#  $L^{2}$ representation of Simpson-Mochizuki’s prolongation of Higgs bundles
and the Kawamata-Viehweg vanishing theorem for semistable parabolic Higgs
bundles

Chen Zhao<EMAIL_ADDRESS>School of Mathematical Sciences, University of
Science and Technology of China, Hefei, 230026, China

###### Abstract.

In this paper, we provide an $L^{2}$ fine resolution of the prolongation of a
nilpotent harmonic bundle in the sense of Simpson-Mochizuki (an analytic
analogue of the Kashiwara-Malgrange filtrations). This is the logarithmic
analogue of Cattani-Kaplan-Schmid’s and Kashiwara-Kawai’s results on the
$L^{2}$ interpretation of the intersection complex. As an application, we give
an $L^{2}$-theoretic proof to the Nadel-Kawamata-Viehweg vanishing theorem
with coefficients in a nilpotent Higgs bundle.

###### Contents

  1. 1 Introduction
    1. 1.1 Main result
    2. 1.2 Application: vanishing theorems
      1. 1.2.1 Kawamata-Viehweg vanishing theorem and its $(i,j)$ version
      2. 1.2.2 Saito-type vanishing theorems
  2. 2 Preliminary
    1. 2.1 Nilpotent harmonic bundles
    2. 2.2 Boundedness for the Higgs field
  3. 3 $L^{2}$ cohomology and $L^{2}$ complex
  4. 4 $L^{2}$ representation of the prolongation of Higgs bundles
    1. 4.1 Prolongation and parabolic structure
    2. 4.2 $L^{2}$ representation
    3. 4.3 Local behavior of the metrics
    4. 4.4 Fine bi-complex resolution
    5. 4.5 Proof of Theorem 4.5
  5. 5 Application: Kawamata-Viehweg type vanishing theorem for Higgs bundles
    1. 5.1 Construction of the relavant metrics
    2. 5.2 Fine resolution
    3. 5.3 Vanishing theorem on weakly pseudoconvex manifolds
    4. 5.4 Generalization of Suh’s vanishing theorem

##  1\. Introduction

###  1.1. Main result

It has been long realized that the (nonabelian) Hodge theory is closely
related to the $L^{2}$-differential forms. Besides the proof of the classical
Hodge decomposition theorem using $L^{2}$ methods, there are many interesting
and deep relations between the various extensions of the polarized variation
of Hodge structure and the $L^{2}$-de Rham complexes. For the $L^{2}$
interpretation of the intersection complex, readers may see Cattani-Kaplan-
Schmid [Cattani_Kaplan_Schmid1987], Kashiwara-Kawai [Kashiwara_Kawai1987], E.
Looijenga [Looijenga1988], Saper-Stern [Saper_Stern1990], S. Zucker
[Zucker1979] and Shentu-Zhao [SC2021_CGM]. The main purpose of the present
paper is to investigate the relavant problem in the context of the nonabelian
Hodge theory. The main result is an $L^{2}$-fine resolution of the Simpson-
Mochizuki’s prolongation (an analytic analogue of the Kashiwara-Malgrange
filtrations [Kashiwara1983]) of nilpotent Higgs bundles (e.g. polarized
complex variation of Hodge structure). Before stating the main result, let us
fix some notations.

Let $X$ be a pre-compact open subset of a hermitian manifold $(M,\omega_{M})$
and let $D=\cup_{i=1}^{l}D_{i}$ be a simple normal crossing divisor on $M$.
Let $\sigma_{i}\in H^{0}(M,\mathscr{O}_{M}(D_{i}))$ be the defining section of
$D_{i}$ and $h_{i}$ an arbitrary hermitian metric on $\mathscr{O}_{M}(D_{i})$.
We denote $\varphi_{i}=|\sigma_{i}|_{h_{i}}$. Let $\phi_{P}\in
C^{\infty}(X\backslash D)$ such that
$\omega_{P}:=\sqrt{-1}\partial\bar{\partial}\phi_{P}+\omega_{M}|_{X\backslash
D}$ is a hermitian metric on $X\backslash D$ which has Poincaré type growth
near $D\cap X$. Let $(E,\theta,h)$ be a nilpotent harmonic bundle on
$M\backslash D$ together with $\bar{\partial}_{E}$ its holomorphic structure.
For every indices ${\bf a}=(a_{1},\dots,a_{l})\in\mathbb{R}^{l}$, denote
${}_{\bf a}E$ to be the associated prolongation of $(E,\theta,h)$ in the sense
of Simpson [Simpson1990] and Mochizuki [Mochizuki20072, Mochizuki20071]. It is
a locally free $\mathscr{O}_{X}$-module consisting of the holomorphic local
sections $s$ of $E$ such that
$|s|_{h}\lesssim\prod_{i=1}^{l}\varphi_{i}^{-a_{i}-\epsilon}$ for every
$\epsilon>0$. The restricted Higgs field $\theta|_{{}_{\bf a}E}$ has at most
logarithmic poles along $D$. We denote by

${\rm Dol}(_{\bf a}E,\theta):={{}_{\bf a}}E\to{{}_{\bf
a}}E\otimes\Omega_{X}(\log D)\to\cdots$

the associated logarithmic Dolbeault complex. The main purpose of the present
paper is to construct a complex of fine sheaves (consisting of certain locally
square integrable differential forms) that is canonically quasi-isomorphic to
${\rm Dol}(_{\bf a}E,\theta)$.

