$W_{.}$ (weight filtration). The weight filtration $W_{.}$ will be defined in
Section 2. $F^{.}$ and $W_{.}$ togethe will define a mixed Hodge structure on
$\text{DR}_{X}(D,E)$.

Then, we will prove a more refined version of the above theorem

######  Theorem 2.

Let $L$ be an ample line bundle on $X$, then

$H^{q}(X,\text{Gr}^{W}_{.}(\Omega_{X}^{p}\otimes E)\otimes L)=0$

Acknowledgement The author is thankful for D. Arapura for explaining mixed
Hodge theory and many key suggestions.

##  1\. Residue map

In this section we will define a residue map $\text{Res}(E)$ on the complex
$\text{DR}_{X}(D,E)$. Similar to the usual residue map on the holomorphic de
Rham complex, $\Omega^{\boldsymbol{\cdot}}_{X}$, $\text{Res}(E)$ will define a
weight filtration on $\text{DR}_{X}(D,E)$. $\text{Res}(E)$ has been defined
and studied in [8].

For $m=1,\cdots n$, let $D_{m}$ be the union of $m$-fold intersection of
components of $D$; Let $\tilde{D}_{m}$ be the disjoint union of components of
$D_{m}$; Let $v_{m}:\tilde{D}_{m}\rightarrow X$ be the composition of the
projection map onto $D_{m}$ and the inclusion map.
$\tilde{C}_{m}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}:=}{v}^{*}_{m}D_{m+1}$
is either empty or a normal

######  Theorem 3.

[8]

  1. (1)

$V_{m}:={j}_{*}V|_{D_{m}-D_{m+1}}$ is a unitary local system on
$D_{m}-D_{m+1}$.

  2. (2)

There exist a unique subvectorbundle $E_{m}$ of ${v}^{*}_{m}E$ and a unique
holomorphic integrable connection $\nabla_{m}$ on $E_{m}$ with logarithmic
poles along $C_{m}$ such that

$\ker\nabla_{m}|_{\tilde{D}_{m}-\tilde{C}_{m}}=v_{m}^{-1}V_{m}$

  3. (3)

There exists a unique subvectorbundle $E_{m}^{*}$ of ${v}^{*}_{m}E$ with

$E_{m}\oplus E_{m}^{*}={v}^{*}_{m}E$

###### Proof.

All of the statements above are local. Therefore, we can assume $X$ is a
polydisk. Write $X=\Delta_{1}\times\cdots\times\Delta_{n}$, and let $z_{i}$ be
the coordinate on $\Delta_{i}$. Suppose $D$ is defined by

$z_{1}\times\cdots\times z_{s}=0$

1\. The local system $V$ on $U$ is equivalent to an unitary representation

$T:\pi_{1}(U)\rightarrow\text{GL}(r,\mathbb{C})$

As $\pi_{1}(U)$ is abelian and $T$ is unitary, we can simultaneously
diagonalize all $T(\gamma_{i}$, where $\gamma_{i}$’s form a generating set of
$\pi_{1}(U)$ (see Appendix 1). Therefore, we can assume $V$ is a direct sum of
rank 1 unitary local systems. Write

$V=V^{1}\oplus\cdots\oplus V^{r}$

For each $V^{i}$, let $\lambda_{i,j}$ be its monodromy around $D^{j}$. So
$V^{i}$ extends to $D^{j}$ if and only if $\lambda_{i,j}=1$.

Now let $D^{j1}\cap\cdots\cap D^{jm}$ be one component of $D_{m}$, and let
$x\in D^{j1}\cap\cdots\cap D^{jm}$. Then, near $x$ $V_{m}$ is

$\bigoplus\limits_{\lambda_{i,j1}=\cdots=\lambda_{i,jm}=1}V^{i}$

This shows that $V_{m}$ is a unitary local system.

2\. The uniqueness of the subvectorbundle $E_{m}$ follows from the uniqueness
of canonical connection. Therefore, we only need to show the existence part.
Use the notation from part 1, and assume $V$ decomposes as direct sum of rank
1 unitary local system $V^{i}$. Let $E^{i}$ be the canonical connection of
$V^{i}$. Then, it is clear that

$E_{m}=\bigoplus\limits_{\lambda_{i,j1}=\cdots=\lambda_{i,jm}=1}{v}^{*}_{m}E^{i}$

3\. $E$ inheits a flat Hermitian form from $V$. Define $E_{m}^{*}$ as the
complement of $E_{m}$ with respect to this metric. On $\Delta$, $E_{m}^{*}$ is
the direct sum of ${v}^{*}_{m}E^{i}$ not appearing in the definition of
$E_{m}$. ∎

######  Remark 1.

$E_{m}$ could have different ranks on different component of $\tilde{D}_{m}$.

For each $m\leq p\leq\dim D_{m}$, there exists a residue map crossing divisor
in $\tilde{D}_{m}$.

$\text{Res}_{m}:\Omega_{X}^{p}(\log
D)\rightarrow{v_{m}}_{*}(\Omega_{\tilde{D}_{m}}^{p-m})$

$\text{Res}_{m}$ commutes with exterior derivative $d$, making it a
homomorphism of complexes

$\text{Res}_{m}:\Omega_{X}^{\boldsymbol{\cdot}}(\log
D)\rightarrow{v_{m}}_{*}\Omega_{\tilde{D}_{m}}^{\boldsymbol{\cdot}}(\log\tilde{C}_{m})[-m]$

Consider the following variation of the residue map $\text{Res}_{m}$

$\displaystyle\text{Res}_{m}(E):\Omega_{X}^{p}(\log D)\otimes E$
$\displaystyle\xrightarrow{\text{Res}_{m}\otimes\text{id}}{v_{m}}_{*}(\Omega_{\tilde{D}_{m}}^{p-m}(\log\tilde{C}_{m}))\otimes
E$
$\displaystyle={v_{m}}_{*}(\Omega_{\tilde{D}_{m}}^{p-m}(\log\tilde{C}_{m})\otimes{v}^{*}_{m}E)$
$\displaystyle\xrightarrow{\text{id}\otimes
p_{m}}{v_{m}}_{*}(\Omega_{\tilde{D}_{m}}^{p-m}(\log\tilde{C}_{m})\otimes
E_{m})$

where $p_{m}:{v}^{*}_{m}E\rightarrow\tilde{E}_{m}$ is the projection onto the
$E_{m}$ component.

######  Lemma 1.

[8] $\text{Res}_{m}(E)\circ\nabla=\nabla_{m}\circ\text{Res}_{m}(E)$, _i.e._
$\text{Res}_{m}(E)$ is homomorphism of complexs

$\text{DR}_{X}(D,E)\rightarrow{v_{m}}_{*}\text{DR}_{\tilde{D}_{m}}(\tilde{C}_{m},E_{m})[-m]$

##  2\. Weight filtration on the de Rham Complex

The residue map

$\text{Res}_{m}(E):\text{DR}_{X}(D,E)\rightarrow{v_{m}}_{*}\text{DR}(\tilde{C}_{m},E_{m},\nabla_{m})[-m]$

can be used to define a weight filtration $W_{.}$ on $\text{DR}_{X}(D,E)$ [8]

$\displaystyle W_{m}(\text{DR}_{X}(D,E))=\ker\text{Res}_{m+1}(E)$
$\displaystyle\text{ if }m\geq 0$ $\displaystyle W_{m}(\text{DR}_{X}(D,E))=0$
$\displaystyle\text{ if }m<0$

Local descriptions of $W_{m}(\text{DR}_{X}(D,E))$ have been given in [8]. We
will review them here:

Let $\Delta=\Delta_{1}\times\cdots\times\Delta_{n}$ be a polydisk of $X$ with
coordinate $z_{1},\cdots,z_{n}$. Suppose $D$ is defined as

$z_{1}\times\cdots\times z_{s}=0$

As in part 1 of Theorem 3, we assume $V$ is the direct sum of rank 1 unitary
local systems on $\Delta$, and write

$V=V^{1}\oplus\cdots\oplus V^{r}$

######  Definition 1.

We say $\frac{dz_{j}}{z_{j}}$ acts on $V^{i}$ by identity if
$\lambda_{i,j}=1$, i.e. the monodromy of $V^{i}$ by a small circle around
$D_{j}$ is the identity.

Let $E^{i}$ be be canonical extension of $V^{i}$ on $\Delta$;

Let $\mu_{i}$ be a generator of $E^{i}$, then

$\frac{dz_{j_{1}}}{z_{j_{1}}}\wedge\cdots\wedge\frac{dz_{j_{k}}}{z_{j_{k}}}\wedge
dz_{j_{k+1}}\cdots\wedge dz_{j_{p}}\otimes\mu_{i}$

is in $W_{m}(\text{DR}_{X}(D,E))$ if and only if there are at most $m$ log
forms acting on $V^{i}$ by identity.

######  Proposition 1.

[8]

  1. (1)

$W_{.}(\text{DR}(D,E,\nabla))$ is an increasing filtration.

