[thm] [assumpsion] [cor] [proposition] [dfn] [lem] [main] [rem]

# On the nilpotent orbit theorem of
complex variations of Hodge structure

Ya Deng  CNRS, Institut Élie Cartan de Lorraine, Université de Lorraine, 54000
Nancy, France<EMAIL_ADDRESS>https://ydeng.perso.math.cnrs.fr

###### Abstract.

We prove some results on the nilpotent orbit theorem for complex variations of
Hodge structure.

###### Key words and phrases:

Nilpotent orbit theorem, acceptable bundle, parabolic bundle, $L^{2}$-estimate

###### 1991 Mathematics Subject Classification:

14D07, 14C30

###### Contents

  1. 0 Introduction
  2. 1 Preliminary
  3. 2 Nilpotent orbit theorem

##  0\. Introduction

The nilpotent and ${\rm SL}_{2}$-orbit theorems of Schmid have been
fundamental in understanding the degeneration of Hodge structure, particularly
in the context of integral variation of Hodge structures. However, their
complete generalization to complex variations of Hodge structure remains
unproven. This paper aims to the study of Schmid’s nilpotent orbit theorem for
complex variations of Hodge structure. The main result of this paper is the
main component of the nilpotent orbit theorem.

######  Theorem A.

Let $X$ be a complex manifold and let $D=\sum_{i=1}^{\ell}D_{i}$ be a simple
normal crossing divisor on $X$. Let $(V,\nabla,F^{\bullet},Q)$ be a complex
polarized variation of Hodge structure on $X\backslash D$. Then for any multi-
index $\bm{\alpha}=(\alpha_{1},\ldots,\alpha_{\ell})\in\mathbb{R}^{\ell}$,
$F^{p}_{\bm{\alpha}}:=j_{*}F^{p}\cap V_{\bm{\alpha}}^{\rm Del}$ and
$F^{p}_{\bm{\alpha}}/F^{p+1}_{\bm{\alpha}}$ are both locally free sheaves.
Here $V_{\bm{\alpha}}^{\rm Del}$ is the Deligne extension of the flat bundle
$(V,\nabla)$ with the eigenvalues of the residue of $\nabla$ over $D_{i}$
lying in $[-\alpha_{i},-\alpha_{i}+1)$.

We prove moreover that the grading
$\oplus_{p+q=m}F^{p}_{\bm{\alpha}}/F^{p+1}_{\bm{\alpha}}$ is naturally
identified with $\oplus_{p+q=m}\mathcal{P}_{\\!\bm{\alpha}}E^{p,q}$ where
$\mathcal{P}_{\\!\bm{\alpha}}E^{p,q}$ is the prolongation of the Hodge bundles
$E^{p,q}:=F^{p}/F^{p+1}$ in terms of the norm growth of the Hodge metric (see
§ 1.6 for the definition).

Based on Theorem A, we can generalize main parts of Schmid’s nilpotent orbit
theorem to complex polarized variation of Hodge structure.

######  Theorem B.

Let $(V,\nabla,F^{\bullet},Q)$ be a complex polarized variation of Hodge
structure on $(\Delta^{*})^{p}\times\Delta^{q}$. Denote by
$\Phi:\mathbb{H}^{p}\times\Delta^{q}\to\mathscr{D}$ its period mapping, where
$\mathscr{D}$ is the period domain and $\mathbb{H}=\\{z\in\mathbb{C}\mid\Re
z<0\\}$. Let us denote by $2\pi iR_{i}$ is the logarithm of the monodromy
operator associated to the counter-clockwise generator of the fundamental
group of the $i$-th copy of $\Delta^{*}$ in $(\Delta^{*})^{p}$, whose
eigenvalues lie in $(2\pi i(\alpha_{i}-1),2\pi i\alpha_{i}]$ for some
$\bm{\alpha}\in\mathbb{R}^{p}$. Then for the holomorphic mapping
$\Psi:(\Delta^{*})^{p}\times\Delta^{q}\to\check{\mathscr{D}}$ induced by
$\tilde{\Psi}:=\exp(\sum_{i=1}^{p}z_{i}R_{i})\circ\Phi(z,w)$,

  1. (i)

$\Psi$ extends holomorphically to $\Delta^{p+q}$;

  2. (ii)

the holomorphic mapping

$\displaystyle\vartheta:\mathbb{H}^{p}\times\Delta^{q}$
$\displaystyle\to\check{\mathscr{D}}$ $\displaystyle(z,w)$
$\displaystyle\mapsto\exp(-\sum_{i=1}^{p}z_{i}R_{i})\circ a(w)$

is horizontal, where $a(w):=\Psi(0,w)$, and $\check{\mathscr{D}}$ is the
compact dual of the period domain $\mathscr{D}$.

  3. (iii)

In the one variable case, $\exp(-zR)\circ a$ lies in $\mathscr{D}$ when $\Re
z\leq-C$ for some $C>0$. Moreover, we have the distance estimate

$d_{{\mathscr{D}}}(\exp(-zR)\circ a,\Phi(z))\leq C^{\prime}|\Re
z|^{\beta}e^{\delta\Re z}\quad\mbox{for some}\quad C^{\prime},\delta,\beta>0$

if $\Re z\leq-C$.

When $(V,\nabla)$ has quasi-unipotent monodromies around $D$, Theorems A and B
are contained in Schmid’s nilpotent orbit theorem [Sch73]. Under this
monodromy assumption he proved B.(iii) for the case of several variables.

Theorems A and B were also proved by Sabbah and Schnell [SS22] for the case of
one variable in a different way. Their methods can be extended to prove the
general case.

Our proof of Theorem A is based on Mochizuki’s work on the prolongation of
acceptable bundles [Moc11] and methods in $L^{2}$-estimates. The proof of
B.(ii) and B.(iii) essentially follows Schmid’s method in [Sch73].

We conclude the introduction by explaining applications of the main result in
this paper. One application is related to the work [Wu22] on the injectivity
and vanishing theorem for $\mathbb{R}$-Hodge modules. We remark that in [Wu22,
Lemma 2.10], nilpotent orbit theorem for real variation of Hodge structures
was claimed. It seems to the author that the proof needs some amplification,
and as such, Theorem A complements the proof presented therein.

Another potential further application is on the complex Hodge modules. It is
well-established that Saito’s theory of mixed Hodge modules fundamentally
relies on the nilpotent orbit theorem. Currently, Sabbah and Schnell are
developing the theory of mixed Hodge modules for complex Hodge structures. We
remark that the nilpotent orbit theorem established in this paper should serve
as a foundational component in their work.

##  1\. Preliminary

###  1.1. Complex polarized variations of Hodge structure

Let us briefly recall the definition of _polarized Hodge structure_. We refer
the readers to [Sim92, SS22] for more details. A Hodge structure of weight $w$
on a complex vector space $V$ is a decomposition

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

and a _polarizatio_ n is a hermitian pairing
$Q:V\otimes_{\mathbb{C}}\bar{V}\rightarrow\mathbb{C}$ such that the above
decomposition is orthogonal with respect to $Q$, and such that $(-1)^{q}Q$ is
positive definite on the subspace $V^{p,q}$. For any $v\in V$, its _Hodge
norm_ is defined

$|v|^{2}=\sum_{p+q=w}(-1)^{q}Q\left(v^{p,q},v^{p,q}\right),$

where $v^{p,q}$ is the $(p,q)$-component of $v$ in the Hodge decomposition
$V=\bigoplus_{p+q=w}V^{p,q}$. Such a polarized Hodge structure induces the
_Hodge filtration_ $F^{\bullet}V$ of $V$ defined by

$F^{p}V=\bigoplus_{i\geq p}V^{i,w-i}.$

As introduced by [Sim92], a complex polarized variation of Hodge structure
$(V=\bigoplus_{r+s=w}V^{r,s},\nabla,Q)$ of weight $w$ on a complex manifold
$U$ consists of the following data:

  1. (a)

a smooth vector bundle $V$ with a Hodge decomposition
$V=\bigoplus_{r+s=w}V^{r,s}$;

  2. (b)

a flat connection $\nabla$ satisfies the Griffiths transversality condition

(1.1.1) $\displaystyle\nabla:{V}^{r,s}\rightarrow
A^{0,1}\left(V^{r+1,s+1}\right)\oplus A^{1,0}\left(V^{r,s}\right)\oplus
A^{0,1}\left(V^{r,s}\right)\oplus{A}^{1,0}\left(V^{r-1,s+1}\right)$

  3. (c)

a parallel Hermitian form $Q$ which makes the Hodge decomposition orthogonal
and which on $\mathrm{V}^{r,s}$ is positive definite if $r$ is even and
negative definite if $r$ is odd.

We decompose $\nabla=\bar{\theta}+\partial+\bar{\partial}+\theta$ according to
the above transversality condition in (1.1.1). The most important component of
the connection turns out to be the _Higgs field_ , which is the linear
operator

$\theta:V^{r,s}\to A^{1,0}(V^{r-1,s+1}).$

We decompose $\nabla=\nabla^{\prime}+\nabla^{\prime\prime}$ into its
(1,0)-component $\nabla^{\prime}:A^{0}(V)\rightarrow A^{1,0}(V)$ and its
$(0,1)$-component $\nabla^{\prime\prime}:A^{0}(V)\rightarrow A^{0,1}(V)$. Then
$\nabla^{\prime\prime}$ gives $V$ the structure of a holomorphic vector
bundle, which we denote by the symbol $\mathcal{V}$; and $\nabla^{\prime}$
defines an integrable holomorphic connection
$\nabla^{\prime}:\mathcal{V}\rightarrow\Omega_{U}^{1}\otimes_{\mathscr{O}_{U}}\mathcal{V}$
on this bundle. The condition on $\nabla$ in (1.1.1) is saying that the Hodge
filtration

$F^{p}V=\bigoplus_{i\geq p}V^{i,w-i}$

is a holomorphic subbundles of $\mathcal{V}$, and that the connection
$\nabla^{\prime}$ satisfies Griffiths’ transversality relation

$\nabla^{\prime}\left(F^{p}\mathcal{V}\right)\subseteq\Omega_{U}^{1}\otimes_{\mathscr{O}_{U}}F^{p-1}\mathcal{V}.$

Note that $(V^{p,q},\bar{\partial})$ defines a holomorphic vector bundle,
denoted by $E^{p,q}$. From the above point of view, the Higgs field is simply
the holomorphic operator

$\theta:E^{p,q}\rightarrow\Omega_{U}^{1}\otimes_{\mathscr{O}_{U}}E^{p-1,q+1}.$

Denote by $(E,\theta)=(\oplus_{p+q=w}E^{p,q},\theta)$, which is called the
_system of Hodge bundles_ relative to the $\mathbb{C}$-VHS
$(V=\oplus_{p+q=w}V^{p,q},\nabla,Q)$. Let us define $h_{p,q}:=(-1)^{p}Q$,
which is a hermitian metric for $E^{p,q}$ by Item (c). For the hermitian
metric $h=\oplus_{p+q=w}h_{p,q}$ of $E$, the component
$\bar{\theta}:V^{r,s}\to A^{0,1}(V^{r+1,s-1})$ is the adjoint of $\theta$ with
respect to $h$. Since the primary objective of this paper is to establish the
nilpotent orbit theorem for $\mathbb{C}$-VHS, we will adopt the notation
$(V,\nabla,F^{\bullet}V,Q)$ to represent the $\mathbb{C}$-VHS, as the Hodge
filtration $F^{\bullet}V$ plays a central role in the nilpotent orbit theorem.