The set of prolongations $\\{_{\bf a}E\\}_{{\bf a}\in\mathbb{R}^{l}}$ forms a
parabolic bundle. In particular, the set of jumping indices along each
component $D_{i}$

$\mathscr{J}_{D_{i}}(E):=\left\\{a\in\mathbb{R}\big{|}{\rm Gr}^{(i)}_{{\bf
a}\uparrow_{i}a}E:=_{{\bf a}\uparrow_{i}a}E/\cup_{b<a}{{}_{{\bf
a}\uparrow_{i}b}}E\neq 0,\forall{\bf a}\in\mathbb{R}^{l}\right\\}$

is a discrete subset of $\mathbb{R}$ and
$\mathscr{J}_{D_{i}}(E)+\mathbb{Z}=\mathscr{J}_{D_{i}}(E)$. Here we denote
${\bf a}\uparrow_{i}c$ by the indices obtained by deleting the $i$th component
of $\bm{a}$ and replacing it with the number $c$.

For $N\in\mathbb{Z}$, denote $h_{N}({\bf
a}):=he^{-N\phi_{P}+\sum_{i=1}^{l}2a_{i}\log\varphi_{i}}$ to be the modified
metric on $E$. Define the operator
$D^{\prime\prime}=\bar{\partial}_{E}+\theta$. Denote by
$\mathscr{D}^{k}_{X,\omega_{P}}(E,D^{\prime\prime},h_{N}({\bf a}))$ the sheaf
of measurable $E$-valued $k$-forms $\alpha$ such that $\alpha$ and
$D^{\prime\prime}\alpha$ are locally square integrable near every point of $X$
with respect to $\omega_{P}$ and $h_{N}(\bm{a})$. Denote

$\mathscr{D}^{\bullet}_{X,\omega_{P}}(E,D^{\prime\prime},h_{N}({\bf
a})):=\mathscr{D}^{0}_{X,\omega_{P}}(E,D^{\prime\prime},h_{N}({\bf
a}))\stackrel{{\scriptstyle
D^{\prime\prime}}}{{\to}}\mathscr{D}^{1}_{X,\omega_{P}}(E,D^{\prime\prime},h_{N}({\bf
a}))\stackrel{{\scriptstyle D^{\prime\prime}}}{{\to}}\cdots$

to be the associated $L^{2}$-Dolbeault complex. There is a natural inclusion

${\rm Dol}(_{\bf
a}E,\theta)\to\mathscr{D}^{\bullet}_{X,\omega_{P}}(E,D^{\prime\prime},h_{N}({\bf
a}+\bm{\epsilon}))$

for every $(0,\dots,0)=:\bm{0}<\bm{\epsilon}\in\mathbb{R}^{l}$111By
$(a_{1},\dots,a_{l})<(b_{1},\dots,b_{l})$ we mean that $a_{i}<b_{i}$ for every
$i=1,\dots,l$..

For every ${\bf a}=(a_{1},\dots,a_{l})\in\mathbb{R}^{l}$, define

$\sigma_{E,i}(a_{i})=\min\\{|b-a_{i}||b\in\mathscr{J}_{D_{i}}(E),b\neq
a_{i}\\},\quad\forall 1\leq i\leq l$

and $\bm{\sigma}_{E}({\bf
a})=(\sigma_{E,1}(a_{1}),\dots,\sigma_{E,l}(a_{l}))$. The main result of the
present paper is

######  Theorem 1.1.

There is a constant $N_{0}$, depending only on $(E,\theta,h)$ and $X$
(independent of $\bm{a}$ and $\bm{\epsilon}$ in the following), such that the
inclusion map

${\rm Dol}(_{{\bf
a}}E,\theta)\to\mathscr{D}^{\bullet}_{X,\omega_{P}}(E,D^{\prime\prime},h_{N}({\bf
a}+\bm{\epsilon}))$

is a quasi-isomorphism for every ${\bf a}\in\mathbb{R}^{l}$, every $N>N_{0}$
and every $\bm{0}<\bm{\epsilon}<\bm{\sigma}_{E}({\bf a})$.

###  1.2. Application: vanishing theorems

Besides its own insterests, Theorem 1.1 allows us to prove the following
Kawamata-Viehweg type vanishing theorem for stable parabolic higgs bundles.