  2. (2)

$\text{Res}_{m}(E)$ induces an isomorphism

$\text{Gr}^{W}_{m}(\text{DR}(D,\nabla,E))\rightarrow{v_{m}}_{*}(W_{0}(\text{DR}_{\tilde{D}_{m}}(\tilde{C}_{m},\tilde{E}_{m}))[-m])$

###### Proof.

The statements are local. We can assume $X$ is a polydisk and $V$ is a unitary
local system of rank 1.

1\. From the local description of $W_{m}(\text{DR}_{X}(D,E))$, it is clear
that $W_{.}$ is an increasing filtration.

2\. Let $s$ be a section $W_{m}(\text{DR}_{X}(D,E))$. Use the local
description above, $s$ is of the form

$\omega\otimes\mu$

where

$\omega=\frac{dz_{j_{1}}}{z_{j_{1}}}\wedge\cdots\wedge\frac{dz_{j_{k}}}{z_{j_{k}}}\wedge
dz_{j_{k+1}}\cdots\wedge dz_{j_{p}}$

and $\omega$ has at most $m$ log 1-forms acting on $V$ by identity. $\mu$ is a
generating section of $E$.

First, we show $\text{Res}_{m}(E)(s)\in
W_{0}(\Omega_{\tilde{D}_{m}}^{p-m}(\log\tilde{C}_{m})\otimes E_{m}$.

$\text{Res}_{m}(E)(s)=\text{Res}_{m}(\omega)\otimes\mu_{m}$

By the construction of $\omega$, $\text{Res}_{m}(\omega)$ does not have log
form $\frac{dz_{j}}{z_{j}}$ acting on $V$ by identity. This shows that

$\text{Res}_{m}(E)(s)\in
W_{0}(\Omega^{p-m}_{\tilde{D}_{m}}(\log\tilde{C}_{m})\otimes E_{m})$

If $\omega_{0}\otimes\mu_{m}\in
W_{0}(\Omega^{p-m}_{\tilde{D}_{m}}(\log\tilde{C}_{m})\otimes E_{m})$, to get a
preimage in $W_{m}(\Omega^{p}_{X}(\log D)\otimes E)$, simply take

$\omega_{m}\wedge\omega_{0}\otimes\mu$

where $\omega_{m}$ is any $m$-form. And
$\omega_{m}\wedge\omega_{0}\otimes\mu\in W_{m}(\Omega_{X}^{p}(\log D)\otimes
E)$ by the construction of $\omega_{0}$. This shows that

$\text{Res}_{m}(E):W_{m}(\text{DR}(D,E,\nabla))\rightarrow
W_{0}(\text{DR}(\tilde{C}_{m},\tilde{E}_{m},\tilde{\nabla}_{m}))$

is surjective.

If $\text{Res}_{m}(E)(s)=0$, that means in $\omega$, there are at most $m-1$
log forms acting on $V$ by identity. This is precisely the local description
of $W_{m-1}(\Omega_{X}^{p}(\log D)\otimes E)$. ∎

##  3\. Mixed Hodge Structure on the de Rham Complex

The framework for studying the mixed Hodge structure on $\text{DR}_{X}(D,E)$
has been worked out by Deligne in [3] and [4]. The analysis of the mixed Hodge
structure on $\text{DR}_{X}(D,E)$ was given by Timmerscheidt in [8]. We will
give an overview about the results from both authors. The vanishing theorem in
the following section is a consequence of the mixed Hodge structure on
$\text{DR}_{X}(D,E)$.

Let $A$ denote $\mathbb{Z},\mathbb{Q}$ or and $A\otimes\mathbb{Q}$ the field
$\mathbb{Q}$ or .

Assume $V$ has a $A$-lattice throughout this section, i.e. there exists a
unitary local system $V_{A}$ defined over $A$ such that

$V=V_{A}\otimes_{A}\mathbb{C}$

Let $D^{+}(A)$(resp. $D^{+}(\mathbb{C}))$ denote the derived category of
$A$-modules (resp. $\mathbb{C}$-vector spaces)

The main result of this section is

######  Theorem 4.

[8](Proposition 6.4)

$(\mathbb{R}{j}_{*}V_{A},(\mathbb{R}{j}_{*}V_{A\otimes\mathbb{Q}},\tau),(\text{DR}_{X}(D,E),F,W))$

is an $A$-cohomological mixed Hodge complex.

For readers’ sake, we included all relvant definitions involved in the above
theorem here. They can be found in [4] or [5](Section 5)

######  Definition 2.

(Hodge Structure (HS)) A Hodge structure of weight $n$ is defined by the data:

  1. (1)

A finitely generated abelian group $H_{\mathbb{Z}}$;

  2. (2)

A decomposition by complex subspaces:

$H_{\mathbb{C}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}:=}H_{\mathbb{Z}}\otimes_{\mathbb{Z}}\mathbb{C}=\bigoplus\limits_{p+q=n}H^{p,q}$

satisfying

$H^{p,q}=\overline{H^{q,p}}$

######  Definition 3.

(Hodge Complex (HC)) A Hodge $A$-complex $K$ of weight $n$ consists of

  1. (1)

A complex $K_{A}$ of $A$-modules, such that $H^{k}(K_{A})$ is an $A$-module of
finite type for all $k$;

  2. (2)

A filtered complex $(K_{\mathbb{C}},F)$ of $\mathbb{C}$-vector spaces ;

  3. (3)

Anisomorphism

$\alpha:K_{A}\otimes\mathbb{C}\rightarrow K_{\mathbb{C}}$

in $D^{+}(\mathbb{C})$;

The following axioms must be satisfied

  1. (1)

The spectral sequence defined by $(K_{\mathbb{C}},F)$ degenerates at $E_{1}$;

  2. (2)

for all $k$, the filtration $F$ on $H^{k}(K_{\mathbb{C}})\cong
H^{k}(K_{A})\otimes\mathbb{C}$ defines a $A$-Hodge strucutre of weight $n+k$
on $H^{k}(K_{A})$

######  Definition 4.

Let $X$ be a topological space. An $A$-Cohomological Hodge Complex (CHC) $K$
of weight $n$ on $X$, consists of:

  1. (1)

A complex of sheaves $K_{A}$ of $A$-modules on $X$;

  2. (2)

A filtered complex of sheaves $(K_{\mathbb{C}},F)$ of $\mathbb{C}$-vector
spaces on $X$;

  3. (3)

an isomorphism

$\alpha:K_{A}\otimes\mathbb{C}\rightarrow K_{\mathbb{C}}$

in $D^{+}(X,\mathbb{C})$

Moreover, the triple
$(R\Gamma(K_{A}),R\Gamma(K_{\mathbb{C}},F),R\Gamma(\alpha))$ is a Hodge
Complex of weight $n$

######  Definition 5.

(Mixed Hodge Complex) An $A$-Mixed Hodge Complex (MHC) $K$ consists of:

  1. (1)

A complex $K_{A}$ of $A$-modules such that $H^{k}(K_{A})$ is an $A$-module of
finite type for all $k$;

  2. (2)

A filtered complex $(K_{A\otimes\mathbb{Q}},W)$ of $A\otimes\mathbb{Q}$\-
vector spaces with an increasing filtration $W$;

  3. (3)

An isomorphism $K_{A}\otimes\mathbb{Q}\cong K_{A\otimes\mathbb{Q}}$ in
$D^{+}(A\otimes\mathbb{Q})$;

  4. (4)

A bi-filtered complex $(K_{\mathbb{C}},W,F)$ of $\mathbb{C}$-vector spaces
with an increasing (resp. descreasing) filtration $W$ (resp. $F$) and an
isomorphism:

$\alpha:(K_{A\otimes\mathbb{Q}},W)\otimes\mathbb{C}\cong(K_{\mathbb{C}},W)$

in $D^{+}F(\mathbb{C})$.

Moreover, the following axiom needs to be satisfied: For all n, the system
consisting of

  * •

the complex $\text{Gr}^{W}_{n}(K_{A\otimes\mathbb{Q}})$ of
$A\otimes\mathbb{Q}$\- vector spaces,

  * •

the complex $\text{Gr}^{W}_{n}(K_{\mathbb{C}},F)$ of $\mathbb{C}$-vector
spaces with induced $F$ filtration,

  * •

the isomorphism

$\text{Gr}^{W}_{n}(\alpha):\text{Gr}^{W}_{n}(K_{A\otimes\mathbb{Q}})\otimes\mathbb{C}\rightarrow\text{Gr}^{W}_{n}(K_{\mathbb{C}})$

is an $A\otimes\mathbb{Q}$-Hodge Complex of weight $n$.

######  Definition 6.

(Cohomological Mixed Hodge Complex (CMHC)) An $A$-Cohomological Mixed Hodge
Complex $K$ (CMHC) on a topological space $X$ consists of:

  1. (1)

A complex of sheaves $K_{A}$ of sheaves of $A$-modules on $X$ such that
$H^{k}(X,K_{A})$ are $A$-modules of finite type;

  2. (2)

A filtered complex $(K_{A\otimes\mathbb{Q}},W)$ of sheaves of
$A\otimes\mathbb{Q}$-vector spaces on $X$ with an increasing filtration $W$
and an isomorphism

$K_{A}\otimes\mathbb{Q}\cong K_{A\otimes\mathbb{Q}}$

in $D^{+}(X,A\otimes\mathbb{Q})$;

  3. (3)

A bi-filtered complex of sheaves $(K_{\mathbb{C}},W,F)$ of $\mathbb{C}$-vector
spaces on $X$ with an increasing (resp. descreasing) filtration $W$ (resp.
$F$) and an isomorphism:

$\alpha:(K_{A\otimes\mathbb{Q}},W)\otimes\mathbb{C}\rightarrow(K_{\mathbb{C}},W)$

in $D^{+}F(X,\mathbb{C})$.