###  1.2. Deligne extension

Let $X$ be a complex manifold and let $D$ be a simple normal crossing divisor
on $X$. For the flat bundle $(V,\nabla)$ defined on $U:=X\backslash D$,
Deligne introduced a way to extend it across $D$. We recall this construction
briefly and refer the readers to [Sch73, Moc07, SS22] for more details. For
any point $x\in D$, we choose an admissible coordinate
$(\Omega;z_{1},\ldots,z_{n})$ such that $\Omega\simeq\Delta^{n}$ and
$D\cap\Omega=(z_{1}\ldots z_{p}=0)$ (see Definition 1.1 for the definition).
Write $q=n-p$. The fundamental group
$\pi_{1}\big{(}(\Delta^{*})^{p}\times\Delta^{q}\big{)}$ is generated by
elements $\gamma_{1},\ldots,\gamma_{p}$, where $\gamma_{j}$ may be identified
with the counter-clockwise generator of the fundamental group of the $j$-th
copy of $\Delta^{*}$ in $(\Delta^{*})^{p}$. We denote by $V^{\nabla}$ the
space of multivalued flat sections of $(V,\nabla)$, which is a finite
dimensional $\mathbb{C}$-vector space. Set $T_{j}$ to be the monodromy
transformation with respect to $\gamma_{j}$, which pairwise commute and are
endomorphisms of $V^{\nabla}$; that is, for any multivalued section
$v(t_{1},\ldots,t_{p+q})\in V^{\nabla}$, one has

$v(t_{1},\ldots,e^{2\pi
i}t_{j},\ldots,t_{p+q})=(T_{j}v)(t_{1},\ldots,t_{p+q})$

and $[T_{j},T_{k}]=0$ for any $j,k=1,\ldots,p$. Let us write $Sp(T_{j})$ the
set of eigenvalues of $T_{j}$, and for any $\lambda_{j}\in Sp(T_{j})$, we
denote by $\mathbb{E}(T_{j},\lambda_{j})\subset V^{\nabla}$ the corresponding
eigenspace. We know that all $\lambda_{j}\in Sp(T_{j})$ has norm $1$ (see
_e.g._ [SS22]). Write $Sp:=\prod_{i=1}^{p}Sp(T_{j})$. For
$\bm{\lambda}=(\lambda_{1},\ldots,\lambda_{p})$, we define

$\mathbb{E}_{\bm{\lambda}}:=\cap_{j=1}^{p}\mathbb{E}(T_{j},\lambda_{j}).$

Since $T_{j}$ pairwise commute, one has

$V^{\nabla}=\oplus_{\bm{\lambda}\in Sp}\mathbb{E}_{\bm{\lambda}},$

and $\mathbb{E}_{\bm{\lambda}}$ is an invariant subspace of $T_{j}$ for any
$\bm{\lambda}\in Sp$ and any $j$.

Let us fix a $p$-tuple
$\bm{\alpha}:=(\alpha_{1},\ldots,\alpha_{p})\in\mathbb{R}^{p}$. Then for
$\bm{\lambda}\in Sp$, there exists a unique
$\beta_{i}\in(\alpha_{i}-1,\alpha_{i}]$ such that $\exp(2\pi
i\beta_{i})=\lambda_{i}$. Since
$\lambda_{i}^{-1}T_{i}|_{\mathbb{E}_{\bm{\lambda}}}$ is unipotent, its
logarithm can be defined as

$\log(\lambda_{i}^{-1}T_{i}|_{\mathbb{E}_{\bm{\lambda}}}):=\sum_{k=1}^{\infty}(-1)^{k+1}\frac{(\lambda_{i}^{-1}T_{i}|_{\mathbb{E}_{\bm{\lambda}}}-I)^{k}}{k}.$

We denote
$N_{i}:=\frac{\log(\lambda_{i}^{-1}T_{i}|_{\mathbb{E}_{\bm{\lambda}}})}{2\pi
i}$. Then for any $v\in\mathbb{E}_{\bm{\lambda}}$, we define

(1.2.1)
$\displaystyle\tilde{v}(t):=\exp\big{(}-\sum_{i=1}^{p}(\beta_{i}I+N_{i})\cdot\log
t_{i}\big{)}v(t)=\prod_{i=1}^{p}t_{i}^{-\beta_{i}}\exp\big{(}-\sum_{i=1}^{p}N_{i}\cdot\log
t_{i}\big{)}v(t).$

One can check that $\tilde{v}$ is single valued, and that
$\nabla^{0,1}\tilde{v}=0$. We now fix a basis $v_{1},\ldots,v_{r}$ of
$V^{\nabla}$ such that each $v_{i}$ belongs to some
$\mathbb{E}_{\bm{\lambda}}$. Then the holomorphic sections
$\tilde{v}_{1},\ldots,\tilde{v}_{r}$ of $\mathcal{V}$ defines a prolongation
of $\mathcal{V}$ over $X$ which we denoted by $V^{\rm Del}_{\bm{\alpha}}$. One
can check that this construction does not depend on our choice of the basis.
This is called the _Deligne extension_ of the flat bundle $(V,\nabla)$ with
the eigenvalues of the residue of $\nabla$ over $D_{i}$ lying in
$[-\alpha_{i},-\alpha_{i}+1)$. Note that it is defined for any flat bundle
$(V,\nabla)$ (not necessarily complex variation of Hodge structure).

If $(V,\nabla)$ underlies a complex polarized variation of Hodge structure
$(V,\nabla,F^{\bullet}V,Q)$, we define $F^{p}_{\bm{\alpha}}:=j_{*}F^{p}\cap
V_{\bm{\alpha}}^{\rm Del}.$ It is called the extension of the Hodge
filtration. It should be noted that, a priori, we do not know whether
$F^{p}_{\bm{\alpha}}$ is locally free.

###  1.3. Acceptable bundles

######  Definition 1.1.

(Admissible coordinate) Let $X$ be a complex manifold and let $D$ be a simple
normal crossing divisor. Let $x$ be a point of $X$, and assume that
$\\{D_{j}\\}_{j=1,\ldots,\ell}$ are the components of $D$ containing $p$. An
_admissible coordinate_ around $x$ is the tuple
$(\Omega;z_{1},\ldots,z_{n};\varphi)$ (or simply $(\Omega;z_{1},\ldots,z_{n})$
if no confusion arises) where

  * •

$\Omega$ is an open subset of $X$ containing $x$.

  * •

there is a holomorphic isomorphism $\varphi:\Omega\to\Delta^{n}$ such that
$\varphi(D_{j})=(z_{j}=0)$ for any $j=1,\ldots,\ell$.

We shall write $\Omega^{*}:=\Omega-D$,
$\Omega(r):=\\{z\in\Omega\mid|z_{i}|<r,\,\forall i=1,\ldots,n\\}$ and
$\Omega^{*}(r):=\Omega(r)\cap\Omega^{*}$.

We define a (incomplete) Poincaré-type metric $\omega_{P}$ on
$(\Delta^{*})^{\ell}\times\Delta^{n-\ell}$ by

(1.3.1) $\displaystyle\omega_{P}=\sum_{j=1}^{\ell}\frac{\sqrt{-1}dz_{j}\wedge
d\bar{z}_{j}}{|z_{j}|^{2}(\log|z_{j}|^{2})^{2}}+\sum_{k=\ell+1}^{n}\sqrt{-1}dz_{k}\wedge
d\bar{z}_{k}.$

Note that

$\omega_{P}=i\partial\bar{\partial}\log\big{(}\prod_{j=1}^{\ell}(-\log|z_{j}|^{2})^{-1}\cdot\prod_{k=\ell+1}^{n}\exp(|z_{k}|^{2})\big{)}.$

For any system of Hodge bundles $(E,\theta,h)$, we have the following crucial
norm estimate for its Higgs field $\theta$. The one dimensional case is due to
Simpson [Sim90, Theorem 1] and the general case was proved by Mochizuki in
[Moc02, Proposition 4.1]. Its proof relies on a clever use of Ahlfors-Schwarz
lemma.

######  Theorem 1.2.

Let $(E,\theta,h)$ be a system of Hodge bundle on $X\backslash D$. Then for
any point $x\in D$, it has an admissible coordinate
$(\Omega;z_{1},\ldots,z_{n})$ such that the norm $|\theta|_{h,\omega_{P}}\leq
C$ holds over $\Omega^{*}$ for some constant $C>0$. Here
$|\theta|_{h,\omega_{P}}$ denotes the norm of $\theta$ with respect to $h$ and
$\omega_{P}$. ∎

Here we also recall the following definition in [Moc07, Definition 2.7].

######  Definition 1.3 (Acceptable bundle).

Let $(E,h)$ be a hermitian vector bundle over $X\backslash D$. We say that
$(E,h)$ is an acceptable at $p\in D$, if the following holds: there is an
admissible coordinate $(\Omega;z_{1},\ldots,z_{n})$ around $p$, such that the
norm $\lvert R(E,h)\rvert_{h,\omega_{P}}\leq C$ for some $C>0$. Here $R(E,h)$
is the Chern curvature of $(E,h)$. When $(E_{,}h)$ is acceptable at any point
$p$ of $D$, it is called acceptable.

Hodge filtrations and Hodge bundles endowed with the Hodge metric are all
acceptable.

######  Lemma 1.4.

Let $(V,\nabla,F^{\bullet},Q)$ be a complex polarized variation of Hodge
structure of weight $m$ on $X\backslash D$. Let $h$ (resp. $h_{p,q}$) be the
hermitian metric on $V$ (resp. on $E^{p,q}$) introduced in § 1.1. Consider the
induced hermitian metric $h_{p}:=h|_{F^{p}}$ on the Hodge filtration $F^{p}$.
Then both $(F^{p},h_{p})$ and $(E^{p,q},h_{p,q})$ are acceptable bundles.

###### Proof.

We write $\theta_{p,q}:=\theta|_{E^{p,q}}$ and let
$\theta_{p,q}^{\dagger}:E^{p-1,q+1}\to A^{0,1}(E^{p,q})$ be its adjoint with
respect to $h_{p,q}$. For the hermitian bundle $(F^{p},h_{p})$, its curvature
is

$R_{h_{p}}(F^{p})=-2\theta_{m,0}^{\dagger}\wedge\theta_{m,0}+2\sum_{i=1}^{m-p}(-\theta_{m-i,i}^{\dagger}\wedge\theta_{m-i,i}-\theta_{m-i+1,i-1}\wedge\theta_{m-i+1,i-1}^{\dagger})+\theta_{p,m-p}^{\dagger}\wedge\theta_{p,m-p}.$

The curvature of the bundle $(E^{p,q},h_{p,q})$ is

$R_{h_{p,q}}(E_{p,q})=-\theta_{p,q}^{\dagger}\wedge\theta_{p,q}-\theta_{p+1,q-1}\wedge\theta_{p+1,q-1}^{\dagger}.$

By Theorem 1.2, for any point $x\in D$ there is an admissible coordinate
$(\Omega;z_{1},\ldots,z_{n})$ around $x$ such that the norm

$\sum_{p+q=m}|\theta_{p,q}|_{h_{p,q},\omega_{P}}=|\theta|_{h,\omega_{P}}\leq
C$

holds over $\Omega^{*}$ for some constant $C>0$. Since
$\theta_{p,q}^{\dagger}$ is the adjoint of $\theta_{p,q}$ with respect to
$h_{p,q}$, one has $|\theta^{\dagger}_{p,q}|_{h,\omega_{P}}\leq C$ for any
$p$. It follows that $|R_{h_{p}}(F^{p})|_{h_{p,q},\omega_{P}}\leq C^{\prime}$
and $|R_{h_{p,q}}(E_{p,q})|_{h_{p,q},\omega_{P}}\leq C^{\prime}$ for some
$C^{\prime}>0$. Hence $(F^{p},h_{p})$ and $(E^{p,q},h_{p,q})$ are both
acceptable. ∎

###  1.4. Adapted to log order

We recall some notions in [Moc07, §2.2.2]. Let $X$ be a complex manifold, $D$
be a simple normal crossing divisor on $X$, and $E$ be a holomorphic vector
bundle on $X\backslash D$ such that $E|_{X\backslash D}$ is equipped with a
hermitian metric $h$. Let $\bm{v}=(v_{1},\ldots,v_{r})$ be a smooth frame of
$E|_{X\backslash D}$. We obtain the $H(r)$-valued function $H(h,\bm{v})$
defined over $X\backslash D$,whose $(i,j)$-component is given by
$h(v_{i},v_{j})$.

Let us consider the case $X=\mathbb{D}^{n}$, and $D=\sum_{i=1}^{\ell}D_{i}$
with $D_{i}=(z_{i}=0)$. We have the coordinate $(z_{1},\ldots,z_{n})$. Let
$h$, $E$ and $\bm{v}$ be as above.