######  Theorem 1.2 (=Theorem 5.8).

Let $X$ be a projective manifold of dimension $n$, $Y$ be an analytic space
and $f:X\rightarrow Y$ be a proper surjective holomorphic map. Let
$D=\sum_{i=1}^{l}D_{i}$ be a normal crossing divisor on $X$ and
$(\\{{{}_{\alpha}}E\\},\theta,h)$ a stable parabolic higgs bundles on
$X\backslash D$ where $\theta$ is nilpotent. Let $L$ be a holomorphic line
bundle on $X$. Suppose that some positive multiple $mL=A+F_{1}+F_{2}$ where
$A$ is an ample line bundle, $F_{1}$ and $F_{2}$ are effective divisors
supported in $D$ such that $A+F_{1}$ is nef. Suppose that
$F_{2}=\sum_{i=1}^{l}r_{i}D_{i}$ with $r_{i}\in\mathbb{Z}_{\geq 0},\forall i$.
Denote ${\bf{r}}:=(r_{1},\dots,r_{l})$.

Then

(1.1) $\displaystyle R^{i}f_{\ast}({\rm Dol}(_{-\frac{\bf{r}}{m}+{\bf
a}}E,\theta)\otimes L)=0$

for any $i>n$ and any ${\bf a}=(a_{1},\dots,a_{l})\in\mathbb{R}^{l}$ such that
$|a_{j}|<\sigma_{E,j}(-\frac{r_{j}}{m})$ for every $j=1,\dots,l$.

######  Remark 1.3.

The nilpotentness condition in Theorem 1.2 is not necessary. By using the
trick of [AHL2019] we could deduce the relavant vanishing results for an
arbitrary tame harmonic bundle to Theorem 1.2.

Theorem 1.2 generalizes several recent vanishing results. When $Y$ is a point
and $L$ is ample, Theorem 1.2 is proved by Arapura-Hao-Li [AHL2019] and Deng-
Hao [DF2021], which extends Arapura’s logarithmic Saito-Kodaira vanishing
theorem for complex polarized variations of Hodge structure with unipotency
condition in [Arapura2019]. When $Y$ is a point, ${\bm{a}}={\bm{0}}$ and
$F_{2}=0$, Theorem 1.2 implies J. Suh’s vanishing theorem for the canonical
extension of a polarizable variation of Hodge structure [Suh2018] which
provides many interesting vanishing results on Shimura varieties. We will come
back to this issue in §1.2.2.

The most interesting phenomenon in Theorem 1.2, compared to other vanishing
theorems, is the appearance of the parameter ${\bf a}\in\mathbb{R}^{l}$. There
may exist more than one vanishing result in Theorem 1.2 depending on whether
$-\frac{r_{i}}{m}\in\mathscr{J}_{D_{i}}(E)$ for some $i\in\\{1,\dots,l\\}$.
Assume that

$\mathscr{J}_{D_{i}}(E)=\\{\cdots<c_{i,j-1}<c_{i,j}<c_{i,j+1}<\cdots\\},\quad\forall
1\leq i\leq l.$

For every $1\leq i\leq l$, we denote

$\mathscr{J}_{E,D_{i}}(b)=\begin{cases}\\{c_{i,j-1},c_{i,j}\\},&b=c_{i,j}\\\
\\{c_{i,j}\\},&c_{i,j}<b<c_{i,j+1}.\end{cases}$

Set

$\mathscr{J}_{E,D}({\bf a})=\prod_{i=1}^{l}\mathscr{J}_{E,D_{i}}(a_{i})$

for each ${\bf a}=(a_{1},\dots,a_{l})\in\mathbb{R}^{l}$. The vanishing result
(1.1) is equivalent to the family of vanishing results

(1.2) $\displaystyle R^{i}f_{\ast}({\rm Dol}(_{{\bf a}}E,\theta)\otimes
L)=0,\quad\forall i>n,\quad{\bf a}\in\mathscr{J}_{E,D}(-\frac{\bf{r}}{m}).$

####  1.2.1. Kawamata-Viehweg vanishing theorem and its $(i,j)$ version

Notations as in Theorem 1.2. For an easy example, let us consider the trivial
Higgs bundle $\mathscr{O}_{X\backslash D}$ with vanishing Higgs field. In this
case $\mathscr{J}_{D_{i}}(\mathscr{O}_{X\setminus D})=\mathbb{Z}$ for every
$i=1,\dots,l$. Hence

(1.3)
$\displaystyle\mathscr{J}_{E,D_{i}}(-\frac{r_{i}}{m})=\begin{cases}\\{-\frac{r_{i}}{m}-1,-\frac{r_{i}}{m}\\},&\frac{r_{i}}{m}\in\mathbb{Z}\\\
\\{\lfloor-\frac{r_{i}}{m}\rfloor\\}&\frac{r_{i}}{m}\notin\mathbb{Z}\end{cases}.$