Moreover, the following axiom needs to be satisfied: For all $n$, the system
consisting of:

  * •

the complex $\text{Gr}^{W}_{n}(K_{A\otimes\mathbb{Q}})$ of sheaves of
$A\otimes\mathbb{Q}$-vector spaces on $X$,

  * •

the complex $\text{Gr}^{W}_{n}(K_{\mathbb{C}},F)$ of sheaves of
$\mathbb{C}$-vector spaces with induced $F$ filtration,

  * •

the isomorphism

$\text{Gr}^{W}_{n}(\alpha):\text{Gr}^{W}_{n}(K_{A\otimes\mathbb{Q}})\otimes\mathbb{C}\rightarrow\text{Gr}^{W}_{n}(K_{\mathbb{C}})$

is an $A\otimes\mathbb{Q}$-Cohomological Hodge Complex of weight $n$.

The following example of Cohomological Mixed Hodge Complex can be found in [4]
and [5]

######  Example 1.

Let $X$ be a smooth projective variety over $\mathbb{C}$, $D\subset X$ a
simple normal crossing divisor. Let
$U{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}:=}X-D$
and let

$j:U\rightarrow X$

be the inclusion map.

Let $\mathbb{Q}_{U}$ be the constant sheaf with $\mathbb{Q}$-coefficient on
$U$.
$\mathbb{R}{j}_{*}\mathbb{Q}_{U}\otimes\mathbb{C}=\mathbb{R}{\mathbb{C}_{U}}_{*}$
is quasi-isomorphic to the logarithmic de Rham complex

$O_{X}\xrightarrow{d}\Omega_{X}(\log
D)\xrightarrow{d}\cdots\xrightarrow{d}\Omega_{X}^{n}(\log D)$

For any complex $K^{\boldsymbol{\cdot}}$ of sheaves on $X$, let $\tau$ be the
canonical increasing filtration

$\displaystyle\tau_{m}K^{q}=\begin{cases}K^{q}&\text{ if }q<m\\\
\ker{d^{q}}\subset K^{q}&\text{ if }q=m\\\ 0&\text{ if }q>m\end{cases}$

See [5](Corollary 6.4) for the following result:

The system consisting of

  1. (1)

$(\mathbb{R}{j}_{*}\mathbb{Q}_{U},\tau)$;

  2. (2)

$(\Omega_{X}^{\boldsymbol{\cdot}}(\log D),W,F)$ with usual weight and Hodge
filtration $W$ and $F$;

  3. (3)

The quasi-isomorphism

$(\mathbb{R}{j}_{*}\mathbb{Q}_{U},\tau)\otimes\mathbb{C}\cong(\Omega_{X}(\log
D)^{\boldsymbol{\cdot}},W)$

is a Cohomological Mixed Hodge Complex on $X$.

##  4\. Vanishing Theorem for the de Rham Complex

we have seen in the previous section that if $V$ has a real lattice, then

$(\mathbb{R}{j}_{*}V_{A},(\mathbb{R}{j}_{*}V_{A\otimes\mathbb{Q}},\tau),(\text{DR}_{X}(D,E),F,W))$

is an $A$-cohomological mixed Hodge complex. As a result of the general theory
developed in [4], we have

######  Theorem 5.

Assume there is a real-valued unitary locayl system $V$ defined on $U$ such
that

$V=V\otimes\mathbb{C}$

Let $V$ and $\text{DR}_{X}(D,E)$ be as above. The spectral sequence associated
to the Hodge filtration on $\text{DR}_{X}(D,E)$.

$E^{p,q}_{1}=H^{q}(X,\Omega_{X}^{p}(\log D)\otimes
E)=>\mathbb{H}^{p+q}(X,\text{DR}_{X}(D,E))$

degenerates at $E_{1}$

If $V$ does not have an $A$-lattice with $A\subset$, then we cannot expect
$\text{DR}_{X}(D,E)$ to carry a mixed Hodge structure. However, the
degeneration of Hodge spectral sequence still holds true. One can proofs of
this in [8]. We will give a simpler proof here:

Let $\bar{V}$ denote the conjugate of $V$, i.e. the monodromy representation
of $\bar{V}$ is the complex conjugate of the monodromy representation of $V$

######  Lemma 2.

There exists a real unitary local system $W$ of rank $2r$ such that

$V\oplus\bar{V}\cong W\otimes\mathbb{C}$

###### Proof.

We will construct $W$ locally, and show it is canonically determined by $V$.
Over a polydisk, we can assume $V$ is diagonal, and we write

$V=\bigoplus_{j=1}^{r}V^{j}$

where $V^{i}$ is a unitary local system of rank 1 with monodromy

$\lambda_{j}=\cos\theta_{j}+i\sin\theta_{j}$

We will construct $W^{j}$ for each $j$. The monodromy of $\bar{V}^{j}$ is
$\bar{\lambda}_{j}$ and the monodromy of $V^{j}\oplus\bar{V}^{j}$ is

$\begin{bmatrix}\cos\theta_{j}+i\sin\theta_{j}&0\\\
0&\cos\theta_{j}-i\sin\theta_{j}\\\ \end{bmatrix}$

Since

$\begin{bmatrix}\cos\theta_{j}+i\sin\theta_{j}&0\\\
0&\cos\theta_{j}-i\sin\theta_{j}\\\ \end{bmatrix}\text{ and
}\begin{bmatrix}\cos\theta_{j}&\sin\theta_{j}\\\
-\sin\theta_{j}&\cos\theta_{j}\\\ \end{bmatrix}$

have the same characteristic polynomial over $\mathbb{C}$, they must be
conjugate over $\mathbb{C}$. Therefore, we can take $W^{j}$ to be

$\begin{bmatrix}\cos\theta_{j}&\sin\theta_{j}\\\
-\sin\theta_{j}&\cos\theta_{j}\\\ \end{bmatrix}$

Then,

$W=\bigoplus_{j=1}^{r}W^{j}$

∎

Now, let $V$ be any unitary local system on $X-D$, and let
$\text{DR}_{X}(D,E)$ be its de Rham complex

######  Corollary 1.

The Hodge spectral sequence

$E^{p,q}_{1}:=H^{q}(X,\Omega^{p}_{X}(\log D)\otimes
E)=>\mathbb{H}^{p+q}(X,\text{DR}(D,E))=H^{p+q}(X,\mathbb{R}{j}_{*}V)$

degenerates at $E_{1}$.

###### Proof.

Direct sum and taking cohomology commutes ∎

######  Theorem 6.

[1](Corollary 3.5) Suppose $U$ is an affine variety of complex dimension $n$.
Then, for any constructible sheaf $\mathcal{L}$ on $U$

$H^{k}(U,\mathcal{L})=0$

for $k>n$

######  Corollary 2.

Let $V$ and $\text{DR}_{X}(D,E)$ be as above. Suppose $U$ is affine, then

$H^{q}(X,\Omega_{X}^{p}(\log D)\otimes E)=0$

for $p+q>\dim X$

######  Lemma 3.

Suppose $B$ is a smooth divisor transversal to $D$. Then, there is short exact
sequence

$0\rightarrow\Omega_{X}^{p}(\log D+B)\otimes
O_{X}(-B)\xrightarrow{i}\Omega_{X}^{p}(\log
D)\xrightarrow{r}\Omega^{p}_{B}(\log D\cap B)\rightarrow 0$

where $i$ is the inclusion map, and $r$ is the restriction map.

###### Proof.

For simplicity, we prove the case for $p=1$. We may also assume $X$ is affine.
Let $X=\text{Spec}A$, and let $f_{1},\cdots,f_{s}$ be the regular sequence
corresponding to $D$, and let $b$ be the defining equation of $B$.

The basis of $\Omega_{X}^{1}(\log D+B)\otimes O_{X}(-B)$ as an $A$-module is

$\frac{df_{1}}{f_{1}}\otimes b,\cdots,\frac{df_{s}}{f_{s}}\otimes
b,\frac{db}{b}\otimes b$

The basis of $\Omega_{X}(\log D)$ as an $A$-module is

$\frac{df_{1}}{f_{1}},\cdots,\frac{df_{s}}{f_{s}}$

The basis of $\Omega_{B}(\log D\cap B)$ as an $\frac{A}{b}$-module is

$\frac{df_{1}}{f_{1}},\cdots,\frac{df_{s}}{f_{s}}$

where by abuse of notation $f_{i}$ are regarded as their image in
$\frac{A}{b}$.

Then, it is clear how to define $i$ and $r$ show that the above sequence is
exact ∎

######  Lemma 4.

Suppose $B$ is a smooth divisor transversal to $D$. Then, $E_{B}:=E\otimes
O_{B}$ is the canonical extension of $V_{B}:=V|_{B-B\cap D}$.