######  Definition 1.5.

A smooth frame $\bm{v}$ on $X\backslash D$ is called _adapted up to log order_
, if the following inequalities hold over $X\backslash D$:

$C^{-1}(-\sum_{i=1}^{\ell}\log|z_{i}|)^{-M}\leq H(h,\bm{v})\leq
C(-\sum_{i=1}^{\ell}\log|z_{i}|)^{M}$

for some positive numbers $M$ and $C$.

###  1.5. Parabolic vector bundles

In this subsection, we recall the notions of parabolic (vector) bundles. For
more details we refer to [Moc06]. Let $X$ be a complex manifold,
$D=\sum_{i=1}^{\ell}D_{i}$ be a reduced simple normal crossing divisor,
$U=X\backslash D$ be the complement of $D$ and $j:U\to X$ be the inclusion.

######  Definition 1.6.

A parabolic bundle $\mathcal{P}_{*}E$ on $(X,D)$ is a holomorphic vector
bundle $E$ on $U$, together with an $\mathbb{R}^{\ell}$-indexed filtration
$\mathcal{P}_{\\!\bm{\alpha}}E$ (parabolic structure) by locally free
subsheaves of $j_{*}E$ such that

  1. (i)

$\bm{\alpha}\in\mathbb{R}^{\ell}$ and $\mathcal{P}_{\\!\bm{\alpha}}E|_{U}=E$.

  2. (ii)

$\mathcal{P}_{\\!\bm{\alpha}}E\subset\mathcal{P}_{\\!\bm{\beta}}E$ if
$\alpha_{i}\leq\beta_{i}$ for all $i$.

  3. (iii)

For $1\leq i\leq\ell$,
$\mathcal{P}_{\bm{\alpha}+\bm{1}_{i}}E=\mathcal{P}_{\\!\bm{\alpha}}E\otimes\mathcal{O}_{X}(D_{i})$,
where $\bm{1}_{i}=(0,\ldots,1,\ldots,0)$ with $1$ in the $i$-th component.

  4. (iv)

$\mathcal{P}_{\bm{\alpha}+\bm{\varepsilon}}E=\mathcal{P}_{\\!\bm{\alpha}}E$
for any vector $\bm{\epsilon}=(\epsilon,\ldots,\epsilon)$ with $0<\epsilon\ll
1$.

  5. (v)

The set of weights
$\\{\bm{\alpha}\mid\mathcal{P}_{\\!\bm{\alpha}}E/\mathcal{P}_{\\!<\bm{\alpha}}E\\}\not=0$
is discrete in $\mathbb{R}^{\ell}$.

###  1.6. Prolongation via norm growth

Let $X$ be a complex manifold, $D=\sum_{i=1}^{\ell}D_{i}$ be a simple normal
crossing divisor, $U=X\backslash D$ be the complement of $D$ and $j:U\to X$ be
the inclusion. Let $(E,h)$ be a hermitian vector bundle on $U$. For any
$\bm{\alpha}=(a_{1},\ldots,a_{\ell})\in\mathbb{R}^{\ell}$, we can prolong $E$
over $X$ by a sheaf of $\mathcal{O}_{X}$-module
$\mathcal{P}_{\\!\bm{\alpha}}E$ as follows:

$\displaystyle\mathcal{P}_{\\!\bm{\alpha}}E(U)=\\{\sigma\in\Gamma(U\backslash
D,E|_{U\backslash
D})\mid|\sigma|_{h}\lesssim{\prod_{i=1}^{\ell}|z_{i}|^{-\alpha_{i}-\varepsilon}}\
\ \mbox{for all}\ \varepsilon>0\\}.$

In [Moc11, Theorem 21.3.1] Mochizuki proved that the prolongation of
acceptable bundles defined above are parabolic bundles.

######  Theorem 1.7 (Mochizuki).

Let $(E,h)$ be an acceptable bundle over $X\backslash D$. Then
$\mathcal{P}_{*}E$ defined above is a parabolic bundle. ∎

###  1.7. Period domain and period mapping

In this subsection we quickly review the definitions of period domain and
period mapping. We refer the readers to [CMP17, KKM11, SS22] for more details.

Let $(V=\oplus_{p+q=m}V^{p,q},Q)$ be a polarized complex Hodge structure of
weight $m$ defined in § 1.1. Recall that the Hodge filtration is defined to be
$F^{p}:=\oplus_{i\geq p}V^{i,m-i}$. After fixing $m$ and
$\dim_{\mathbb{C}}F^{p}$, the set of all such filtration $F^{\bullet}$ is a
complex flag manifold, which is denoted by $\check{\mathscr{D}}$. It is a
closed submanifold of a product of Grassmannians, and is thus a projective
manifold. The subset $\mathscr{D}$ of all complex polarized Hodge structures
are charcterized by

  1. (a)

$F^{p}=F^{p}\cap(F^{p+1})^{\perp}\oplus F^{p+1}$.

  2. (b)

$(-1)^{p}Q$ is positive definite over $F^{p}\cap(F^{p+1})^{\perp}$.

It is an open submanifold of $\check{\mathscr{D}}$. We usually write $F$
instead of $F^{\bullet}$ to lighten the notation. Since the groups ${\rm
GL}(V)$ and $G:=U(V,Q)$ act transitively on $\check{\mathscr{D}}$ and
$\mathscr{D}$ respectively, $\check{\mathscr{D}}$ and $\mathscr{D}$ are thus
homogeneous spaces.

For any Hodge structure $F\in\check{\mathscr{D}}$, the holomorphic tangent
space $T_{\check{\mathscr{D}},F}$ of $\check{\mathscr{D}}$ at $F$ is
identified with

$\displaystyle{\rm\small End}(V)/\\{A\in{\rm\small End}(V)\mid A(F^{p})\subset
F^{p}\ \mbox{for all}\ p\\}.$

For any $A\in{\rm\small End}(V)$, we denote by $[A]_{F}$ its image in
$T_{\check{\mathscr{D}},F}$.

A tangent vector $[A]_{F}$ in $T_{\check{\mathscr{D}},F}$ is called
_horizontal_ if $A(F^{p})\subset F^{p-1}$ for all $p$. The subbundle of
$T_{\check{\mathscr{D}}}$ consisting of horizontal vectors is denoted by
$T_{\check{\mathscr{D}}}^{-1,1}$ and one can show that it is a holomorphic
subbundle of $T_{\check{\mathscr{D}}}$. A holomorphic map
$f:\Omega\to\check{\mathscr{D}}$ from a complex manifold $\Omega$ is called
_horizontal_ if $df:T_{\Omega}\to f^{*}T_{\check{\mathscr{D}}}$ factors
through $f^{*}T_{\check{\mathscr{D}}}^{-1,1}$.

A complex (unpolarized) variation of Hodge structure
$(V=\oplus_{p+q=m},\nabla)$ over a complex manifold $\Omega$ induces a
horizontal holomorphic map $\Phi:\tilde{\Omega}\to\check{\mathscr{D}}$ by the
Griffiths transversality, where $\tilde{\Omega}$ is the universal cover of
$\Omega$. Here we choose the reference space of $\check{\mathscr{D}}$ to be
the space of multivalued flat sections $V^{\nabla}$. $\Phi$ is called the
period mapping associated to $(V,\nabla,F^{\bullet})$. When this complex
variation of Hodge structure is moreover polarized, $\Phi$ factors through
$\mathscr{D}$.

##  2\. Nilpotent orbit theorem

###  2.1. Two results of $L^{2}$-estimate

Set $X=\Delta^{n}$ and $D=(z_{1}\cdots z_{\ell}=0)$. We equip the complement
$U:=X\backslash D$ with the Poincaré metric $\omega_{P}$ defined in (1.3.1).
Write

(2.1.1) $\displaystyle X(r):=\\{z\in X\mid|z_{i}|<r\ \mbox{for}\
i=1,\ldots,\ell\\}\quad\mbox{and}\quad U(r)=X(r)\cap U.$

######  Lemma 2.1.

Let $(F,h_{F})$ be a hermitian vector bundle on $U$ such that
$|R_{h_{F}}(F)|\leq C\omega_{P}$ for some constant $C>0$. Then for any section
$\eta\in\mathscr{C}^{\infty}(U,\Lambda^{0,1}T_{U}^{*}\otimes F)$ such that
$|\eta|_{h_{F},\omega_{P}}\lesssim\prod_{j=1}^{\ell}|z_{j}|^{\varepsilon}$ for
some $\varepsilon>0$, and $\bar{\partial}\eta=0$, there exists
$\sigma\in\mathscr{C}^{\infty}(U,F)$ such that $\bar{\partial}\sigma=\eta$ and

$\displaystyle\int_{U}|\sigma|^{2}_{h_{F}}\prod_{j=1}^{\ell}(-\log|z_{j}|^{2})^{N}d\mbox{vol}_{\omega_{P}}<\infty$

for some $N\gg 1$.

###### Proof.

For the line bundle $K_{U}^{-1}$ endowed with the natural metric $g$ induced
by $\omega_{P}$, it is acceptable. Hence for the hermitian vector bundle
$(E,h):=(K_{U}^{-1}\otimes F,g\cdot h_{F})$, it is also acceptable. It follows
from [DH19, Lemma 1.10] that one can choose $N\gg 1$ such that

$iR_{h}(E)\geq_{Nak}-(N-1)\omega_{P}\otimes{\rm Id}_{E},$

where “$\geq_{Nak}$" stands for Nakano semipositive (see [Dem82, Définition
2.2]). For the function

(2.1.2)
$\displaystyle\varphi:=\log\big{(}\prod_{j=1}^{\ell}(-\log|z_{j}|^{2})^{-1}\cdot\prod_{k=\ell+1}^{n}\exp(|z_{k}|^{2})\big{)},$

one has $i\partial\bar{\partial}\log\varphi=\omega_{P}.$ For any
$k\in\mathbb{Z}$ we define a new metric $h(k)=h\cdot e^{-k\varphi}$ for $E$.
Therefore,

$iR_{h(N)}(E)=iR_{h}(E)+N\omega_{P}\otimes{\rm
Id}_{E}\geq_{Nak}\omega_{P}\otimes{\rm Id}_{E}.$

Note that $\mathscr{C}^{\infty}(U,\Lambda^{n,1}T_{U}^{*}\otimes
E)=\mathscr{C}^{\infty}(U,\Lambda^{0,1}T_{U}^{*}\otimes F)$ with
$|\eta|_{h,\omega_{P}}=|\eta|_{h_{F},\omega_{P}}$. Since
$|\eta|_{h_{F},\omega_{P}}\lesssim\prod_{j=1}^{\ell}|z_{j}|^{\varepsilon}$,
$|\eta|_{h(N),\omega_{P}}\leq C^{\prime}$ for some $C^{\prime}>0$. Hence
$\lVert\eta\rVert_{h(N),\omega_{P}}<\infty$. Thanks to the Demailly-Hörmander
$L^{2}$-estimate [Dem82, Théorème 4.1 and Remarque 4.2], there exists
$\sigma\in\mathscr{C}^{\infty}(U,K_{U}\otimes E)=\mathscr{C}^{\infty}(U,F)$
such that $\bar{\partial}\sigma=\eta$ and $\lVert\sigma\rVert_{h(N)}<\infty$.
Here we note that the smoothness of $\sigma$ follows from the elliptic
regularity of the Laplacian. The lemma is proved. ∎

######  Lemma 2.2.

Let $(E,h)$ be a hermitian vector bundle on $U$ such that $|R_{h}(E)|\leq
C\omega_{P}$ for some constant $C>0$. Assume that $\sigma\in H^{0}(U,E)$ such
that $\lVert\sigma\rVert_{h(N)}<\infty$ for some integer $N\geq 1$, where
$h(N):=h\cdot e^{-N\varphi}$ with $\varphi$ defined in (2.1.2), then over
$U(\frac{1}{2})$,
$|\sigma|_{h}\lesssim\prod_{j=1}^{\ell}|z_{j}|^{-\varepsilon}$ for any
$\varepsilon>0$ .