Let $B$ be a big and nef $\mathbb{Q}$-divisor such that $L=\lceil B\rceil$ and
$F_{2}=\lceil B\rceil-B$. In this case $0\leq\frac{r_{i}}{m}<1$ for every
$i=1,\dots,l$. Since

${\rm Dol}({{}_{\bf b}}E,\theta)\simeq\bigoplus_{j=0}^{n}\Omega^{j}_{X}(\log
D)(\sum_{i=1}^{l}\lfloor b_{i}\rfloor
D_{i})[-j],\quad\forall\bm{b}=(b_{1},\dots,b_{l})\in\mathbb{R}^{l},$

we obtain the following Kawamata-Viehweg type vanishing theorem.

######  Corollary 1.4.

Let $X$ be a projective manifold of dimension $n$, $Y$ be an analytic space
and $f:X\rightarrow Y$ be a proper surjective holomorphic map. Let
$D=\sum_{i=1}^{l}D_{i}$ be a normal crossing divisor on $X$. Let $B=A+F$ be a
$\mathbb{Q}$-divisor where $B$ is nef, $A$ is an ample $\mathbb{Q}$-divisor,
${\rm supp}(F)\subset D$ and ${\rm supp}(\lceil B\rceil-B)\subset D$. Then

(1.4) $\displaystyle R^{i}f_{\ast}(\Omega^{j}_{X}(\log
D)\otimes{\mathscr{O}_{X}}(\lceil B\rceil-G))=0,\quad\forall i+j>n$

for every reduced effective divisor $G$ such that ${\rm supp}(\lceil
B\rceil-B)\subset G\subset D$.

When $j=n$, by standard arguments Corollary 1.4 implies the classical
Kawamata-Viehweg vanishing theorem for $\mathbb{Q}$-divisors [Mori1998,
Theorem 2.64] which is widely used in birational geometry. When $B$ is ample
and $Y$ is algebraic, (1.4) has been recently proved by Arapura-Matsuki-Patel-
Włodarczyk [Arapura2018].

####  1.2.2. Saito-type vanishing theorems

Notations as in Theorem 1.2. Assume that $(E,\theta,h)$ is associated with a
polarized $\mathbb{C}$-variation of Hodge structure
$(\mathcal{V},\nabla,\\{F^{\bullet}\\},h)$ via Simpson’s correspondence
[Simpson1988]. For every ${\bm{a}}=(a_{1},\dots,a_{l})\in\mathbb{R}^{l}$, let
$\mathcal{V}_{\geq\bf{a}}$ be the unique locally free $\mathscr{O}_{X}$-module
extending $\mathcal{V}$ such that $\nabla$ induces a connection with
logarithmic singularities
$\nabla:\mathcal{V}_{\geq{\bm{a}}}\to\mathcal{V}_{\geq{\bm{a}}}\otimes\Omega_{X}(\log
D)$ whose real parts of the eigenvalues of the residue of $\nabla$ along
$D_{i}$ belong to $[a_{i},a_{i}+1)$. Let $j:X\backslash D\to X$ be the open
immersion. Denote
$F^{p}_{\geq{\bm{a}}}:=j_{\ast}F^{p}\cap\mathcal{V}_{\geq{\bm{a}}}$. The
recent work of Deng [Deng2022] shows that $F^{p}_{\geq{\bm{a}}}$ is a
subbundle of $\mathcal{V}_{\geq{\bm{a}}}$ and $\nabla$ induces a complex

$F^{p}_{\geq{\bm{a}}}\stackrel{{\scriptstyle\nabla}}{{\to}}F^{p-1}_{\geq{\bm{a}}}\otimes\Omega_{X}(\log
D)\stackrel{{\scriptstyle\nabla}}{{\to}}F^{p-2}_{\geq{\bm{a}}}\otimes\Omega^{2}_{X}(\log
D)\stackrel{{\scriptstyle\nabla}}{{\to}}\cdots,\quad\forall p.$

Denote the graded quotient complex as

${}_{\bf a}{\rm
DR}_{(X,D)}(\mathcal{V},F^{\bullet}):=\bigoplus_{p}F^{p}_{\geq{\bm{a}}}/F^{p+1}_{\geq{\bm{a}}}\stackrel{{\scriptstyle{\rm
Gr}\nabla}}{{\to}}\bigoplus_{p}F^{p-1}_{\geq{\bm{a}}}/F^{p}_{\geq{\bm{a}}}\otimes\Omega_{X}(\log
D)\stackrel{{\scriptstyle{\rm Gr}\nabla}}{{\to}}\cdots.$

By [Simpson1988] one has $E=Gr_{\mathscr{F}}(\mathcal{V})$ and
$\theta=Gr_{\mathscr{F}}(\nabla)$. We obtain that

(1.5) ${}_{\bf a}{\rm DR}_{(X,D)}(\mathcal{V},F^{\bullet})\simeq{\rm
Dol}(_{\bm{a}}E,\theta),\quad\forall\bm{a}\in\mathbb{R}^{l}.$

Thus Theorem 1.2 implies the following Saito-type vanishing result.