###### Proof.

The statement is local, therefore we may assume $X$ is a polydisk

$\Delta_{1}\times\cdots\times\Delta_{n}$

such that the analytic coordinate of $\Delta_{i}$, for $i=1,\cdots,s$, are
defining equation of $D_{i}$, and the analytic coordinate of $\Delta_{n}$ is
the defining equation of $B$.

First, we study $V_{B}$ by computing its monodromy representation:

Let $T:\pi_{1}(X-D,x)\rightarrow\text{GL}(r,\mathbb{C})$ be the monodromy
representation of $V$. For each generator $\gamma_{i}$ of $\pi_{1}(X-D,x)$,
let $\Gamma_{i}=T(\gamma_{i})$. As $\Gamma_{i}$ are commuting and unitary, we
can use one matrix to diagolize all of them. Therefore, we can assume all
$\Gamma_{i}$ are diagonal matrices. Moreover, as $V$ is undefined only on $D$,
so for each $i$, $\Gamma_{i}^{jj}=1$, for $j=s+1,\cdots,n$.

Now, $B=\Delta_{1}\times\cdots\times\Delta_{n-1}$, and the monodromy
reprentation of $V|_{B-B\cap D}$ is given by

$\pi_{1}(B-B\cap
D)\xrightarrow{i}\pi_{1}(X-D)\xrightarrow{T}\text{GL}(r,\mathbb{C})$

where $i$ is the natural inclusion map. It is clear that one can choose the
basis of $\pi_{1}(B-B\cap D)$ and $\pi_{1}(X-D)$ such that $i$ can be realized
as the identity map. Therefore, the monodromy representations of $V_{B-B\cap
D}$ are also $\Gamma_{i}$, for $i=1,\cdots,s$.

To show $E|_{B}$ is the canonical extension of $V_{B-B\cap D}$, we compute the
connection matrix of $E|_{B}$ and relate it to the monodromy representations
of $V|_{B-B\cap D}$.

One can assume $E$ is trivial over $X$. Choose a local frame of $V$ on $X$,
and use it as a trivialization of $E$. With respect to this trivialization,
the connection $\nabla$ can be realized as

$d+N_{1}\frac{dz_{1}}{z_{1}}+\cdots+N_{s}\frac{dz_{s}}{z_{s}}$

where $N_{1},\cdots,N_{s}$ are commuting matrices with eigenvalues in the
stripe

$\\{z\in\mathbb{C}|0\leq\text{Re}z<1\\}$

such that $e^{-2\pi iN_{i}}=\Gamma_{i}$.

Now, restrict $E$ to $B$, we see that the connection $\nabla|_{B}$ can still
be realized as

$d+N_{1}\frac{dz_{1}}{z_{1}}+\cdots+N_{s}\frac{dz_{s}}{z_{s}}$

As monodromy representations of $V_{B-B\cap D}$ are $\Gamma_{i}$, it follows
that $E|_{B}$ is the canonical extension of $V_{B-B\cap D}$. ∎

######  Theorem 7.

Suppose $L$ is very ample on $X$. Then

$H^{q}(X,E\otimes\Omega_{X}^{p}(\log D)\otimes L)=0$

for $p+q>\dim X$

###### Proof.

Let $B$ be a smooth divisor transversal to $D$ such that $L\cong O_{X}(B)$. By
Lemma 3 we have the following exact sequence

$0\rightarrow\Omega_{X}^{p}(\log D+B)\xrightarrow{i}\Omega_{X}^{p}(\log
D)\otimes O_{X}(B)\xrightarrow{r}\Omega^{p}_{B}(\log D\cap B)\otimes
O_{X}(B)\rightarrow 0$

Tensor it by $E$ and take the cohomology sequence, we get:

$\displaystyle\cdots$ $\displaystyle H^{q}(X,\Omega^{p}_{X}(\log D+B)\otimes
E)\rightarrow H^{q}(X,\Omega^{p}_{X}(\log D)\otimes O_{X}(B)\otimes E)$
$\displaystyle\rightarrow$ $\displaystyle H^{q}(X,\Omega^{p}_{B}(\log B\cap
D)\otimes O_{X}(B)\otimes E)\cdots$

Therefore, to prove the theorem, it is enough to show

Claim 1: $H^{q}(X,\Omega^{p}_{X}(\log D+B)\otimes E)=0$

Claim 2: $H^{q}(X,\Omega^{p}_{B}(\log B\cap D)\otimes O_{X}(B)\otimes E)=0$
for $p+q>\dim X$.

Proof of claim 1: Consider the maps

$X-(B+D)\xrightarrow{f}X-B\xrightarrow{h}X$

Let $V^{o}$ be the restriction of $V$ on $X-(B+D)$. The complex
$\text{DR}(D+B,E,\nabla)$ is quasi-isomorphic to $\mathbb{R}{(h\circ
f)}_{*}V^{o}$. Therefore,

$H^{k}(X-(B+D),V^{o})=\mathbb{H}^{k}(X,\text{DR}(D+B,E,\nabla)$

The claim then follows from Corollary 2.

End of Proof

Claim 2 follows from induction on the dimension of the variety.

Now to finish the proof, it remains to show the base case of Claim 2. One may
assume now that $X$ is a smooth projective curve over $\mathbb{C}$,

We need to show that

$H^{1}(X,\Omega_{X}(\log D)\otimes E\otimes L)=0$

But for the curve case, $\Omega_{X}(\log D)\otimes O_{X}(B)=\Omega_{X}(\log
D+B)$. So the result follows again from Theorem 2 ∎

Now suppose $L$ is any ample line bundle. Let $m$ be an integer such that
${L}^{\otimes m}$ is very ample. Take a smooth divisor $B$ transversal to $D$
such that ${L}^{\otimes m}\cong O_{X}(B)$. Let $\varphi$ be the local equation
of $B$ on some affine open set, and let $\pi:X^{\prime}\rightarrow X$ be the
normalization of $X$ in $\mathbb{C}(X)(\varphi^{\frac{1}{m}})$.

######  Proposition 2.

Let $\pi:X^{\prime}\rightarrow X$, $B$ and $L$ be as above

  1. (1)

$X^{\prime}$ is smooth.

  2. (2)

${\pi}^{*}B=m\tilde{B}$, where $\tilde{B}=({\pi}^{*}B)_{red}$.

  3. (3)

$D^{\prime}:={\pi}^{*}D$ is a normal crossing divisor on $X^{\prime}$.

  4. (4)

$\tilde{B}$ is transversal to ${\pi}^{*}D$.

  5. (5)

${\pi}^{*}\Omega^{p}_{X}(\log D)=\Omega^{p}_{X^{\prime}}(\log D^{\prime})$.

  6. (6)

${\pi}^{*}E$ is the canonical extension of $\pi^{-1}V$.

###### Proof.

1\. We will construct $X^{\prime}$ by constructing its affine cover and
specefiying the gluing morphisms. Let $U_{i}=\text{Spec}A_{i}$ be an affine
cover of $X$, and let $f_{i}$ be the defining equation of $D$ in $A_{i}$.

For each $A_{i}$, $\frac{A_{i}[Y]}{(Y^{m}-f_{i})}$ is integrally closed in
$\mathbb{C}(X)(f_{i}^{1/m})$. Therefore,

$U_{i}^{\prime}:=\text{Spec}\frac{A_{i}[Y]}{(Y^{m}-f_{i})}$

is the normalization of $U_{i}$ in $\mathbb{C}(X)(f_{i}^{1/m})$

The same morphisms used to glue $U_{i}$ into $X$ can be used to glue
$U_{i}^{\prime}$ into $X^{\prime}$. Therefore, to show $X^{\prime}$ is smooth,
it is enough to show $\frac{A_{i}[Y]}{(Y^{m}-f_{i})}$ is a regular ring.

2\. The local defining equation of $\tilde{B}$ is $Y$, and
${\pi}^{*}(f_{i})=Y^{m}$

3\. To see this, we describe ${\pi}^{*}D$ in ${\pi}^{*}U$ for any polydisk
$U=\Delta_{1}\times\cdots\times\Delta_{n}$. If $B\cap U\neq\varnothing$, then
construct $\Delta_{i}$ such that defining equation of $D_{i}$, for
$i=1,\cdots,s$, are coordinates of $D_{i}$, for $i=1,\cdots,s$; and the
defining equation of $B$ is the coordinate of $D_{n}$. Then,

${\pi}^{*}U=\Delta_{1}\times\Delta_{1}\cdots\Delta_{n-1}\times\Sigma^{m}$

where $\Sigma^{m}$ is the $m$-sheeted cover over a complex disk branched over
the origin. In this case, ${\pi}^{*}D$ is still defined by $z_{1}\times
z_{2}\times\cdots z_{s}$.

If $B\cap U=\varnothing$, then ${\pi}^{*}U$ is etale over $U$. Therefore,
${\pi}^{*}D$ is etale over $D$. So ${\pi}^{*}D$ is again a simple normal
crossing divisor.

4.This is clear from the case 1 of part 3.

5\. Straighforward computation.