###### Proof.

Since $|R_{h}(E)|\leq C\omega_{P}$ for some constant $C>0$, it follows from
[DH19, Lemma 1.10] that $(E,h(-N^{\prime}))$ is Griffiths semi-negative for
some $N^{\prime}\gg 1$, where $h(-N^{\prime}):=h\cdot e^{N^{\prime}\varphi}$
with $\varphi$ defined in (2.1.2). One can show that
$\log|\sigma|^{2}_{h(-N^{\prime})}$ is a plurisubharmonic function. For any
$z\in U^{*}(\frac{1}{2})$, one has

$\displaystyle\log|\sigma(z)|^{2}_{h(-N^{\prime})}$
$\displaystyle\leqslant\frac{4^{n}}{\pi^{n}\prod_{i=1}^{\ell}|z_{i}|^{2}}\int_{\Omega_{z}}\log|\sigma(w)|^{2}_{h(-N^{\prime})}d\mbox{vol}_{g}$
$\displaystyle\leqslant\log\big{(}\frac{4^{n}}{\pi^{n}\prod_{i=1}^{\ell}|z_{i}|^{2}}\cdot\int_{\Omega_{z}}|\sigma(w)|^{2}_{h(-N^{\prime})}d\mbox{vol}_{g}\big{)}$
$\displaystyle\leqslant\log\big{(}C\int_{\Omega_{z}}\frac{1}{\prod_{i=1}^{\ell}|w_{i}|^{2}}|\sigma(w)|^{2}_{h(-N^{\prime})}d\mbox{vol}_{g}\big{)}$
$\displaystyle\leqslant
C_{1}+\log\int_{\Omega_{z}}|\sigma(w)|^{2}_{h(-N^{\prime})}\cdot|\prod_{i=1}^{\ell}(\log|w_{i}|^{2})^{2}|d\mbox{vol}_{\omega_{P}}$
$\displaystyle\leqslant
C_{2}+\log\int_{\Omega_{z}}|\sigma(w)|^{2}_{h(N)}d\mbox{vol}_{\omega_{P}}$
$\displaystyle\leqslant C_{2}+\log\lVert\sigma\rVert_{h(N)}^{2}$

where $\Omega_{z}:=\\{w\in U^{*}\mid|w_{i}-z_{i}|\leq\frac{|z_{i}|}{2}\mbox{
for }i\leq\ell;|w_{i}-z_{i}|\leq\frac{1}{2}\mbox{ for }i>\ell\\}$ and $g$ is
the Euclidean metric. $C_{1},C_{2}$ are two positive constants which do not
depend on $z\in U^{*}(\frac{1}{2})$. The first inequality is due to the mean
value inequality, and the second one follows from the Jensen inequality. It
follows that

$\displaystyle|\sigma(z)|_{h}$
$\displaystyle=|\sigma(z)|_{h(-N^{\prime})}\cdot\big{(}\prod_{j=1}^{\ell}(-\log|z_{j}|^{2})^{\frac{N^{\prime}}{2}}\cdot\prod_{k=\ell+1}^{n}\exp(|z_{k}|^{2})^{-\frac{N^{\prime}}{2}}\big{)}$
$\displaystyle\leq\exp(\frac{C_{2}}{2})\cdot\lVert\sigma\rVert_{h(N)}\cdot\big{(}\prod_{j=1}^{\ell}(-\log|z_{j}|^{2})^{\frac{N^{\prime}}{2}}\cdot\prod_{k=\ell+1}^{n}\exp(|z_{k}|^{2})^{-\frac{N^{\prime}}{2}}\big{)}\lesssim\prod_{i=1}^{\ell}|z_{i}|^{-\varepsilon}$

for any $\varepsilon>0$. ∎

###  2.2. Proof of Theorem A

We first prove that the Deligne extension of the flat bundle unerlying a
complex variation of Hodge structure coincides with the prolongation defined
in § 1.6.

######  Proposition 2.3.

Let $X$ be a complex manifold, $D=\sum_{i=1}^{\ell}D_{i}$ be a simple normal
crossing divisor. For a complex variation of Hodge structure
$(V,\nabla,F^{\bullet},h)$ defined on $U:=X\backslash D$, one has
$V_{\bm{\alpha}}^{\rm Del}=\mathcal{P}_{\\!\bm{\alpha}}V$ for any multi-index
$\bm{\alpha}\in\mathbb{R}^{\ell}$, where $\mathcal{P}_{\\!\bm{\alpha}}V$ is
the prologation of $V$ defined in § 1.6.

###### Proof.

Step 1: we prove that $V_{\bm{\alpha}}^{\rm
Del}\subset\mathcal{P}_{\\!\bm{\alpha}}V$.  We will use the notation in § 1.2.
Since this is a local problem, we can assume that $X=\Delta^{n}$ and
$D=(t_{1}\cdots t_{p}=0)$. By the construction of $V_{\bm{\alpha}}^{\rm Del}$
one can take a basis $v_{1},\ldots,v_{r}$ of $V^{\nabla}$ with each
$v_{i}\in\mathbb{E}_{\bm{\lambda}(v_{i})}$ for some $\bm{\lambda}(v_{i})\in
Sp$ such that $\\{\tilde{v}_{1},\ldots,\tilde{v}_{r}\\}$ defined in (1.2.1)
forms a basis of $V_{\bm{\alpha}}^{\rm Del}$. It thus suffices to estimate the
norm

$\displaystyle\tilde{v}(t):=\exp\big{(}-\sum_{i=1}^{p}(\beta_{i}I+N_{i})\cdot\log
t_{i}\big{)}v(t)=\prod_{i=1}^{p}t_{i}^{-\beta_{i}}\exp\big{(}-\sum_{i=1}^{p}N_{i}\cdot\log
t_{i}\big{)}v(t)$

for any $\bm{\lambda}$ and $v\in\mathbb{E}_{\bm{\lambda}}$.

By the weaker norm estimate in [Moc07, Lemma 9.31] for general harmonic
bundles, there exists a frame $v_{1},\ldots,v_{r}$ of $V^{\nabla}$ with each
$v_{i}\in\mathbb{E}_{\bm{\lambda}(v_{i})}$ and
$\\{a_{ij}\\}_{i=1,\ldots,r;j=1,\ldots,p}\subset\mathbb{R}$ such that if we
put ${v}^{\prime}_{i}:=v_{i}\cdot\prod_{j=1}^{p}|t_{j}|^{-a_{ij}},$ then for
the multivalued smooth sections
${\bm{v}}^{\prime}=({v}^{\prime}_{1},\ldots,{v}^{\prime}_{r})$, over a given
sector of $U$ one has the norm estimate

(2.2.1) $\displaystyle(\prod_{i=1}^{p}|\log|t_{i}||)^{-M}\lesssim
H(h,{\bm{v}^{\prime}})\lesssim(\prod_{i=1}^{p}|\log|t_{i}||)^{M}$

for some $M>0$. Here $h$ is the Hodge metric and $H(h,{\bm{v}^{\prime}})$ is
the $H(r)$-valued function defined in § 1.4.

Fix some $\\{t_{1},\ldots,t_{p}\\}\subset\Delta^{*}$. For each
$\ell=1,\ldots,p$, over any sector of $\Delta^{*}$, we have

$|\log|t||)^{-M}\lesssim|v_{i}|^{2}(t_{1},\ldots,t_{\ell-1},t,t_{\ell+1},\ldots,t_{n})\lesssim|\log|t||^{M}$

by the Hodge norm estimate in one dimensional case in [SS22, Sim90]. Together
with (2.2.1), it implies that $a_{ij}=0$ for all
${i=1,\ldots,r;j=1,\ldots,\ell}$. Therfore, we conclude that over a given
sector of $U$ one has the norm estimate

(2.2.2) $\displaystyle(\prod_{i=1}^{p}|\log|t_{i}||)^{-M}\lesssim
H(h,{\bm{v}})\lesssim(\prod_{i=1}^{p}|\log|t_{i}||)^{M}.$

Then for any multivalued flat section $v\in V^{\nabla}$, over any given sector
of $U$ one has

$(\prod_{i=1}^{p}|\log|t_{i}||)^{-M}\lesssim|v(t)|_{h}\lesssim(\prod_{i=1}^{p}|\log|t_{i}||)^{M}$

for some $M>0$. Since all $N_{i}$ are nilpotent and pairwise commute,

$\exp\big{(}-\sum_{i=1}^{p}N_{i}\cdot\log
t_{i}\big{)}v(t)=\sum_{i=1}^{p}\sum_{k=0}^{N}\frac{1}{k!}(\log
t_{i})^{k}(N_{i}^{k}v)(t)$

for some integer $N>0$. Note that if $v\in\mathbb{E}_{\bm{\lambda}}$, we also
have $N_{i}^{k}v\in\mathbb{E}_{\bm{\lambda}}$ for any $k\geq 0$. Since one can
cover $X\backslash D$ by finitely many sectors, this proves the norm estimate

$|\exp\big{(}-\sum_{i=1}^{p}N_{i}\cdot\log
t_{i}\big{)}v(t)|_{h}\lesssim(\prod_{i=1}^{p}|\log|t_{i}||)^{M^{\prime}}$

for some $M^{\prime}>0$. Hence

$|\tilde{v}(t)|_{h}\lesssim\prod_{i=1}^{p}|t_{i}|^{-\beta_{i}-\varepsilon}\lesssim\prod_{i=1}^{p}|t_{i}|^{-\alpha_{i}-\varepsilon}$

for any $\varepsilon>0.$ This proves the inclusion $V_{\bm{\alpha}}^{\rm
Del}\subset\mathcal{P}_{\\!\bm{\alpha}}V$ by the definition of
$\mathcal{P}_{\\!\bm{\alpha}}V$ in § 1.6.

Step 2: we prove that $\mathcal{P}_{\\!\bm{\alpha}}V\subset
V_{\bm{\alpha}}^{\rm Del}$. First we note that the decomposition
$V^{\nabla}=\oplus_{\bm{\lambda}\in Sp}\mathbb{E}_{\bm{\lambda}}$ induces a
decomposition of the flat bundle $(V,\nabla)$ into

(2.2.3) $\displaystyle(V,\nabla)=\oplus_{\bm{\lambda}\in
Sp}(V(\bm{\lambda}),\nabla|_{V(\bm{\lambda})}),$

where $(V(\bm{\lambda}),\nabla|_{V(\bm{\lambda})})$ is the flat subbundle
induced by $\mathbb{E}_{\bm{\lambda}}$. We fix a basis
$(v_{1},\ldots,v_{r})\in V^{\nabla}$ such that
$v_{i}\in\mathbb{E}_{\bm{\lambda}(v_{i})}$ for some $\bm{\lambda}(v_{i})\in
Sp$. This means that such basis is compatible with the above decomposition
(2.2.3); namely $v_{j}$ is a mutivalued flat section of
$(V(\bm{\lambda}(v_{j})),\nabla|_{V(\bm{\lambda}(v_{j}))})$. Consider the dual
bundle $V^{*}$ of $V$, and it can endowed with the natural connection $\nabla$
defined by

$(\nabla\mu)v=d(\mu(v))-\mu(\nabla(v))$

for $\mu$ and $v$ sections in $V^{*}$ and $V$ respectively. $(V^{*},\nabla)$
is thus also a flat bundle. Moreover, $(V^{*})^{\nabla}$ is the dual space of
$(V^{\nabla})$. Consider the dual basis $(v_{1}^{*},\ldots,v^{*}_{r})$ of
$(v_{1},\ldots,v_{r})$. Since

$(\nabla v_{i}^{*})v_{j}=d(v_{i}^{*}(v_{j}))-v_{i}^{*}(\nabla v_{j})=0,$

one has $v_{i}^{*}\in(V^{*})^{\nabla}$. Recall that $T_{j}$ is the monodromy
transformation of $(V,\nabla)$ with respect to $\gamma_{j}$ defined by

$v(t_{1},\ldots,e^{2\pi
i}t_{j},\ldots,t_{p+q})=(T_{j}v)(t_{1},\ldots,t_{p+q})$

for any $v\in V^{\nabla}$. Let us denote by $\tilde{T}_{j}$ the monodromy
transformation of $(V,\nabla)$ with respect to $\gamma_{j}$ defined by