######  Corollary 1.5.

Let $X$ be a projective manifold of dimension $n$, $Y$ be an analytic space
and $f:X\rightarrow Y$ be a proper surjective holomorphic map. Let
$D=\sum_{i=1}^{l}D_{i}$ be a normal crossing divisor on $X$. Let
$(\mathcal{V},\nabla,\\{F^{\bullet}\\},h)$ be a polarizable
$\mathbb{C}$-variation of Hodge structure on $X\backslash D$. Let $L$ be a
holomorphic line bundle on $X$. Suppose that some positive multiple
$mL=A+F_{1}+F_{2}$ where $A$ is an ample line bundle, $F_{1}$ and $F_{2}$ are
effective divisors supported in $D$ such that $A+F_{1}$ is a nef holomorphic
line bundle. Suppose that $F_{2}=\sum_{i=1}^{l}r_{i}D_{i}$ with
$r_{i}\in\mathbb{Z}_{\geq 0},\forall i$. Denote
${\bf{r}}:=(r_{1},\dots,r_{l})$.

Then

$R^{i}f_{\ast}\left({}_{-\frac{\bf{r}}{m}+{\bf a}}{\rm
DR}_{(X,D)}(\mathcal{V},F^{\bullet})\otimes L\right)=0$

for any $i>n$ and any ${\bf a}=(a_{1},\dots,a_{l})\in\mathbb{R}^{l}$ such that
$|a_{j}|<\sigma_{E,j}(-\frac{r_{j}}{m})$ for every $j=1,\dots,l$.

When $Y$ is a point, $F_{2}\simeq\mathscr{O}_{X}$, ${\bf a}={\bf 0}$ and
$(\mathcal{V},\nabla,\\{F^{\bullet}\\},h)$ is an $\mathbb{R}$-polarized
variation of Hodge structure, it is proved by J. Suh [Suh2018] by using the
theory of Hodge modules. As noted above, there are actually many other
vanishing theorems in the setting of Corollary 1.5 when $-\frac{r_{i}}{m}$ is
the jumping index with respect to $D_{i}$ for some $i$. The jumping indices of
the set of prolongations $\\{{{}_{\bm{a}}}E|{\bm{a}}\in\mathbb{R}^{l}\\}$ are
closely related to the monodromies of the flat bundle $(\mathcal{V},\nabla)$.
Let ${\rm Res}_{i}(\nabla)$ be a residue of the monodromy of
$(\mathcal{V},\nabla)$ along $D_{i}$ whose real parts of the eigenvalues lie
in $[0,1)$. Let

$J_{i}=\\{\textrm{the real part of an eigenvalue of the residue of
}(\mathcal{V},\nabla)\textrm{ along }D_{i}\\}$

By Simpson’s table [Simpson1990, p. 720], one obtains that

(1.6) $\displaystyle\mathscr{J}_{D_{i}}(E)=J_{i}+\mathbb{Z}.$

Notations:

  * •

For a holomorphic vector bundle $E$ on a complex manifold $X$, we denote
$\bar{\partial}_{E}$ to be its holomorphic structure. We write it as
$\bar{\partial}$ if no ambiguity appears.

  * •

We say that two metrics $ds_{1}^{2}$ and $ds_{2}^{2}$ are quasi-isometric,
i.e., $ds_{1}^{2}\sim ds_{2}^{2}$, if there exists a constant $C>0$ such that
$C^{-1}ds_{1}^{2}\leq ds_{2}^{2}\leq Cds_{1}^{2}$. Two hermitian metrics $h$
and $h^{\prime}$ of a holomorphic vector bundle on $X$ are called quasi-
isometric if there exits a contant $C>0$ such that $C^{-1}h<h^{\prime}<Ch$. We
denote it by $h\sim h^{\prime}$.

  * •

Let $\alpha$ and $\beta$ be functions, metrics or $(1,1)$-forms. We denote
$\alpha\lesssim\beta$ if $\alpha\leq C\beta$ for some $C\in\mathbb{R}_{>0}$.
We say that $\alpha$ and $\beta$ are quasi-isometric if $\alpha\lesssim\beta$
and $\beta\lesssim\alpha$. We denote it by $\alpha\sim\beta$.

  * •

For a hermitian vector bundle $(E,h)$ on a complex manifold $X$, we always
denote $R(E,h)$ (or simply $R(h)$) to be its Chern curvature.