6\. We compute the monodromy representation of $\pi^{-1}V$ first:

let $T:\pi_{1}(U-D,x)\rightarrow\text{GL}(r,\mathbb{C})$ be the representation
corresponding to the local system $V$.

Case 1: Suppose $x\notin B$, then $\pi^{-1}(U)$ is etale over $U$. Let
$U^{\prime}$ be an component of $\pi^{-1}(U)$, and let $x^{\prime}\in
U^{\prime}$ be a preimage of $x$. Then,

$T^{\prime}:\pi_{1}(U^{\prime}-D^{\prime},x^{\prime})\xrightarrow{{\pi}_{*}}\pi_{1}(U-D,x)\xrightarrow{T}\text{GL}(r,\mathbb{C})$

is the representation corresponding to $\pi^{-1}V$.

Case 2: Suppose $x\in B$, then use the description from part 3, we know that

$\pi^{-1}U=\Delta_{1}\times\Delta_{2}\times\cdots\times\Sigma^{m}$

In both cases, $\pi^{-1}U-D^{\prime}$ is homotopic to $S_{1}\times
S_{2}\times\cdots\times S_{s}$ So we can define generators of
$\pi_{1}(U^{\prime}-D^{\prime},x^{\prime})$ and $\pi_{1}(U-D,x)$ such that
${\pi}_{*}$ is the identity map.

To show ${\pi}^{*}E$ is the canonical extension of $\pi^{-1}V$, we only need
to compute the connection matrix of ${\pi}^{*}E$ and relate it to the
monodromies of $\pi^{-1}V$:

Let $\gamma_{i}$ be a small circle around $D_{i}$, and let $\Gamma_{i}$ be the
monodromy $T(\gamma_{i})$. As

${\pi}_{*}:\pi_{1}(U^{\prime}-D^{\prime},x^{\prime})\rightarrow\pi_{1}(U-D,x)$

is the identity map, $\Gamma_{i}$ are also the monodromy representations of
$\pi^{-1}V$. Next, we compute the connection matrix of $E$. Let $U$ be small
enough so that $E$ is trivial over it. Choose a local frame of $V$, and use it
as a trivialization of $E$. With respect to this trivialization, the
connection $\nabla$ can be realized as

$d+N_{1}\frac{dz_{1}}{z_{1}}+\cdots+N_{s}\frac{dz_{s}}{z_{s}}$

where $N_{1},\cdots,N_{s}$ are commuting matrices with eigenvalue in the
stripe

$\\{z\in\mathbb{C}|0\leq\text{Re}z<1\\}$

such that $e^{-2\pi iN_{i}}=\Gamma_{i}$.

As ${\pi}^{*}z_{i}=z_{i}$, for $i=1,\cdots,s$, we see that the
${\pi}^{*}\nabla$ over $\pi^{-1}U$ can be realized as:

$d+N_{1}\frac{dz_{1}}{z_{1}}+\cdots+N_{s}\frac{dz_{s}}{z_{s}}$

This shows that ${\pi}^{*}E$ is the canonical extension of $\pi^{-1}V$.

∎

######  Corollary 3.

For any ample line bundle $L$ on $X$,

$H^{q}(X,E\otimes\Omega_{X}^{p}(\log D)\otimes L)=0$

for $p+q>\dim X$

###### Proof.

Let $m$, $B$ and $\pi:X^{\prime}\rightarrow X$ be as above. By Theorem 7

$H^{q}(X^{\prime},{\pi}^{*}(E\otimes\Omega_{X}^{p}(\log D)\otimes L))=0$

for $p+q>\dim X^{\prime}=\dim X$.

$\pi:X^{\prime}\rightarrow X$ is a finite morphism, so for $i>0$,
$R^{i}{\pi}_{*}\mathscr{F}=0$ for any coherent sheaf $\mathscr{F}$ on
$X^{\prime}$. This implies

$\displaystyle H^{q}(X^{\prime},{\pi}^{*}(E\otimes\Omega_{X}^{p}(\log
D)\otimes L))$
$\displaystyle=H^{q}(X,{\pi}_{*}({\pi}^{*}(E\otimes\Omega_{X}^{p}(\log
D)\otimes L)))$ $\displaystyle=H^{q}(X,{\pi}_{*}(O_{Y})\otimes
E\otimes\Omega_{X}^{p}(\log D)\otimes L)$ $\displaystyle=0$

for $p+q>\dim X$. The second equality follows from the projection formula.

As ${\pi}_{*}(O_{Y})\cong\bigoplus\limits_{i=0}^{m-1}O_{X}(-{L}^{\otimes i})$,
the result follows. ∎

##  5\. Partial weight filtration

In the previous section, we proved the vanishing theorem for the complex

$\text{DR}_{X}(D,E)\otimes O_{X}(B)$

where $B$ is a smooth very ample divisor transversal to $D$. The intermediate
step for the proof is the vanishing theorem for the complex

$\text{DR}(D+B,E)$

In this section, we define a partial weight filtration on the The complex

$\text{DR}_{X}(D+B,E)$

It is a more refined weight filtration than the one defined in Section 2, and
it will be used to prove the vanishing theorem for the graded complex

$\text{Gr}^{W}_{.}\text{DR}_{X}(D,E)$

For simplicity, suppose $V$ is a rank 1 unitary local system. We will define
partial weight filtration by giving local description of forms. Then, we will
show it is a well-defined global notion after Theorem 8. Let $\mu$ be a
section of $E$. Recall that $W_{m}\Omega^{p}_{X}(\log D)\otimes E$ consists of
sections of the form

$\omega\otimes\mu$

where $\omega$ can be written as

$\frac{dz_{j_{1}}}{z_{j_{1}}}\wedge\frac{dz_{j_{2}}}{z_{j_{2}}}\wedge\cdots\frac{dz_{j_{k}}}{z_{j_{k}}}\wedge
dz_{j_{k+1}}\wedge\cdots\wedge dz_{j_{p}}$

Moreover, $\omega$ has at most $m$ log forms acting on $V$ by identity.

Now let $W_{m}^{D^{1}}\text{DR}(D,E)$ be the set of form that can be written
as $\omega\otimes\mu$, where $\omega$ can be written as

$\frac{dz_{j_{1}}}{z_{j_{1}}}\wedge\frac{dz_{j_{2}}}{z_{j_{2}}}\wedge\cdots\frac{dz_{j_{k}}}{z_{j_{k}}}\wedge
dz_{j_{k+1}}\wedge\cdots\wedge dz_{j_{p}}$

Moreover, let $g$ be the cardinality of the following set

$\\{\text{log forms in $\omega$ acting on $V$ by
identity}\\}\cap\\{\frac{dz_{2}}{z_{2}},\cdots,\frac{dz_{s}}{z_{s}}\\}$

Then $g\leq m$.

To generalize, $W_{m}^{D^{i_{1}}+\cdots D^{i_{l}}}\text{DR}(D,E)$ is the set
of forms that can be written as $\omega\otimes\mu$, where $\omega$ can be
written as

$\frac{dz_{j_{1}}}{z_{j_{1}}}\wedge\frac{dz_{j_{2}}}{z_{j_{2}}}\wedge\cdots\frac{dz_{j_{k}}}{z_{j_{k}}}\wedge
dz_{j_{k+1}}\wedge\cdots\wedge dz_{j_{p}}$

Moreover, let $g$ be the cardinality of the following set

$\\{\text{log forms in $\omega$ acting on $V$ by
identity}\\}\cap(\\{\frac{dz_{1}}{z_{1}},\cdots,\frac{dz_{s}}{z_{s}}\\}-\\{\frac{dz_{i_{1}}}{z_{i_{1}}},\cdots\frac{dz_{i_{l}}}{z_{i_{l}}}\\})$

Write $T$ for $D+B$. Let $T_{2}$ be the union of 2-fold intersections of
components of $T$.

Let $v_{1}:\tilde{D}_{1}\rightarrow D$ be the normalization map, i.e.
$\tilde{D}_{1}$ is the disjoint union of components of $D$. Let $F_{1}$ be
${v_{1}}^{*}T_{2}$. Then, $F_{1}$ is a normal crossing divisor in $D_{1}$.

We have seen in Section 1 that the restiction of ${j}_{*}V$ on $D_{1}-T_{2}$
is a unitary local system, denote it by $V_{1}$; and let $E_{1}$ be the
subbundle of ${v}^{*}_{1}E$ which is the canonical extension of $V_{1}$.

######  Proposition 3.

There is an exact sequence

$0\rightarrow
W^{B}_{0}\text{DR}(D+B,E,\nabla)\xrightarrow{i}\text{DR}(D+B,E,\nabla)\xrightarrow{res}{v_{1}}_{*}\text{DR}(F_{1},E_{1},\nabla_{1})\rightarrow
0$

where $i$ is the inclusion map, and $res$ is the residue map.

###### Proof.

Suppose for simplicity $D$ is smooth, i.e. $D$ has only one component. Also,
suppose $V$ is a unitary local system of rank 1. Let $\mu$ be a local section
of $E$.