$\mu(t_{1},\ldots,e^{2\pi
i}t_{j},\ldots,t_{p+q})=(\tilde{T}_{j}\mu)(t_{1},\ldots,t_{p+q})$

for any $\mu\in(V^{*})^{\nabla}$. Then for any $v\in V^{\nabla}$ and any
$\mu\in(V^{*})^{\nabla}$ one has

$\displaystyle\mu(t)(v(t))$ $\displaystyle=\mu(t_{1},\ldots,e^{2\pi
i}t_{j},\ldots,t_{p+q})(v(t_{1},\ldots,e^{2\pi i}t_{j},\ldots,t_{p+q}))$
$\displaystyle=(\tilde{T}_{j}\mu(t))(T_{j}v(t))=(\tilde{T}_{j}\mu)(T_{j}v)=(T_{j}^{*}\tilde{T}_{j}\mu)(v),$

where $T_{j}^{*}:(V^{*})^{\nabla}\to(V^{*})^{\nabla}$ is the adjoint of
$T_{j}$. Hence

(2.2.4) $\displaystyle\tilde{T}_{j}=(T_{j}^{*})^{-1}.$

It follows that $Sp(\tilde{T}_{j})=\\{\lambda^{-1}\\}_{\lambda\in Sp(T_{i})}$.
Set $\mathbb{E}(\tilde{T}_{j},\lambda_{j})\subset(V^{*})^{\nabla}$ to be the
corresponding eigenspace of $\lambda_{j}\in Sp(\tilde{T}_{j})$. We know that
all $\lambda_{j}\in Sp(\tilde{T}_{j})$ have norm $1$ since $(V^{*},\nabla)$
admits a complex variation of Hodge structure. For
$\bm{\lambda}=(\lambda_{1},\ldots,\lambda_{p})\in Sp$, we define

$\tilde{\mathbb{E}}_{\bm{\lambda}}:=\cap_{j=1}^{p}\mathbb{E}(\tilde{T}_{j},\lambda^{-1}_{j})\subset(V^{*})^{\nabla}$

Since $T_{j}^{\prime}s$ are pairwise commute, one has

$(V^{*})^{\nabla}=\oplus_{\bm{\lambda}\in
Sp}\tilde{\mathbb{E}}_{\bm{\lambda}},$

and $\tilde{E}_{\bm{\lambda}}$ is an invariant subspace of $\tilde{T}_{j}$ for
any $\bm{\lambda}\in Sp$ and any $j$.

By Lemma 2.4 below, one can show that for any
$\mu\in\tilde{\mathbb{E}}_{\bm{\lambda^{\prime}}}$ and
$v\in\mathbb{E}_{\bm{\lambda}}$, $\mu(v)=0$ if
$\bm{\lambda}\neq\bm{\lambda}^{\prime}$, which implies that
$v_{j}^{*}\in\tilde{\mathbb{E}}_{\bm{\lambda}(v_{j})}.$

For $\bm{\lambda}\in Sp$, there exists a unique
$\beta_{i}\in(\alpha_{i}-1,\alpha_{i}]$ such that $\exp(2\pi
i\beta_{i})=\lambda_{i}$. Denote
$N_{i}:=\frac{\log(\lambda_{i}^{-1}T_{i}|_{\mathbb{E}_{\bm{\lambda}}})}{2\pi
i}$. Recall that for any $v\in\mathbb{E}_{\bm{\lambda}}$, we define

$\displaystyle\tilde{v}(t):=\exp\big{(}-\sum_{i=1}^{p}(\beta_{i}I+N_{i})\cdot\log
t_{i}\big{)}v(t)=\prod_{i=1}^{p}t_{i}^{-\beta_{i}}\exp\big{(}-\sum_{i=1}^{p}N_{i}\cdot\log
t_{i}\big{)}v(t).$

Likewise, since
$\lambda_{i}\tilde{T}_{i}|_{\tilde{\mathbb{E}}_{\bm{\lambda}}}$ is unipotent,
its logarithm can be defined as

$\log(\lambda_{i}\tilde{T}_{i}|_{\tilde{\mathbb{E}}_{\bm{\lambda}}}):=\sum_{k=1}^{\infty}(-1)^{k+1}\frac{(\lambda_{i}\tilde{T}_{i}|_{\tilde{\mathbb{E}}_{\bm{\lambda}}}-I)^{k}}{k}.$

Write
$\tilde{N}_{i}:=\frac{\log(\lambda_{i}\tilde{T}_{i}|_{\tilde{\mathbb{E}}_{\bm{\lambda}}})}{2\pi
i}$. Then for any $\mu\in\tilde{\mathbb{E}}_{\bm{\lambda}}$, we define

(2.2.5)
$\displaystyle\tilde{\mu}(t):=\exp\big{(}-\sum_{i=1}^{p}(-\beta_{i}I+\tilde{N}_{i})\cdot\log
t_{i}\big{)}\mu(t)=\prod_{i=1}^{p}t_{i}^{\beta_{i}}\exp\big{(}-\sum_{i=1}^{p}\tilde{N}_{i}\cdot\log
t_{i}\big{)}\mu(t).$

Since $\tilde{T}_{i}=(T_{i}^{*})^{-1}$, one has $\tilde{N}_{j}=-N_{j}^{*}$.
Therefore,

$\displaystyle\tilde{\mu}(t)(\tilde{v}(t))$
$\displaystyle=\exp\big{(}-\sum_{i=1}^{p}\tilde{N}_{i}\cdot\log
t_{i}\big{)}\mu(t)\Big{(}\exp\big{(}-\sum_{i=1}^{p}N_{i}\cdot\log
t_{i}\big{)}v(t)\Big{)}$
$\displaystyle=\exp\big{(}\sum_{i=1}^{p}{N}^{*}_{i}\cdot\log
t_{i}\big{)}\mu(t)\Big{(}\exp\big{(}-\sum_{i=1}^{p}N_{i}\cdot\log
t_{i}\big{)}v(t)\Big{)}$ $\displaystyle=\mu(v)=\mbox{constant}.$

This implies that
$\tilde{v}_{i}^{*}(t)(\tilde{v}_{j}(t))=v_{i}^{*}(v_{j})\equiv\delta_{ij}$ if
$\bm{\lambda}=\bm{\lambda}^{\prime}$, where $\tilde{v}^{*}_{i}$ is defined in
(2.2.5) in terms of $v_{i}^{*}\in\tilde{\mathbb{E}}_{\bm{\lambda}(v_{i})}.$

If $\mu\in\tilde{\mathbb{E}}_{\bm{\lambda}}$ and
$v\in\mathbb{E}_{\bm{\lambda}^{\prime}}$ with
$\bm{\lambda}\neq\bm{\lambda}^{\prime}$, the above construction shows that
$\tilde{\mu}$ and $\tilde{v}$ are holomorphic sections of
$V^{*}(\bm{\lambda})$ and $V(\bm{\lambda}^{\prime})$. Here
$V(\bm{\lambda}^{\prime})$ is the invariant flat subbundle of $(V,\nabla)$
defined in (2.2.3), and $V^{*}(\bm{\lambda})$ is defined to be the invariant
flat subbundle of $(V^{*},\nabla)$ generated by $\tilde{E}_{\bm{\lambda}}$.
Hence $\tilde{v}^{*}_{i}$ and $\tilde{v}_{j}$ are holomorphic sections of
$V^{*}(\bm{\lambda}(v_{i}))$ and $V(\bm{\lambda}(v_{j}))$ respectively. This
shows that $\tilde{v}_{i}^{*}(t)(\tilde{v}_{j}(t))\equiv 0$ for
$\bm{\lambda}\neq\bm{\lambda}^{\prime}$ by Lemma 2.4.

In conclusion, we prove that $\tilde{v}^{*}_{1},\ldots,\tilde{v}^{*}_{r}$ is
the dual frame of $\tilde{v}_{1},\ldots,\tilde{v}_{r}$.

Recall that $v_{j}\in\mathbb{E}_{\bm{\lambda}(v_{j})}$ for some
$\bm{\lambda}(v_{j})\in Sp$. There exists a unique
$\beta(v_{j})_{i}\in(\alpha_{i}-1,\alpha_{i}]$ such that $\exp(2\pi
i\beta(v_{j})_{i})=\bm{\lambda}(v_{j})_{i}$. Define a smooth section
$\tilde{v}_{j}^{\prime}=\tilde{v}_{j}\cdot\prod_{i=1}^{p}|t_{i}|^{\beta(v_{j})_{i}}$.
By the norm estimate in the first step, for all $\tilde{v}_{j}^{\prime}$ one
has the norm estimate

$|\tilde{v}_{j}^{\prime}|_{h}\lesssim(\prod_{i=1}^{p}|\log|t_{i}||)^{M}$

for some $M>0$. It follows that

$H(h;\tilde{v}^{\prime}_{1},\ldots,\tilde{v}_{r}^{\prime})\lesssim(\prod_{i=1}^{p}|\log|t_{i}||)^{M}$

Here $H(h;\tilde{v}_{1}^{\prime},\ldots,\tilde{v}_{r}^{\prime})$ is a $r\times
r$-matrix function whose $(i,j)$-component is
$h(\tilde{v}_{i}^{\prime},\tilde{v}_{j}^{\prime})$. On the other hand, we put
$\mu^{\prime}_{i}=\tilde{v}^{*}_{i}\cdot\prod_{j=1}^{p}|t_{j}|^{-\beta(v_{i})_{j}}$.
Since complex polarized variation of Hodge structure is functorial by taking
dual, $(V^{*},\nabla)$ admits a complex polarized variation of Hodge structure
whose Hodge metric is the dual metric $h^{*}$ of the Hodge metric $h$ for
$(V,\nabla,F^{\bullet},Q)$. In the same manner we obtain

$|\mu_{i}^{\prime}|_{h^{*}}\lesssim(\prod_{i=1}^{p}|\log|t_{i}||)^{M^{\prime}}$

for every $\mu_{i}^{\prime}$ and some $M^{\prime}>0$. This implies that

$H(h^{*};\mu^{\prime}_{1},\ldots,\mu^{\prime}_{r})\lesssim(\prod_{i=1}^{p}|\log|t_{i}||)^{M^{\prime}}$

By our construction, $\mu^{\prime}_{1},\ldots,\mu^{\prime}_{r}$ is the dual of
the smooth frame $\tilde{v}^{\prime}_{1},\ldots,\tilde{v}^{\prime}_{r}$. It
follows that

$(\prod_{i=1}^{p}|\log|t_{i}||)^{-M^{\prime}}\lesssim
H(h^{*};\mu^{\prime}_{1},\ldots,\mu^{\prime}_{r})^{-1}=H(h;\tilde{v}^{\prime}_{1},\ldots,\tilde{v}_{r}^{\prime}).$

Hence

(2.2.6) $\displaystyle(\prod_{i=1}^{p}|\log|t_{i}||)^{-M^{\prime}}\lesssim
H(h;\tilde{v}^{\prime}_{1},\ldots,\tilde{v}_{r}^{\prime})\lesssim(\prod_{i=1}^{p}|\log|t_{i}||)^{M}.$

Now we are ready to prove the inclusion $\mathcal{P}_{\\!\bm{\alpha}}V\subset
V_{\bm{\alpha}}^{\rm Del}$. For any $s\in\mathcal{P}_{\\!\bm{\alpha}}V(U)$, it
can be written as $s=\sum_{i=1}^{r}f_{i}\tilde{v}_{i}$ where $f_{i}$ is a
holomorphic function on $U$. By (2.2.6) one has

$\sum_{i=1}^{r}|f_{i}|^{2}\cdot\prod_{j=1}^{p}|t_{j}|^{-2\beta(v_{i})_{j}}\cdot(\prod_{k=1}^{p}|\log|t_{k}||)^{-2M^{\prime}}\lesssim|s|^{2}_{h}\lesssim\prod_{i=1}^{p}|t_{j}|^{-2\alpha_{j}-\varepsilon}$

for any $\varepsilon>0$. Since $\beta(v_{i})_{j}\in(\alpha_{j}-1,\alpha_{j}]$,
one can choose $\delta>0$ such that $\beta(v_{i})_{j}-\alpha+1>\delta$ for all
$v_{i}$ and every $j=1,\ldots,p$. The above inequality implies that for every
$f_{i}$,

$|f_{i}|\lesssim\prod_{i=1}^{p}|t_{j}|^{-1+\delta}.$

Hence all $f_{i}$ extend to holomorphic functions over $X$. This proves that
$s\in V_{\bm{\alpha}}^{\rm Del}(X)$ since $\tilde{v}_{1},\ldots,\tilde{v}_{r}$
is a holomorphic basis of $V_{\bm{\alpha}}^{\rm Del}$ by the definition of
Deligne extension in § 1.2. The inclusion
$\mathcal{P}_{\\!\bm{\alpha}}V\subset V_{\bm{\alpha}}^{\rm Del}$ is proved. We
complete the proof of the proposition. ∎

We leave the proof of the following lemma of linear algebra to the reader.