  * •

Let $\Theta\in A^{1,1}(X,{\rm End}(E))$ be a real form. Assume locally that

$\Theta=\sqrt{-1}\sum_{i,j}\omega_{ij}e_{i}\otimes e_{j}^{\ast}$

where $\omega_{ij}\in A^{1,1}_{X}$, $(e_{1},\dots,e_{r})\in E$ is an
orthogonal local frame of $E$ and $(e^{\ast}_{1},\dots,e^{\ast}_{r})\in
E^{\ast}$ is the dual frame. We say that $\Theta$ is Nakano semi-positive,
denoting $\Theta\geq_{{\rm Nak}}0$, if the bilinear form

$\displaystyle\theta(u_{1},u_{2}):=\sum_{i,j}\omega_{ij}(u_{1i},\overline{u_{2j}}),\quad{\textrm{where}}\quad
u_{l}=\sum_{i}u_{lk}\otimes e_{k}\in T_{X}\otimes E,\quad\forall l=1,2$

is semi-positive definite.

Let $\Theta_{1},\Theta_{2}\in A^{1,1}(X,{\rm End}(E))$ be two real forms. We
denote $\Theta_{1}\geq_{{\rm Nak}}\Theta_{2}$ if
$\Theta_{1}-\Theta_{2}\geq_{{\rm Nak}}0$. $(E,h)$ is Nakano semi-positive if
$\sqrt{-1}R(E,h)\geq_{{\rm Nak}}0$.

##  2\. Preliminary

In this section, we review the basic knowledge of Higgs bundles and nilpotent
harmonic bundles. References include [Simpson1988, Simpson1990, Simpson1992,
Mochizuki20071, Mochizuki20072] and so on.

###  2.1. Nilpotent harmonic bundles

######  Definition 2.1.

Let $X$ be a complex manifold. A Higgs bundle on $X$ is a pair $(E,\theta)$
where $E$ is a holomorphic vector bundle on $X$ together with
$\bar{\partial}_{E}$ its holomorphic structure, and $\theta:E\rightarrow
E\otimes\Omega_{X}^{1}$ is a holomorphic one form such that
$\theta\wedge\theta=0$ in $\rm End$$(E)\otimes\Omega_{X}^{2}$. Here $\theta$
is called the Higgs field.

Let $X$ be a complex manifold and let $(E,\theta)$ be a Higgs bundle on $X$.
Define an operator $D^{\prime\prime}=\bar{\partial}_{E}+\theta$. Then
$D^{\prime\prime 2}=0$. Consider a smooth hermitian metric $h$ on $E$. Let
$\partial_{h}+\bar{\partial}_{E}$ be the Chern connection associated to $h$
and denote $\theta_{h}^{\ast}$ to be the adjoint of $\theta$ with respect to
$h$. Denote $D_{h}^{\prime}:=\partial_{h}+\theta_{h}^{\ast}$.

######  Definition 2.2.

A smooth hermitian metric $h$ on a Higgs bundle $(E,\theta)$ is called
harmonic if the operator $D_{h}:=D_{h}^{\prime}+D^{\prime\prime}$ is
integrable, that is, $D_{h}^{2}=0$. A harmonic bundle is a Higgs bundle
endowed with a harmonic metric.

Let $(E,\theta,h)$ be a harmonic bundle. Then one has the self-dual equation

(2.1) $\displaystyle R(h)+[\theta,\theta_{h}^{\ast}]=0.$

Let $X$ be an $n$-dimensional complex manifold and $D=\cup_{i\in I}D_{i}$ a
simple normal crossing divisor on $X$.

######  Definition 2.3 (Admissible coordinate).

Let $p$ be a point of $X$ and $\\{D_{j}\\}_{j=1,\dots,l}$ the components of
$D$ containing $p$. An admissible coordinate around $p$ is the tuple
$(U,\varphi)$ (or $(U;z_{1},\dots,z_{n})$ if no ambiguity appears) such that

  1. (1)

$U$ is an open subset of $X$ containing $p$.

  2. (2)

$\varphi$ is a biholomorphic morphism
$U\rightarrow\Delta^{n}=\\{(z_{1},\dots,z_{n})\in\mathbb{C}^{n}\big{|}|z_{i}|<1,\forall
1\leq i\leq n\\}$ such that $\varphi(p)=(0,\dots,0)$ and
$\varphi(D_{j})=\\{z_{j}=0\\}$ for any $j=1,\dots,l$.

######  Definition 2.4.

Let $(E,\theta,h)$ be a harmonic bundle of rank $r$ defined on $X\backslash
D$. Let $p$ be any point of $X$ and $(U,\varphi)$ an admissible coordinate
around $p$. On $U$, we have the description:

(2.2) $\displaystyle\theta=\sum_{j=1}^{l}f_{j}\cdot d\log
z_{j}+\sum_{k=l+1}^{n}g_{k}\cdot dz_{k}.$

  1. (1)

(tameness) Let $t$ be a formal variable. We have the polynomials
$\textrm{det}(t-f_{j})$ and $\textrm{det}(t-g_{k})$ of $t$, whose coefficients
are holomorphic functions defined over $U\backslash{\cup_{j=1}^{l}D_{j}}$.
When the functions are extended to the holomorphic functions over $U$, the
harmonic bundle is called tame at $p$. A harmonic bundle is called tame if it
is tame at every point $p\in X$.