Let $z_{1}$ be the local equation of $D$. Suppose $\frac{dz_{1}}{z_{1}}$ acts
on $V$ by identity, then $V$ extends to a unitary local system on $D-D\cap B$.
In this case, $D_{1}=D$, and $F_{1}=D\cap B$. Let $z_{n}$ be the local
equation for $B$. Then, locally over a polydisk

1\. $W^{B}_{0}\Omega^{p}_{X}(\log D+B)\otimes E$ is generated by sections of
the form

$\frac{dz_{n}}{z_{n}}\wedge\omega\otimes\mu$

where $\omega\in\Omega^{p-1}_{X}$.

2\. $\Omega^{p}_{X}(\log D+B)$ is generated by sections of the form

$\frac{dz_{n}}{z_{n}}\wedge\omega\otimes\mu$

where $\omega\in\Omega^{p-1}_{X}(\log D)$.

3\. $\Omega^{p-1}_{D_{1}}(\log F_{1})\otimes E_{1}$ is generated by sections
of the form

$\frac{dz_{n}}{z_{n}}\wedge\omega\otimes\mu_{1}$

where $\omega\in\Omega_{D_{1}}^{p-2}(\log F_{1})$.

Use the local description, it is clear that the sequence is exact.

∎

######  Theorem 8.

Let $(E_{B},\nabla_{B})$ be the restriction of $(E,\nabla)$ on $B$, and let
$\text{DR}(B\cap D,E_{B},\nabla_{B})$ be the complex

$0\rightarrow E_{B}\rightarrow\Omega_{B}^{1}(B\cap D)\otimes E_{B}\cdots$

then there is an exact sequence of complexes

$\displaystyle 0\rightarrow
W^{B}_{m}\text{DR}_{X}(D+B,E)\xrightarrow{i}W_{m}\text{DR}_{X}(D,E)\otimes
O_{X}(B)$ $\displaystyle\xrightarrow{r}W_{m}\text{DR}_{B}(D\cap
B,E_{B})\otimes O_{X}(B)\rightarrow 0$

$i$ is the inclusion map, and $r$ is the restriction map.

###### Proof.

For simplicity, we assume $E$ has rank 1. The statement is local, so we work
on a polydisk, and we use the notation from above. Let $\mu$ be a generating
section of $E$, then

1\. $W^{B}_{m}\text{DR}_{X}(D+B,E)\otimes O_{X}(-B)$ is generated by

$\omega\otimes\mu\otimes z_{n}$

where $\omega\in\Omega^{p}_{X}(\log D+B)$ is a $p$-form that has at most $m$
log forms coming from

$\\{\frac{dz_{1}}{z_{1}},\cdots,\frac{dz_{s}}{z_{s}}\\}$

acting on $E$ by identity.

2\. $W_{m}\text{DR}_{X}(D,E)$ is generated by

$\omega\otimes\mu$

where $\omega\in\Omega^{p}_{X}(\log D)$ is a $p$-form that has at most $m$ log
forms acting on $E$ by identity.

3\. $W_{m}\text{DR}_{B}(D\cap B,E_{B})$ is generated by

$\omega\otimes\mu$

where $\omega\in\Omega_{B}^{p}(\log B\cap D)$ is a $p$-form that has at most
$m$ log forms acting on $\mu_{B}$ by identity.

The map $i$ is the natural inclusion map, i.e. $\frac{dz_{n}}{z_{n}}\otimes
z_{n}\mapsto dz_{n}$; The map $r$ is the restriction on $B$.

∎

The above theorem also gives a description of

$W_{m}^{B}\text{DR}_{X}(D+B,E)$

as the kernel of the restriction map

$r:\text{DR}_{X}(D+B,E)\otimes O_{X}(B)\rightarrow W_{m}\text{DR}_{B}(B\cap
D,E_{B})\otimes O_{X}(B)$

It means that $W_{m}\text{DR}_{X}(D+B,E)$ is indeed globally well-defined.

##  6\. Mixed Hodge Structure on the Complex
$W^{B}_{0}\text{DR}(D+B,E,\nabla)$

Throughout this section, we assume the unitary local system $V$ has a real
lattice $V$ such that

$V=V\otimes\mathbb{C}$

We will study the mixed Hodge structure on the complex

$W^{B}_{0}\text{DR}_{X}(D+B,E)$

Consider the maps

$X-(D+B)\xrightarrow{f}X-B\xrightarrow{h}X$

Write $V^{o}$ (resp. $V^{o}$) for the restriction of $V$ (resp. $V$) on
$X-(D+B)$.

Let $\tau$ be the canonical filtration on $\mathbb{R}{h}_{*}{f}_{*}V^{o}$; let
$W$ be the increasing filtration on $W^{B}_{0}\text{DR}_{X}(D+B,E)$ defined as

$\displaystyle W_{m}W^{B}_{0}\text{DR}_{X}(D+B,E)=\begin{cases}0&\text{ if
}m<0\\\ W_{0}\text{DR}_{X}(D+B,E)&\text{ if }m=0\\\
W^{B}_{0}\text{DR}_{X}(D+B,E)&\text{ if }m>0\end{cases}$

The main result of this section is

######  Theorem 9.

$(\mathbb{R}{h}_{*}{f}_{*}V^{o},(\mathbb{R}{h}_{*}{f}_{*}V^{o},\tau),(W^{B}_{0}\text{DR}_{X}(D+B,E),F^{.},W_{.}))$

is a -cohomological mixed Hodge complex.

######  Proposition 4.

$\mathbb{R}{h}_{*}({f}_{*}V^{o})$ is quasi-isomoprhic to

$W^{B}_{0}\text{DR}_{X}(D+B,E)$

###### Proof.

The statement is local, so we can assume $X$ is a polydisk. For the basic
case, one can assume $V$ is of rank 1, $D$ has two components $D^{1}$ and
$D^{2}$ such that the monodromy of $V$ around $D^{1}$ is trivial, and the
monodromy of $V$ around $D^{2}$ is nontrivial.

Let $Y=X-B$. Then, $\Omega^{\boldsymbol{\cdot}}_{Y}(\log
D^{2})\otimes{h}^{*}E$ is a resolution of ${f}_{*}V^{o}$(see [8]).

Let $g:Y-D^{2}\rightarrow Y$ be the inclusion map. By a theorem of
Griffith[Griffith69] and Deligne[3], the inclusion map

$i:\Omega^{\boldsymbol{\cdot}}_{Y}(\log
D^{2})\rightarrow{g}_{*}\mathscr{A}^{\boldsymbol{\cdot}}_{Y-D^{2}}$

is a quasi-isomorphism. Therefore, ${f}_{*}V$ is quasi-isomorphic to

${g}_{*}\mathscr{A}^{\boldsymbol{\cdot}}_{Y-D^{2}}$

As ${g}_{*}\mathscr{A}^{\boldsymbol{\cdot}}_{Y-D^{2}}$ is a complex of flasque
sheaves, $\mathbb{R}{h}_{*}{f}_{*}V$ is quasi-isomorphic to

${h}_{*}{g}_{*}\mathscr{A}^{\boldsymbol{\cdot}}_{Y-D^{2}}$

Now,

$W^{B}_{0}\Omega^{\boldsymbol{\cdot}}_{X}(\log D+B)\otimes
E=\Omega^{\boldsymbol{\cdot}}_{X}(\log D^{2}+B)\otimes E$

But as we have seen the complex $\Omega^{\boldsymbol{\cdot}}_{X}(\log
D^{2}+B)$ is quasi-isomorphic to

${(h\circ g)}_{*}\mathscr{A}^{\boldsymbol{\cdot}}_{Y-D^{2}}$

So the result for the basic case follows.

Now, let $V$ be of rank $r$. For each $i=1,2$, let $\Gamma_{i}$ be the
monodromy of $V$ around $D^{i}$. As $V$ is unitary, we can simultaneously
diagonalize all $\Gamma_{1}$ and $\Gamma_{2}$. Therefore, we can assume $V$ is
the direct sum of two rank 1 unitary local systems. As $\mathbb{R}{h}_{*}$ and
${f}_{*}$ commutes with direct sum. The result follows.

Now, let $V$ be of rank 1 and let $D^{1},\cdots,D^{s}$ be components of $D$.
Now let $D_{1}$ be the subdivisor of $D$ over which $V$ has identity
monodromy; and let $D_{2}$ be the subdivisor of $D$ over which $V$ has
nontrivial monodromy. Then, the result follows after the same steps in the
basic case. ∎

######  Proposition 5.

The inclusion map

$i:(W^{B}_{0}\text{DR}_{X}(D+B,E),\tau_{.})\rightarrow(W^{B}_{0}\text{DR}_{X}(D+B,E),W)$

is a quasi-isomorphism of filtered complexes.

###### Proof.

This is again a local statement, so we can assume $X$ is a polydisk and $V$ is
of rank 1. We need to show that the induced maps of $i$

$H^{k}(i):H^{k}(\text{Gr}^{\tau}_{m}W^{B}_{0}\text{DR}_{X}(D+B,E))\rightarrow
H^{k}(\text{Gr}^{W}_{m}W^{B}_{0}\text{DR}_{X}(D+B,E))$

are isomorphisms.