######  Lemma 2.4.

Let $T:V\to V$ be an isomorphism of a finite dimensional $\mathbb{C}$-vector
space $V$. Decompose $V=V_{\lambda_{1}}\oplus\ldots\oplus V_{\lambda_{k}}$
into eigenspaces of $T$, where $\lambda_{i}$ is a eigenvalue of $T$ and
$V_{\lambda_{i}}$ is the corresponding eigenspace. Denote by $V^{*}$ the dual
vector space. Then for the isomorphism $(T^{*})^{-1}:V^{*}\to V^{*}$, its
eigenvalues are $\lambda_{1}^{-1},\ldots,\lambda_{k}^{-1}$ and its eigenspace
decomposition is $V^{*}=V^{*}_{\lambda^{-1}_{1}}\oplus\ldots\oplus
V^{*}_{\lambda^{-1}_{k}}$, where $V^{*}_{\lambda^{-1}_{j}}$ is the
corresponding eigenspace of $\lambda_{j}^{-1}$. Moreover, one has $\mu(v)=0$
if $\mu\in V^{*}_{\lambda^{-1}_{i}}$ and $v\in V_{\lambda_{j}}$ with $i\neq
j$. ∎

######  Theorem 2.5.

Let $X$ be a complex manifold and let $D=\sum_{i=1}^{\ell}D_{i}$ be a simple
normal crossing divisor on $X$. Let $(V,\nabla,F^{\bullet},Q)$ be a complex
polarized variation of Hodge structure of weight $m$ on $X\backslash D$. Let
$\mathcal{P}_{*}F^{p}$ and $\mathcal{P}_{*}E_{p,m-p}$ be the induced filtered
bundle of hermitian bundles $(F^{p},h_{p})$ and $(E_{p,m-p},h_{p,m-p})$
defined in § 1.6. Then for every $p$, $\mathcal{P}_{*}F^{p}$ and
$\mathcal{P}_{*}E_{p,m-p}$ are parabolic bundles and for each multi-index
$\alpha\in\mathbb{R}^{\ell}$, there is a natural exact sequence

(2.2.7) $\displaystyle
0\to\mathcal{P}_{\\!\bm{\alpha}}F^{p+1}\to\mathcal{P}_{\\!\bm{\alpha}}F^{p}\to\mathcal{P}_{\\!\bm{\alpha}}E_{p,m-p}\to
0.$

###### Proof.

By Lemma 1.4, $(F^{p},h_{p})$ and $(E_{p,m-p},h_{p,m-p})$ are acceptable
bundles for every $p$. It follows from Theorem 1.7 that $\mathcal{P}_{*}F^{p}$
and $\mathcal{P}_{*}E_{p,m-p}$ defined in § 1.6 are parabolic ones.

We first show that we can define

(2.2.8) $\displaystyle
0\to\mathcal{P}_{\\!\bm{\alpha}}F^{p+1}\to\mathcal{P}_{\\!\bm{\alpha}}F^{p}\stackrel{{\scriptstyle
q}}{{\to}}\mathcal{P}_{\\!\bm{\alpha}}E_{p,m-p},$

which is exact. Note that we have the following exact sequence

(2.2.9) $\displaystyle 0\to F^{p+1}\to F^{p}\stackrel{{\scriptstyle
q}}{{\to}}E_{p,m-p}\to 0$

on $X\backslash D$ by the definition of $\mathbb{C}$-VHS. Pick any $x\in D$
and any admissible coordinate $(\Omega;z_{1},\ldots,z_{n})$ centering at $x$
such that $D\cap\Omega=(z_{1}\cdots z_{k}=0)$. By the prolongation via norm
growth defined in § 1.6, any section
$s\in\mathcal{P}_{\\!\bm{\alpha}}F^{p+1}(\Omega)$ satisfies that $s\in
F^{p+1}(\Omega\backslash D)$ and
$|s|_{h^{p+1}}\lesssim\prod_{i=1}^{k}|z_{i}|^{-\alpha_{i}-\varepsilon}$ for
any $\varepsilon>0$. Since $h_{p+1}$ is the induced metric of $h_{p}$ on
$F^{p+1}$, it follows that

(2.2.10)
$\displaystyle|s|_{h_{p+1}}=|s|_{h_{p}}\lesssim\prod_{i=1}^{k}|z_{i}|^{-\alpha_{i}-\varepsilon}$

for any $\varepsilon>0$. Hence the inclusion $F^{p+1}\subset F^{p}$ also
results in the inclusion
$\mathcal{P}_{\\!\bm{\alpha}}F^{p+1}\subset\mathcal{P}_{\\!\bm{\alpha}}F^{p}$.
We proved that (2.2.8) is exact in the left.

Note that the metric $h_{p,m-p}$ on $E_{p,m-p}$ is the quotient metric of
$h_{p}$. It follows that for any section
$s\in\mathcal{P}_{\\!\bm{\alpha}}F^{p}(\Omega)$ , we have

$|q(s)|_{h_{p,m-p}}\leq|s|_{h_{p}}\lesssim\prod_{i=1}^{k}|z_{i}|^{-\alpha_{i}-\varepsilon}$

for any $\varepsilon>0$. Hence the quotient $q:F^{p}\to E_{p,m-p}$ induces the
morphism
$\mathcal{P}_{\\!\bm{\alpha}}F^{p}\to\mathcal{P}_{\\!\bm{\alpha}}E_{p,m-p}$
and thus (2.2.8) can be defined. Next we will show that (2.2.8) is exact in
the middle.

Take any section $s\in\mathcal{P}_{\\!\bm{\alpha}}F^{p}(\Omega)$ such that
$q(s)=0$. Thanks to the exactness in (2.2.9), we have $s\in
F^{p+1}(\Omega\backslash D)$. By (2.2.10), we conclude that
$s\in\mathcal{P}_{\\!\bm{\alpha}}F^{p+1}(\Omega)$. This implies the exactness
of (2.2.8).

In what follows we will prove that (2.2.8) is exact on the right. It suffices
to prove that for any point $x\in D$ and any section
$s\in\mathcal{P}_{\\!\bm{\alpha}}E_{p,m-p}(\Omega)$ where $\Omega$ is a
neighborhood of $x$, there is a section
$\tilde{s}\in\mathcal{P}_{\\!\bm{\alpha}}F^{p}(\Omega^{\prime})$ for some
smaller neighborhood $\Omega^{\prime}$ of $x$ such that
$q(\tilde{s})=s|_{\Omega^{\prime}}$. We shall construct such $\tilde{s}$ by
utilizing the previous results on $L^{2}$-estimate in Lemmas 2.1 and 2.2.

Since this is a local problem, we can assume that $X=\Delta^{n}$,
$D=(z_{1}\cdots z_{\ell}=0)$ and $x$ is the origin. We equip the complement
$U:=X\backslash D$ with the Poincaré metric $\omega_{P}$. Let $X(r)$ and
$U(r)$ be defined as in (2.1.1). By the semicontinuity of the parabolic bundle
in 1.6.(iv), we can choose $\bm{\beta}\in\mathbb{R}^{\ell}$ such that
$\beta_{i}>\alpha_{i}$ and

(2.2.11)
$\displaystyle\mathcal{P}_{\\!\bm{\beta}}F^{p}=\mathcal{P}_{\\!\bm{\alpha}}F^{p}.$

Pick any $s\in\mathcal{P}_{\\!\bm{\alpha}}E_{p,m-p}(X)$. Then $s\in
H^{0}(U,E_{p,m-p})$ with
$|s|_{h_{p,m-p}}\lesssim\prod_{i=1}^{\ell}|z_{i}|^{-\alpha_{i}-\varepsilon}$
for any $\varepsilon>0$. We will construct a section $\tilde{s}\in
H^{0}(U(r),F^{p})$ for some $0<r<1$ such that $q(\tilde{s})=s|_{U(r)}$ and
$|\tilde{s}|_{h_{p}}\lesssim\prod_{i=1}^{\ell}|z_{i}|^{-\beta_{i}-\varepsilon}$
for any $\varepsilon>0$. Note that there is a canonical smooth isomorphism
(and isometry)

$\Phi:(F^{p},h_{p})\to(F^{p+1},h_{p+1})\oplus(E_{p,m-p},h_{p,m-p})$

such that the holomorphic structure of $F^{p}$ via $\Phi$ is defined by

$\begin{bmatrix}\bar{\partial}_{F^{p+1}}&\theta_{p+1,m-p-1}^{\dagger}\\\
0&\bar{\partial}_{E_{p,m-p}}\end{bmatrix},$

where $\theta_{p+1,m-p-1}^{\dagger}$ is the adjoint of $\theta_{p+1,m-p-1}$
with respect to $h_{p+1,m-p-1}$. If $q(\tilde{s})=s$, then
$\Phi(\tilde{s})=[\sigma,s]$ for some
$\sigma\in\mathscr{C}^{\infty}(U,F^{p+1})$ such that

$\begin{bmatrix}\bar{\partial}_{F^{p+1}}&\theta_{p+1,m-p-1}^{\dagger}\\\
0&\bar{\partial}_{E_{p,m-p}}\end{bmatrix}\begin{bmatrix}\sigma\\\
s\end{bmatrix}=0$

Hence $\bar{\partial}_{F^{p+1}}\sigma=-\theta_{p+1,m-p-1}^{\dagger}s$. We will
solve this $\bar{\partial}$-equation with proper norm estimate.

By Theorem 1.2, after we replace $U$ by $U(r)$ for some $0<r<1$, we have
$|\theta_{p+1,m-p-1}|_{h,\omega_{P}}\leq C$ over $U$. This implies that
$|\theta_{p+1,m-p-1}^{\dagger}|_{h,\omega_{P}}\leq C$ over $U$. Hence,

$\displaystyle|\theta_{p+1,m-p-1}^{\dagger}s|_{h_{p+1},\omega_{P}}\leq|\theta_{p+1,m-p-1}^{\dagger}|_{h,\omega_{P}}\cdot|s|_{h_{p,m-p}}\lesssim\prod_{i=1}^{\ell}|z_{i}|^{-\alpha_{i}-\varepsilon}$

for any $\varepsilon>0$. We now introduce a new metric for $F^{p}$ defined by

$h_{p}(\bm{\beta}):=h_{p}\cdot\prod_{i=1}^{\ell}|z_{i}|^{\beta_{i}}.$

Since $\beta_{i}>\alpha_{i}$ for each $i$, we have

$|\theta_{p+1,m-p-1}^{\dagger}s|_{h_{p+1}(\bm{\beta}),\omega_{P}}\lesssim\prod_{j=1}^{\ell}|z_{j}|^{\delta}$

for some $\delta>0$. Note that $\theta_{p+1,m-p-1}^{\dagger}s\in
A^{0,1}(E_{p+1,m-p-1})$. We have

$\displaystyle\bar{\partial}_{F^{p+1}}(\theta_{p+1,m-p-1}^{\dagger}s)$
$\displaystyle=(\bar{\partial}_{E_{p+1,m-p-1}}+\theta_{p,m-p}^{\dagger})(\theta_{p+1,m-p-1}^{\dagger}s)$
$\displaystyle=\bar{\partial}_{E_{p+1,m-p-1}}(\theta_{p+1,m-p-1}^{\dagger}s)$
$\displaystyle=(D_{h}^{0,1}\theta_{p+1,m-p-1}^{\dagger})s-\theta_{p+1,m-p-1}^{\dagger}(\bar{\partial}_{E_{p,m-p}}s)=0,$

where the second equality follows from
$\theta_{p+1,m-p-1}^{\dagger}\wedge\theta_{m-p}^{\dagger}=0$, and the last one
follows from $D_{h}^{0,1}(\theta^{\dagger})=0$. Here $D_{h}$ is the Chern
connection for the Hodge bundle $(E=\oplus_{p+q=m}E_{p,q},h)$. Since
$(F^{p+1},h_{p+1}(\bm{\beta}))$ is also acceptable by Lemma 1.4, we can invoke
Lemma 2.1 to find some $\sigma\in\mathscr{C}^{\infty}(U,F^{p+1})$ such that