  2. (2)

(nilpotentness) When $\textrm{det}(t-f_{j})|_{U\cap D_{j}}=t^{r}$, the
harmonic bundle is called nilpotent at $p$. When $(E,h,\theta)$ is nilpotent
at any point $p\in X$, it is called a nilpotent harmonic bundle.

###  2.2. Boundedness for the Higgs field

Denote $\Delta$ (resp. $\Delta^{\ast}$) to be the unit disc (resp. punctured
unit disc) in $\mathbb{C}$. A Poincaré metric $\omega_{P}$ on
$(\Delta^{\ast})^{l}\times\Delta^{n-l}$ is defined as

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

It can be defined by a potential function as

$\omega_{P}=-\sqrt{-1}\partial\bar{\partial}\log(\prod_{j=1}^{l}(-\log|{z}_{j}|^{2}))+\sqrt{-1}\partial\bar{\partial}\sum_{k=l+1}^{n}|z_{k}|^{2}.$

######  Definition 2.5 (Poincaré type metric).

A metric $ds^{2}$ on $X\backslash D$ is said to have Poincaré type growth near
the divisor $D$ if, for every point $p\in D$ there is a coordinate
neighborhood $U_{p}\subset X$ of $p$ with $U_{p}\cap(X\backslash
D)\simeq(\Delta^{\ast})^{l}\times\Delta^{n-l}$ for some $1\leq l\leq n$ such
that in these coordinates, $ds^{2}$ is quasi-isometric to $\omega_{P}$.

We will always denote $\omega_{P}$ to be the Poincaré type metric if no
ambiguity causes.

######  Proposition 2.6 ([CG1975, Zucker1979]).

The Poincaré type metric has finite volume, bounded curvature tensor and
bounded covariant derivatives.

For nilpotent harmonic bundles, the following important norm estimates lead to
the boundedness of the Higgs field $\theta$ with respect to the Poincaré type
metric.

######  Theorem 2.7 ([Simpson1990], Theorem 1 and [Mochizuki2002], Proposition
4.1).

Let $(E,\theta,h)$ be a nilpotent harmonic bundle on $X\backslash D$. Let
$f_{j},g_{k}$ be the matrix valued holomorphic functions as in (2.2). Then
there exists a positive constant $C>0$ such that

$|f_{j}|_{h}\leq C(-\log|z_{j}|^{2})^{-1},\quad\quad\textrm{for}\quad
j=1,\dots,l;$ $|g_{k}|_{h}\leq C,\quad\quad\textrm{for}\quad k=l+1,\dots,n.$

Therefore

$|\theta|_{h,\omega_{P}}\leq C$

holds on $U\backslash D$ for some admissible neighborhood $U$.

##  3\. $L^{2}$ cohomology and $L^{2}$ complex

Let $(X,ds^{2})$ be a complex hermitian manifold of dimension $n$ and $D$ a
normal crossing divisor on $X$. Let $(E,h)$ be a hermitian holomorphic vector
bundle on $X^{\ast}:=X\backslash D$. It gives us an inner product on the
vector space of $E$-valued $(p,q)$-forms

$\langle u,v\rangle_{h,ds^{2}}:=\int_{X^{\ast}}(u,v)_{h,ds^{2}}{\rm
vol}_{ds^{2}}$

where $(u,v)_{h,ds^{2}}$ is the pointwise inner product of $u$ and $v$ with
respect to $h$ and $ds^{2}$. The $L^{2}$ norm of $u$ is defined as

$\|u\|_{h,ds^{2}}=\sqrt{\langle u,u\rangle_{h,ds^{2}}}.$

Let $\mathscr{A}^{p,q}_{X^{\ast}}$ denote the sheaf of smooth $(p,q)$-forms on
$X^{\ast}$ for every $0\leq p,q\leq n$. Denote $\bar{\partial}:E\to
E\otimes\mathscr{A}^{0,1}_{X^{\ast}}$ to be the canonical $\bar{\partial}$
operator. Let $L^{p,q}_{(2)}(X^{\ast},E;ds^{2},h)$ (resp.
$L^{m}_{(2)}(X^{\ast},E;ds^{2},h)$) be the space of square integrable
$E$-valued $(p,q)$-forms (resp. $m$-forms) on $X^{\ast}$ with respect to the
metrics $ds^{2}$ and $h$. Denote $\bar{\partial}_{\rm max}$ to be the maximal
extension of the $\bar{\partial}$ operator defined on the domains

$D^{p,q}_{X^{\ast},ds^{2}}(E,h):=\textrm{Dom}^{p,q}(\bar{\partial}_{\rm
max})=\\{\phi\in L_{(2)}^{p,q}(X^{\ast},E;ds^{2},h)|\bar{\partial}\phi\in
L_{(2)}^{p,q+1}(X^{\ast},E;ds^{2},h)\\}.$

Here $\bar{\partial}$ is taken in the sense of distribution.