$\displaystyle
H^{k}(\text{Gr}^{\tau}_{m}W^{B}_{0}\text{DR}_{X}(D+B,E))=\begin{cases}H^{m}(W^{B}_{0}\text{DR}_{X}(D+B,E))&\text{
if }i=m\\\ 0&\text{ otherwise}\end{cases}$

Claim 1 If $m>1$, then $H^{m}(W^{B}_{0}\text{DR}_{X}(D+B,E))=0$.

Proof of Claim 1 We have a short exact sequence of complexes

$0\rightarrow W_{0}\text{DR}_{X}(D+B,E)\rightarrow
W^{B}_{0}\text{DR}_{X}(D+B,E)\xrightarrow{res}W_{0}\text{DR}_{B}(B\cap
D,E_{B})[-1]\rightarrow 0$

where the $\text{DR}(D\cap B,E_{B},\nabla_{B})$ is the complex

$\cdots\rightarrow\Omega^{m}_{B}(\log B\cap D)\otimes
E_{B}\xrightarrow{\nabla_{B}}\Omega^{m+1}_{B}(\log B\cap D)\otimes
E_{B}\rightarrow\cdots$

and the map $res$ is the residue map.

Taking cohomology, we get

$\displaystyle\cdots$ $\displaystyle\rightarrow
H^{k}(W_{0}\text{DR}_{X}(D+B,E))\rightarrow
H^{k}(W^{B}_{0}\text{DR}_{X}(D+B,E))$ $\displaystyle\rightarrow
H^{k-1}(W_{0}\text{DR}_{B}(B\cap D,E_{B}))\rightarrow\cdots$

Timmerscheidt proved in the Appendix D of [7] that $W_{0}\text{DR}_{X}(D+B,E)$
is a resolution of $(h\circ f)_{*}V$. Therefore,
$W^{B}_{0}\text{DR}_{X}(D+B,E)$ is exact. Likewise,

$W_{0}\text{DR}_{B}(B\cap D,E_{B})$

is also exact.

So the conlusion follows.

End of Proof

The above proof also shows that

$\displaystyle
H^{k}(\text{Gr}^{W}_{1}W^{B}_{0}\text{DR}_{X}(D+B,E))=\begin{cases}H^{1}(\text{Gr}^{W}_{1}W^{B}_{0}\text{DR}_{X}(D+B,E))&\text{
if }k=1\\\ 0&\text{ if }k>1\end{cases}$

$\displaystyle
H^{k}(\text{Gr}^{W}_{0}W^{B}_{0}\text{DR}_{X}(D+B,E))=\begin{cases}H^{0}(W_{0}\text{DR}(D+B,E))&\text{
if }k=0\\\ 0&\text{ if }k>0\end{cases}$

Therefore, to prove

$i:(W^{B}_{0}\text{DR}_{X}(D+B,E),\tau)\rightarrow(W^{B}_{0}\text{DR}_{X}(D+B,E),W)$

is a quasi-isomorphism of filtered complexes, it remains to prove that both

$H^{0}(i):H^{0}(\text{Gr}^{\tau}_{0}W^{B}_{0}\text{DR}_{X}(D+B,E))\rightarrow
H^{0}(\text{Gr}^{W}_{0}W^{B}_{0}\text{DR}_{X}(D+B,E))$

and

$H^{1}(i):H^{1}(\text{Gr}^{\tau}_{1}W^{B}_{0}\text{DR}_{X}(D+B,E))\rightarrow
H^{1}(\text{Gr}^{W}_{1}W^{B}_{0}\text{DR}_{X}(D+B,E))$

are isomorphisms.

Now,

$H^{0}(\text{Gr}^{\tau}_{0}W^{B}_{0}\text{DR}_{X}(D+B,E))=\ker(E\xrightarrow{\nabla}W^{B}_{0}(\Omega^{1}_{X}(\log
D+B)\otimes E)$

and

$H^{0}(\text{Gr}^{W}_{0}W^{B}_{0}\text{DR}_{X}(D+B,E))=\ker(E\xrightarrow{\nabla}W_{0}(\Omega^{1}_{X}(\log
D+B)\otimes E)$

It is clear that the map $H^{0}(i)$ is an isomorphism.

To simplify notations, write $K^{\boldsymbol{\cdot}}$ for
$W^{B}_{0}\text{DR}_{X}(D+B,E)$, from the proof of Claim 1, we have a
commutative diagram

${H^{1}(\text{Gr}^{\tau}_{1}K^{\boldsymbol{\cdot}})}$${H^{1}(\text{Gr}^{W}_{1}K^{\boldsymbol{\cdot}})}$${H^{1}(W_{0}\text{DR}_{B}(B\cap
D,E_{B})[-1])}$${H^{1}(K^{\boldsymbol{\cdot}})}$${H^{1}(W_{0}\text{DR}_{B}(B\cap
D,E_{B})[-1])}$$\scriptstyle{H^{1}(i)}$$\scriptstyle{res}$$\scriptstyle{res}$

and the residue map on the second row is an isomorphism. As the residue map on
the first row is an isomorphism (even on the complex level), we see that the
map $H^{1}(i)$ is an isomorphism.

∎

For reader’s sake, we restate the main theorem of this Section:

######  Theorem 10.

$(\mathbb{R}{h}_{*}{f}_{*}V^{o},(\mathbb{R}{h}_{*}{f}_{*}V^{o},\tau),(W^{B}_{0}\text{DR}_{X}(D+B,E),F^{.},W_{.}))$

is a cohomological mixed -Hodge complex

###### Proof.

The quasi-isomorphism

$(\mathbb{R}{h}_{*}{f}_{*}V^{o},\tau)\otimes\mathbb{C}\rightarrow(W^{B}_{0}\text{DR}_{X}(D+B,E),W_{.})$

was proved in the previous proposition.

It remains to show

$(\text{Gr}^{\tau}_{m}\mathbb{R}{h}_{*}{f}_{*}V^{o},(\text{Gr}^{W}_{m}W^{B}_{0}\text{DR}_{X}(D+B,E),F))$

is a cohomological -complex of weight $m$, i.e. the Hodge spectral sequence of
$(\text{Gr}^{W}_{m}W^{B}_{0}\text{DR}_{X}(D+B,E),F)$ degenerates at $E_{1}$,
and the induced filtration on

$\mathbb{H}^{k}(X,\text{Gr}^{W}_{m}W^{B}_{0}\text{DR}_{X}(D+B,E))=\mathbb{H}^{k}(X,\text{Gr}^{\tau}_{m}\mathbb{R}{h}_{*}{f}_{*}V^{o})\otimes\mathbb{C}$

defines a pure -Hodge structure of weight $k+m$ on

$\mathbb{H}^{k}(X,\text{Gr}^{\tau}_{m}\mathbb{R}{h}_{*}{f}_{*}V^{o})$

i.e. the induced filtration $F$ on
$\mathbb{H}^{k}(X,\text{Gr}^{W}_{m}W^{B}_{0}\text{DR}_{X}(D+B,E))$ is $m+k$
opposed to its conjugate.

For $m>1$, all $\text{Gr}^{W}_{m}W^{B}_{0}\text{DR}_{X}(D+B,E)$ are 0, so we
only need to show the case for $m=0,1$.

For $m=0$,

$(\text{Gr}^{W}_{m}W^{B}_{0}\text{DR}_{X}(D+B,E),F)=(W_{0}\text{DR}_{X}(D+B,E),F)$

Timmerscheidt showed that it is a cohomological -complex of weight $0$ in
[7](Appendix D).

For $m=1$, we have seen that

$\text{Gr}^{W}_{1}W^{B}_{0}\text{DR}_{X}(D+B,E)\cong W_{0}\text{DR}(B\cap
D,E_{B},\nabla_{B})[-1]$

Let $F$ be the induced Hodge filtration on
$\text{Gr}^{W}_{1}W^{B}_{0}\text{DR}_{X}(D+B,E)$, and let $F_{B}$ be the usual
Hodge filtration on $W_{0}\text{DR}(B\cap D,E_{B},\nabla_{B})$. let $\bar{F}$
and $\bar{F}_{B}$ be their conjugates.

To show $F$ and $\bar{F}$ are $k+1$ opposed on
$\mathbb{H}^{k}(X,\text{Gr}^{W}_{1}W^{B}_{0}\text{DR}_{X}(D+B,E))$, we show
that

$\text{Gr}^{\bar{F}}_{q}\text{Gr}^{F}_{p}\mathbb{H}^{k}(X,\text{Gr}^{W}_{1}W^{B}_{0}\text{DR}_{X}(D+B,E))=0\text{
if }p+q\neq k+1$

As $\text{Gr}^{W}_{1}W^{B}_{0}\text{DR}_{X}(D+B,E)\cong
W_{0}\text{DR}_{B}(B\cap D,E_{B})[-1]$,

$\displaystyle\text{Gr}^{F}_{p}\mathbb{H}^{k}(X,\text{Gr}^{W}_{1}W^{B}_{0}\text{DR}_{X}(D+B,E))$
$\displaystyle=\text{Gr}^{F_{B}}_{p-1}\mathbb{H}^{k-1}(B,W_{0}\text{DR}_{B}(B\cap
D,E_{B}))$
$\displaystyle\text{Gr}^{\bar{F}}_{q}\mathbb{H}^{k}(X,\text{Gr}^{W}_{1}W^{B}_{0}\text{DR}_{X}(D+B,E))$
$\displaystyle=\text{Gr}^{\bar{F}}_{q-1}\mathbb{H}^{k-1}(B,W_{0}\text{DR}_{B}(B\cap
D,E_{B}))$

Therefore,
$\text{Gr}^{\bar{F}}_{q}\text{Gr}^{F}_{p}\mathbb{H}^{k}(X,\text{Gr}^{W}_{1}W^{B}_{0}\text{DR}_{X}(D+B,E))=0$
if $p-1+q-1\neq k-1$.