$\bar{\partial}_{F^{p+1}}(\sigma)=-\theta_{p+1,m-p-1}^{\dagger}s$

and

$\displaystyle\int_{U}|\sigma|^{2}_{h_{p+1}(\bm{\beta},N)}d\mbox{vol}_{\omega_{P}}<\infty$

for some $N\gg 1$. Here $h_{p+1}(\bm{\beta},N)$ is a new metric for $F^{p+1}$
define by

$h_{p+1}(\bm{\beta},N)=h_{p+1}\cdot\prod_{i=1}^{\ell}|z_{i}|^{\beta_{i}}\cdot
e^{-N\varphi}$

with $\varphi$ defined in (2.1.2). Since
$|s|_{h_{p,m-p}}\lesssim\prod_{i=1}^{\ell}|z_{i}|^{-\alpha_{i}-\varepsilon}$
for any $\varepsilon>0$, it follows that $|s|_{h_{p,m-p}(\bm{\beta},N)}<C$ for
some constant $C>0$, where we define

$h_{p,m-p}(\bm{\beta},N)=h_{p,m-p}\cdot\prod_{i=1}^{\ell}|z_{i}|^{\beta_{i}}\cdot
e^{-N\varphi}.$

Thus the section $\tilde{s}:=\Phi^{-1}([\sigma,s])$ is a holomorphic section
of $F^{p}$ such that

$\displaystyle\int_{U}|\tilde{s}|^{2}_{h_{p}(\bm{\beta},N)}d\mbox{vol}_{\omega_{P}}=\int_{U}|\sigma|^{2}_{h_{p+1}(\bm{\beta},N)}d\mbox{vol}_{\omega_{P}}+\int_{U}|s|^{2}_{h_{p,m-p}(\bm{\beta},N)}d\mbox{vol}_{\omega_{P}}<\infty$

Since $(F^{p},h_{p}(\bm{\beta}))$ is also acceptable by Lemma 1.4, thanks to
Lemma 2.2, over some $U(r)$ for $0<r<1$ we have
$|\tilde{s}|_{h_{p}(\bm{\beta})}\lesssim\prod_{j=1}^{\ell}|z_{j}|^{-\varepsilon}$
for any $\varepsilon>0$. Therefore,
$|\tilde{s}|_{h_{p}}\lesssim\prod_{j=1}^{\ell}|z_{j}|^{-\beta_{j}-\varepsilon}$
for any $\varepsilon>0$. It follows that
$\tilde{s}\in\mathcal{P}_{\bm{\beta}}F^{p}(X(r)).$ By (2.2.11) we conclude
that $\tilde{s}\in\mathcal{P}_{\bm{\alpha}}F^{p}(X(r^{\prime}))$ for some
$0<r^{\prime}<1$. This implies the right exactness of (2.2.7) as
$q(\tilde{s})=s$. The theorem is proved. ∎

Let us prove Theorem A.

###### Proof of Theorem A.

Thanks to Proposition 2.3, we have $V^{\rm
Del}_{\bm{\alpha}}=\mathcal{P}_{\\!\bm{\alpha}}V$. By Lemma 1.4,
$(F^{p},h_{p})$ and $(E_{p,m-p},h_{p,m-p})$ are acceptable bundles for any
$p$. It follows from Theorem 1.7 that the induced filtered bundle
$\mathcal{P}_{*}F^{p}$ and $\mathcal{P}_{*}E_{p,m-p}$ defined in § 1.6 are
parabolic ones. In particular, $\mathcal{P}_{\\!\bm{\alpha}}F^{p}$ and
$\mathcal{P}_{\\!\bm{\alpha}}E_{p,m-p}$ are locally free sheaves. Denote by
$j:X\backslash D\to X$ the inclusion map. Note that

(2.2.12)
$\displaystyle\mathcal{P}_{\\!\bm{\alpha}}F^{p}=j_{*}(F^{p})\cap\mathcal{P}_{\\!\bm{\alpha}}V\stackrel{{\scriptstyle\lx@cref{creftype~refnum}{lem:equal}}}{{=}}j_{*}F^{p}\cap
V_{\bm{\alpha}}^{\rm Del}=:F^{p}_{\bm{\alpha}}.$

Hence the exact sequence (2.2.7) in Theorem 2.5 implies the following one

$0\to F^{p+1}_{\bm{\alpha}}\to
F^{p}_{\bm{\alpha}}\to\mathcal{P}_{\\!\bm{\alpha}}E_{p,m-p}\to 0.$

In particular, $F^{p}_{\bm{\alpha}}/F^{p+1}_{\bm{\alpha}}$ is isomorphic to
$\mathcal{P}_{\\!\bm{\alpha}}E_{p,m-p}$, which is thus locally free. The
theorem is proved. ∎

###  2.3. On the nilpotent orbit theorem

In this subsection we apply Theorem A to prove Theorem B following closely
Schmid’s original method [Sch73, p. 288-289]. We will use the notations and
conventions in § 1.7.

Let $(V,\nabla,F^{\bullet},Q)$ be a complex polarized variation of Hodge
structure on $(\Delta^{*})^{p}\times\Delta^{q}$. Denote by
$\Phi:\mathbb{H}^{p}\times\Delta^{q}\to\mathscr{D}$ its period mapping, where
we set

$\displaystyle\mathbb{H}^{p}\times\Delta^{q}$ $\displaystyle\to\Delta^{n}$
$\displaystyle(z,w)$ $\displaystyle\mapsto(e^{z_{1}},\ldots,e^{z_{p}},w)$

to be the uniformizing map. Let $T_{j}$ be the monodromy transformation
defined in § 1.2. For some fixed $\bm{\alpha}\in\mathbb{R}^{p}$, there exist
$S_{i},N_{i}\in{\rm\small End}(V^{\nabla})$ such that

  * •

$T_{i}=\exp(2\pi i(S_{i}+N_{i}))$;

  * •

$[S_{i},S_{j}]=0$, $[S_{i},N_{j}]=0$, and $[N_{i},N_{j}]=0$;

  * •

$S_{i}$ is semisimple whose eigenvalues lying in $(\alpha_{i}-1,\alpha_{i}]$
and $N_{i}$ is nilpotent.

Let us define

$\tilde{\Psi}(z,w):=\exp(\sum_{i=1}^{p}(S_{i}+N_{i})z_{i})\Phi(z,w),$

which satisfies $\tilde{\Psi}(z_{1},\ldots,z_{i}+2\pi
i,\ldots,z_{p},w)=\tilde{\Psi}(z,w)$ for $i=1,\ldots,p$. It thus descends to a
single valued map
$\Psi:(\Delta^{*})^{p}\times\Delta^{q}\to\check{\mathscr{D}}$ such that
$\Psi(e^{z_{1}},\ldots,e^{z_{p}},w)=\tilde{\Psi}(z,w)$.

######  Lemma 2.6.

The twisted holomorphic map $\Psi$ extends holomorphically to $\Delta^{p+q}$.

###### Proof.

We use the notations in § 1.2. We fix a basis $v_{1},\ldots,v_{r}$ of
$V^{\nabla}$ such that each $v_{i}$ belongs to some
$\mathbb{E}_{\bm{\lambda}}$. Then the sections
$\tilde{v}_{1},\ldots,\tilde{v}_{r}$ defined in (1.2.1) induces a
trivialization

$V^{\nabla}\otimes_{\mathbb{C}}\Delta^{p+q}\simeq V^{\rm Del}_{\bm{\alpha}},$

where $V^{\rm Del}_{\bm{\alpha}}$ is the Deligne extension. Under such
trivialization, the Hodge filtration
$F^{\bullet}_{\bm{\alpha}}(t_{1},\ldots,t_{p+1})$ becomes
$\Psi(t_{1},\ldots,t_{p+1})$. Thanks to Theorem A, the Hodge filtration
$F^{\bullet}_{\bm{\alpha}}$ extends to locally free sheaves over
$\Delta^{p+q}$ such that $F^{p}_{\bm{\alpha}}/F^{p+1}_{\bm{\alpha}}$ is also
locally free. Therefore, $\Psi$ extends holomorphic maps over $\Delta^{p+q}$.
∎

This lemma thus proves B.(i). We write $a(w):=\Psi(0,w)$. In general it does
not lie in $\mathscr{D}$.

The following well-known result follows from the fact that $GL(V^{\nabla})$
acts transitively on $\check{\mathscr{D}}$.

######  Lemma 2.7.

For any $g\in GL(V^{\nabla})$, consider the left translation
$L_{g}:\check{\mathscr{D}}\to\check{\mathscr{D}}$ with $L_{g}(F):=gF$. Then

$\displaystyle(L_{g})_{*}:T_{\check{\mathscr{D}},F}^{-1,1}\stackrel{{\scriptstyle\sim}}{{\to}}T_{\check{\mathscr{D}},gF}^{-1,1}.$

∎

Recall that for any $A\in{\rm\small End}(V^{\nabla})$ and any
$F\in\check{\mathscr{D}}$, we denote by $[A]_{F}$ the image of $A$ under the
natural map ${\rm\small End}(V^{\nabla})\to T_{\check{\mathscr{D}},F}$.

######  Lemma 2.8.

For each $i=1,\ldots,p$, $[S_{i}+N_{i}]_{a(w)}\subset
T_{\check{\mathscr{D}},a(w)}^{-1,1}$.

###### Proof.

Since

$\tilde{\Psi}_{*}(\frac{\partial}{\partial
z_{i}})(z,w)=[S_{i}+N_{i}]_{\tilde{\Psi}(z,w)}+(L_{\exp(\sum_{i=1}^{p}(S_{i}+N_{i})z_{i})})_{*}\Phi_{*}(\frac{\partial}{\partial
z_{i}})(z,w)$

$\Phi_{*}(\frac{\partial}{\partial z_{i}})$ is horizontal since $\Phi$ is a
horizontal mapping by § 1.7. By Lemma 2.7
$(L_{\exp(\sum_{i=1}^{p}(S_{i}+N_{i})z_{i})})_{*}\Phi_{*}(\frac{\partial}{\partial
z_{i}})(z,w)$ is horizontal. On the other hand,

$\tilde{\Psi}_{*}(\frac{\partial}{\partial
z_{i}})(z,w)=\Psi_{*}(\frac{\partial}{\partial
t_{i}})(e^{z_{1}},\ldots,e^{z_{p}},w)\cdot e^{z_{i}}$

which tends to zero if $\Re z_{i}\to-\infty$ and $\Re z_{j}\leq C$ for other
$j$. By the continuity, this implies that

$[S_{i}+N_{i}]_{a(w)}\in T_{\check{\mathscr{D}},a(w)}^{-1,1}.$

∎

We are ready to prove B.(ii).

######  Lemma 2.9.

The holomorphic mapping

$\displaystyle\vartheta:\mathbb{H}^{p}\times\Delta^{q}$
$\displaystyle\to\check{\mathscr{D}}$ $\displaystyle(z,w)$
$\displaystyle\mapsto\exp(-\sum_{i=1}^{p}z_{i}(S_{i}+N_{i}))\circ a(w)$

is horizontal.

###### Proof.

Note that $\vartheta_{*}(\frac{\partial}{\partial
z_{i}})=[S_{i}+N_{i}]_{\vartheta(z,w)}$. Since $[S_{i},N_{i}]=0$, one has

$(L_{\exp(\sum_{i=1}^{p}(S_{i}+N_{i})z_{i})})_{*}\big{(}[S_{i}+N_{i}]_{\vartheta(z,w)}\big{)}=\Big{[}{\rm
Ad}\,_{\exp(\sum_{i=1}^{p}(S_{i}+N_{i})z_{i})}(S_{i}+N_{i})\Big{]}_{a(w)}=[S_{i}+N_{i}]_{a(w)}.$

It then follows from Lemmas 2.7 and 2.8 that
$[S_{i}+N_{i}]_{\vartheta(z,w)}\in
T_{\check{\mathscr{D}},\vartheta(z,w)}^{-1,1}$. We conclude that
$\vartheta_{*}(\frac{\partial}{\partial z_{i}})$ is horizontal.