The $E_{1}$-degeneration of
$(\text{Gr}^{W}_{1}W^{B}_{0}\text{DR}_{X}(D+B,E),F)$ follows from the
$E_{1}$-degneration of $(W_{0}\text{DR}(B\cap D,E_{B},\nabla_{B}),F_{B})$.

∎

So far, we have shown that if $V$ has a real lattice $V$, i.e.

$V=V\otimes\mathbb{C}$

Then, the Hodge spectral sequence

$E_{1}^{p,q}=H^{q}(X,W^{B}_{0}\Omega^{p}_{X}(\log D+B)\otimes
E)=>\mathbb{H}^{p+q}(X,W^{B}_{0}\text{DR}_{X}(D+B,E))$

degenerates at $E_{1}$.

Now, consider the case when $V$ does not have a real-lattice.

##  7\. Vanishing Theorem for the complex
$\text{Gr}^{W}_{.}\text{DR}_{X}(D,E)$

Now, let $V$ be any unitary local system over $\mathbb{C}$. We have seen in
Section 4 that even if $V$ does not have a real lattice, the spectral sequence
of $(\text{DR}_{X}(D,E),F)$ still have $E_{1}$-degeneration. Similarly, we
have

######  Lemma 5.

Let $B$ be a smooth divisor transversal to $D$, then The spectral sequence of
$(W^{B}_{0}\text{DR}(D+B,E),F)$:

$E^{p,q}_{1}=H^{q}(X,W^{B}_{0}(\Omega_{X}^{p}(\log
D+B)))=>\mathbb{H}^{p+q}(X,W^{B}_{0}(\text{DR}_{X}(D+B,E)))$

degenerates at $E_{1}$

######  Theorem 11.

Let $B$ be a smooth very ample divisor transveral to $D$, then for
$m=0,\cdots,n-1$

$H^{q}(X,\text{Gr}^{W}_{m}\text{DR}^{p}(D,E,\nabla)\otimes O_{X}(B))=0$

for $p+q>n+1$.

###### Proof.

We show first that

$H^{q}(X,W_{0}\text{DR}^{p}(D,E,\nabla)\otimes O_{X}(B))=0$

for $p+q>n+1$.

By Theorem 8, we have the exact sequence

$0\rightarrow W^{B}_{0}\text{DR}_{X}(D+B,E)\rightarrow
W_{0}\text{DR}_{X}(D,E)\otimes O_{X}(B)\rightarrow W_{0}\text{DR}_{B}(B\cap
D,E_{B})\otimes O_{X}(B)\rightarrow 0$

Take cohomology sequence, we get

$\cdots\rightarrow H^{q}(X,W^{B}_{0}(\Omega^{p}_{X}(\log D+B)\otimes
E))\rightarrow H^{q}(X,W_{0}(\Omega^{p}_{X}(\log D)\otimes E)\otimes
O_{X}(B))\rightarrow$ $\rightarrow H^{q}(B,W_{0}(\Omega^{p}_{B}(\log B\cap
D)\otimes E)\otimes O_{X}(B))\rightarrow\cdots$

Therefore, it is enough to show

Claim 1: $H^{q}(X,W_{0}^{B}(\Omega^{p}_{X}(\log D+B)\otimes E))=0$ for
$p+q>n$.

Claim 2: $H^{q}(B,W_{0}(\Omega^{p}_{B}(\log B\cap D)\otimes E_{B}))\otimes
O_{X}(B))=0$ for $p+q>n$.

Proof of Claim 1: Consider the maps

$X-(B+D)\xrightarrow{f}X-B\xrightarrow{h}X$

Write $V^{o}$ for the restriction of $V$ on $X-B$. We have seen in Lemma 5
that the spectral sequence

$\displaystyle E_{1}^{p,q}=H^{q}(X,W^{B}_{0}(\Omega^{p}_{X}(\log D+B)\otimes
E))=>$ $\displaystyle\mathbb{H}^{p+q}(X,W^{B}_{0}\text{DR}_{X}(D+B,E))$
$\displaystyle=$ $\displaystyle H^{p+q}(X,\mathbb{R}{h}_{*}{f}_{*}V^{o})$
$\displaystyle=$ $\displaystyle H^{p+q}(X-B,{f}_{*}V^{o})$

As $X-B$ is affine, it follows from Theorem 6 that

$H^{q}(X,W^{B}_{0}(\Omega^{p}_{X}(\log D+B)\otimes E))=0$

for $p+q>n$.

End

Proof of Claim 2: Induct on dimension of $X$ End

Therefore, it remains to show that if $X$ is a smooth projective curve, then

$H^{1}(X,W_{0}(\Omega_{X}(\log D)\otimes E)\otimes O_{X}(B))=0$

But for the curve case,

$W_{0}(\Omega_{X}(\log D)\otimes E)\otimes O_{X}(B)=W^{B}_{0}(\Omega_{X}(\log
D+B)\otimes E)$

Therefore the result follows again from Theorem 6

To finish the rest of the proof, we use the identification from proposition 1

$(W_{m}/W_{m-1})\text{DR}_{X}(D,E)\cong
W_{0}\text{DR}(\tilde{C}_{m},E_{m},\nabla_{m})[-m]$

and then apply the above argument to $\tilde{D}_{m}$. ∎

######  Corollary 4.

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

$H^{q}(X,W_{m}(\Omega_{X}^{p}(\log D)\otimes E)\otimes O_{X}(B))=0$

for $p+q>\dim X$

######  Corollary 5.

Let $L$ be an ample line bundle on $X$, then

$H^{q}(X,\text{Gr}^{W}_{.}(\Omega^{p}_{X}(\log D)\otimes E)\otimes L)=0$

for $p+q>n$.

###### Proof.

Like in Theorem 11, it is enough to show

$H^{q}(X,W_{0}\text{DR}^{p}(D,E,\nabla)\otimes L)=0$

for $p+q>n$.

Let $m$ be a large enough integer such that ${L}^{\otimes m}$ is very ample.
Let $B$ be a smooth hyperplane divisor transversal to $D$ so that

$L\cong O_{X}(B)$

Use the same idea from Corollary 3, we construct a cyclic cover of degree $m$
branched over $B$

$\pi:X^{\prime}\rightarrow X$

To finish the proof, it remains to show

${\pi}^{*}W_{0}\text{DR}_{X}(D,E)=W_{0}\text{DR}(\tilde{D},\tilde{E},\tilde{\nabla})$

But this is clear from the local description of $W_{0}$. ∎

##  8\. Appendix

###  8.1. Linear algebra

######  Theorem 12.

Let $U$ be an unitary matrix over $\mathbb{C}$, then $U$ is diagonalizable.

######  Theorem 13.

Let $A$ and $B$ be commuting diagonalizable $n\times n$ matrices over any
field $k$, then $A$ and $B$ can be simultaneously diagolized.

###### Proof.

Let $V$ be the vector space $k^{n}$. It is enough to show that $A$ and $B$
share the same eigenvectors.

Claim 1: $A$ and $B$ share at least one eigenvector.
Proof of Claim 1: Let $v$ be an eigenvector of $A$ with eigenvalue $\lambda$,
then

$ABv=BAv=B\lambda v=\lambda Bv$

i.e. $Bv$ is also an eigenvector of $A$ with eigenvalue $\lambda$.

Let $W$ be the subspace spanned by

$v,Bv,\cdots,B^{n}v$

Then, $W$ is invariant under $B$. As $V$ has a basis by eigenvectors of $B$,
one can choose a vector $w\in W$ which is an eigenvector of $B$. Then, from
the construction of $W$, $w$ is also an eigenvector of $A$. End

Let $w$ be as above, with $Bw=\mu w$; Let $e_{1},\cdots,e_{n}$ be the standard
basis of $V$; Let $V^{\prime}$ be the subspace spanned by
$e_{1},\cdots,e_{n-1}$; Let $\phi:V\rightarrow V$ be the linear map such that
$\phi(e_{n})=w$.

$\displaystyle\phi^{-1}\circ A\circ\phi$
$\displaystyle=A^{\prime}\oplus\text{Diag}(\lambda)$
$\displaystyle\phi^{-1}\circ B\circ\phi$
$\displaystyle=B^{\prime}\oplus\text{Diag}(\mu)$

where $A^{\prime}$ and $B^{\prime}$ are $n-1\times n-1$ submatrices of $A$ and
$B$, representing the restriction of $A$ and $B$ on $V^{\prime}$.

Now, $A^{\prime}$ and $B^{\prime}$ are diagonlizable, and they commute,
therefore, by inducting on the size of the matrix, we are done. ∎

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