On the other hand, one has

$\vartheta_{*}(\frac{\partial}{\partial
w_{i}})=(L_{\exp(-\sum_{i=1}^{p}(S_{i}+N_{i})z_{i})})_{*}a_{*}(\frac{\partial}{\partial
w_{i}}),$

and

${\Psi}_{*}(\frac{\partial}{\partial
w_{i}})(e^{z},w)=\tilde{\Psi}_{*}(\frac{\partial}{\partial
w_{i}})(z,w):=(L_{\exp(\sum_{i=1}^{p}(S_{i}+N_{i})z_{i})})_{*}\Phi_{*}(\frac{\partial}{\partial
w_{i}})(z,w).$

Since $\Phi_{*}(\frac{\partial}{\partial w_{i}})$ is horizontal, by Lemma 2.7,
$\tilde{\Psi}_{*}(\frac{\partial}{\partial
w_{i}})(z,w):=(L_{\exp(\sum_{i=1}^{p}(S_{i}+N_{i})z_{i})})_{*}\Phi_{*}(\frac{\partial}{\partial
w_{i}})(z,w)$ is horizontal, and thus ${\Psi}_{*}(\frac{\partial}{\partial
w_{i}})(e^{z},w)$ is horizontal. Letting $\Re z_{i}\to-\infty$ for
$i=1,\ldots,p$, we conclude that

${\Psi}_{*}(\frac{\partial}{\partial
w_{i}})(0,w)=a_{*}(\frac{\partial}{\partial w_{i}})$

is also horizontal. We apply Lemma 2.7 again to conclude that
$\vartheta_{*}(\frac{\partial}{\partial w_{i}})$ is horizontal. In conclusion,
$\vartheta$ is a horizontal mapping. We proved B.(ii). ∎

The rest of the paper is devoted to the proof of B.(iii). We will only deal
with the case of one variable. We first start a lemma in linear algebra whose
proof is direct (cf. [SS22, §7.5] for a detailed proof).

######  Lemma 2.10.

Let $S\in{\rm\small End}(V^{\nabla})$ be semisimple with real eigenvalues.
Then there exists a constant $C>0$ such that

$\lVert{\rm Ad}\,e^{xS}\rVert\leq
C\exp((\lambda_{\max}-\lambda_{\min})\cdot|x|)\quad\mbox{for all}\quad
x\in\mathbb{R},$

where $\lambda_{\max}$ and $\lambda_{\min}$ are the largest and smallest
eigenvalue of $S$. Let $N\in{\rm\small End}(V^{\nabla})$ be nilpotent. Then

$\lVert{\rm Ad}\,e^{xN}\rVert\leq C^{\prime}|x|^{m}$

for some $C^{\prime},m>0$.

Here we fix a reference polarized Hodge structure $o\in\mathscr{D}$ which
induces metrics for $V^{\nabla}$ and ${\rm\small End}(V^{\nabla})$.
$\lVert{\rm Ad}\,e^{xS}\rVert$ is the operator norm with respect to such
metric of ${\rm\small End}(V^{\nabla})$. ∎

The following two lemmas are due to Schmid [Sch73, Lemmas 8.12 & 8.19]. They
are stated for period domains of real Hodge structures. However, their proofs
can be generalized to period domains of complex polarized Hodge structures
verbatim, and we thus omit their proofs here.

######  Lemma 2.11.

If $g\in GL(V^{\nabla})$, then for some natural distance
$d_{\check{\mathscr{D}}}$ of $\check{\mathscr{D}}$, we have

$d_{\check{\mathscr{D}}}(ga,gb)\leq\lVert{\rm Ad}\,g\rVert
d_{\check{\mathscr{D}}}(a,b)$

for any points $a,b\in\check{\mathscr{D}}$. ∎

######  Lemma 2.12.

Let $\Phi:\mathbb{H}\to\mathscr{D}$ be the period map associated to a complex
polarized variation of Hodge structure on $\Delta^{*}$. Fix $\alpha,k>0$ and a
reference point $o\in\mathscr{D}$. Choose $g(z)\in G=U(V^{\nabla},Q)$ such
that $g(z)\cdot o=\Phi(z)$. Then there exist $C,\beta>0$ such that if $|\Im
z|\leq k$, one has

$\lVert{\rm Ad}\,g(z)^{-1}\rVert\leq C|\Re z|^{\beta}$

for $\Re z<-\alpha$. Here $\lVert{\rm Ad}\,g(z)^{-1}\rVert$ is the operator
norm defined in Lemma 2.10. ∎

By [SS22, §4.1], we know that $G:=U(V^{\nabla},Q)$ acts transitively on the
period domain $\mathscr{D}$, and $\mathscr{D}$ admits a natural $G$-invariant
distance $d_{\mathscr{D}}$.

###### Proof of B.(iii).

Let $T\in GL(V^{\nabla})$ be the monodromy operator associated to the counter-
clockwise generator of $\pi_{1}(\Delta^{*})$. Note that $T\in
G:=U(V^{\nabla},Q)$. Recall that there exist commuting $S,N\in GL(V^{\nabla})$
such that

  * •

$\exp(2\pi i(S+N))=T$;

  * •

$S$ is semisimple with eigenvalues lying in $(\alpha-1,\alpha]$;

  * •

$N$ is nilpotent.

Denote by $a=\Psi(0)$. Then for $|t|$ small enough, one has

$d_{\check{\mathscr{D}}}(a,\Psi(t))<C|t|\quad\mbox{for some}\ \ C>0,$

which is equivalent to that

(2.3.1) $\displaystyle d_{\check{\mathscr{D}}}(a,\Psi(e^{z}))<Ce^{x}$

when $x\leq-M$ for some $M>0$. Here we write $z=x+iy$. Assume now $|y|\leq
2\pi$ and $x\leq-M$. Then

$\displaystyle d_{\check{\mathscr{D}}}(\exp(-(S+N)z)a,\Phi(z))$
$\displaystyle\leq\lVert{\rm Ad}\,\exp((S+N)z)\rVert\cdot
d_{\check{\mathscr{D}}}(a,\Psi(e^{z}))$ $\displaystyle\leq\lVert{\rm
Ad}\,\exp(Nx)\rVert\cdot\lVert{\rm Ad}\,\exp(i(S+N)y)\rVert\cdot\lVert{\rm
Ad}\,\exp(Sx)\rVert\cdot d_{\check{\mathscr{D}}}(a,\Psi(e^{z}))$
$\displaystyle\leq C_{1}\lVert{\rm Ad}\,\exp(Nx)\rVert\cdot\lVert{\rm
Ad}\,\exp(Sx)\rVert\cdot d_{\check{\mathscr{D}}}(a,\Psi(e^{z}))$
$\displaystyle\leq
C_{2}|x|^{m}\cdot\exp((\lambda_{\max}-\lambda_{\min})\cdot|x|)\cdot
d_{\check{\mathscr{D}}}(a,\Psi(e^{z}))$ $\displaystyle\leq
C_{3}|x|^{m}\cdot\exp((\lambda_{\max}-\lambda_{\min})\cdot|x|)\cdot e^{x}\leq
C_{3}|x|^{m}e^{\delta x}.$

The first inequality is due to Lemma 2.11, the third one holds since $|y|\leq
2\pi$, the fourth one follows from Lemma 2.10, and the fifth one follows from
(2.3.1). Here $\lambda_{\max}$ and $\lambda_{\min}$ are the largest and
smallest eigenvalue of $S$. Therefore, $\lambda_{\max}-\lambda_{\min}<1$ and
thus the last inequality can be achieved for some $\delta>0$. Here
$C_{1},\ldots,C_{3}>0$ are some positive constants.

Fix a reference point $o\in\mathscr{D}$ and let $g(z)\in G$ such that
$g(z)\cdot o=\Phi(z)$. By Lemmas 2.11 and 2.12 one gets

(2.3.2) $\displaystyle d_{\check{\mathscr{D}}}(g(z)^{-1}\exp(-(S+N)z)a,o)$
$\displaystyle\leq\lVert{\rm Ad}\,g(z)^{-1}\rVert\cdot
d_{\check{\mathscr{D}}}(\exp(-(S+N)z)a,\Phi(z))$ $\displaystyle\leq
C_{4}|x|^{m+\beta}e^{\delta x}$

if $|y|\leq 2\pi$ and $x<-M_{2}$ for some $M_{2}\geq M$ and $C_{4},\beta>0$.
Pick a small neighborhood $U$ of $o$ in $\mathscr{D}$ such that the distance
functions $d_{\mathscr{D}}$ and $d_{\check{\mathscr{D}}}$ are mutually bounded
over $U$. By (2.3.2) when $|y|\leq 2\pi$, $x\leq-M_{3}$ for some $M_{3}\geq
M_{2}$, $g(z)^{-1}\exp(-(S+N)z)a$ will be entirely contained in $U$. Note that
$g(z)\in G$, it follows that $\exp(-(S+N)z)a\in\mathscr{D}$ if $|y|\leq 2\pi$
and $x\leq-M_{3}$. When $|y|>2\pi$ and $x\leq-M_{3}$, we find some integer
$\ell$ such that $|y-2\pi\ell|\leq 2\pi$. Then $\exp(-(S+N)(z-2\pi
i\ell))a\in\mathscr{D}$. Since $\exp(-(S+N)z)a=T^{-\ell}\exp(-(S+N)(z-2\pi
i\ell))a$ and $T\in G$, it follows that $\exp(-(S+N)z)a\in\mathscr{D}$. In
conclusion, $\exp(-(S+N)z)a\in\mathscr{D}$ if $x\leq-M_{3}$. We prove the
first claim in B.(iii).

Recall that the distance functions $d_{\mathscr{D}}$ and
$d_{\check{\mathscr{D}}}$ are mutually bounded over $U$. By (2.3.2) again for
some $C_{5}>0$ we have

$d_{{\mathscr{D}}}(g(z)^{-1}\exp(-(S+N)z)a,o)\leq C_{5}|x|^{m+\beta}e^{\delta
x}.$

for $|y|\leq 2\pi$, $x\leq-M_{3}$. Since the action of $g(z)$ is
$d_{\mathscr{D}}$-distance invariant, we obtain the distance estimate

$d_{{\mathscr{D}}}(\exp(-(S+N)z)a,\Phi(z))\leq C_{5}|x|^{m+\beta}e^{\delta x}$

for $|y|\leq 2\pi$, $x\leq-M_{3}$. When $|y|>2\pi$ and $x\leq-M_{3}$, one
picks some integer $\ell$ such that $|y-2\pi\ell|\leq 2\pi$. Then

$d_{{\mathscr{D}}}(\exp(-(S+N)(z-2\pi i\ell))a,\Phi(z-2\pi i\ell))\leq
C_{5}|x|^{m+\beta}e^{\delta x}.$

In other words,

$d_{{\mathscr{D}}}(T^{\ell}\exp(-(S+N)z)a,T^{\ell}\Phi(z))\leq
C_{5}|x|^{m+\beta}e^{\delta x}.$

As $T$ is also $d_{\mathscr{D}}$-distance invariant, it follows that

$d_{{\mathscr{D}}}(\exp(-(S+N)z)a,\Phi(z))\leq C_{5}|x|^{m+\beta}e^{\delta
x}.$

for $x\leq-M_{3}$. The distance estimate is obtained. ∎

Acknowledgment. This paper is motivated by a question asked by Junchao Shentu,
to whom I express my gratitude. I especially thank Christian Schnell for his
personal notes and discussions on the nilpotent orbit theorem in 2020 and
2022, and sending me the preprint [SS22]. I also thank the referees for their
valuable suggestions, which have greatly improved the paper. This work is
partially supported by the French Agence Nationale de la Recherche (ANR) under
reference ANR-21-CE40-0010.

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