The $L^{2}$-$\bar{\partial}$ cohomology $H_{(2),\rm
max}^{p,\bullet}(X^{\ast},E;ds^{2},h)$ is defined as the cohomology of the
complex

(3.1) $\displaystyle
D^{p,\bullet}_{X^{\ast},ds^{2}}(E,h):=D^{p,0}_{X^{\ast},ds^{2}}(E,h)\stackrel{{\scriptstyle\bar{\partial}_{\rm
max}}}{{\to}}\cdots\stackrel{{\scriptstyle\bar{\partial}_{\rm
max}}}{{\to}}D^{p,n}_{X^{\ast},ds^{2}}(E,h).$

Let $U\subset X$ be an open subset. Define $L_{X,ds^{2}}^{p,q}(E,h)(U)$ (resp.
$L_{X,ds^{2}}^{m}(E,h)(U)$) to be the space of measurable $E$-valued
$(p,q)$-forms (resp. $m$-forms) $\alpha$ on $U^{\ast}:=U\backslash D$ such
that for every point $x\in U$, there is a neighborhood $V_{x}$ of $x$ so that

$\int_{V_{x}\cap U^{\ast}}|\alpha|^{2}_{ds^{2},h}{\rm vol}_{ds^{2}}<\infty.$

For each $p$ and $q$, we define a sheaf $\mathscr{D}_{X,ds^{2}}^{p,q}(E,h)$ on
$X$ by

$\mathscr{D}_{X,ds^{2}}^{p,q}(E,h)(U):=\\{\sigma\in
L_{X,ds^{2}}^{p,q}(E,h)(U)|\bar{\partial}_{\rm max}\sigma\in
L_{X,ds^{2}}^{p,q+1}(E,h)(U)\\}$

for every open subset $U\subset X$. Define the $L^{2}$-Dolbeault complex of
sheaves $\mathscr{D}_{X,ds^{2}}^{p,\bullet}(E,h)$ as

(3.2)
$\displaystyle\mathscr{D}_{X,ds^{2}}^{p,0}(E,h)\stackrel{{\scriptstyle\bar{\partial}}}{{\to}}\mathscr{D}_{X,ds^{2}}^{p,1}(E,h)\stackrel{{\scriptstyle\bar{\partial}}}{{\to}}\cdots\stackrel{{\scriptstyle\bar{\partial}}}{{\to}}\mathscr{D}_{X,ds^{2}}^{p,n}(E,h)$

where $\bar{\partial}$ is defined in the sense of distribution.

######  Lemma 3.1 (fineness of the $L^{2}$ sheaf).

Let $X$ be a compact complex manifold and $D$ a simple normal crossing divisor
on $X$. Let $ds^{2}$ be a hermitian metric on $X^{\ast}:=X\backslash D$ which
has Poincaré type growth along $D$ and let $(E,h)$ be a holomorphic vector
bundle on $X^{\ast}$ with a possibly singular hermitian metric. Then
$\mathscr{D}^{p,q}_{X,ds^{2}}(E,h)$ is a fine sheaf for each $p$ and $q$.

###### Proof.

Take an open subset $U\subset X$. It suffices to show that
$\eta\alpha\in\mathscr{D}_{X,ds^{2}}^{p,q}(E,h)(U)$ for every
$\alpha\in\mathscr{D}_{X,ds^{2}}^{p,q}(E,h)(U)$ and every $\eta\in A^{0}_{\rm
cpt}(U)$. Due to the asymptotic behavior of $ds^{2}$ (Proposition 2.6),
$|\eta|$ and $|\bar{\partial}\eta|_{ds^{2}}$ are $L^{\infty}$ bounded on $U$.
If we assume that $\alpha$ and $\bar{\partial}\alpha$ are locally $L^{2}$
integrable, then

$\eta\alpha\in
L_{X,ds^{2}}^{p,q}(E,h)(U),\quad\bar{\partial}(\eta\alpha)=\bar{\partial}\eta\wedge\alpha+\eta\wedge\bar{\partial}\alpha\in
L_{X,ds^{2}}^{p,q+1}(E,h)(U).$

Thus the lemma is proved. ∎

##  4\. $L^{2}$ representation of the prolongation of Higgs bundles

The purpose of this section is to establish a fine $L^{2}$ bi-complex
resolution of the prolongation of Higgs bundles.

###  4.1. Prolongation and parabolic structure

Let $X$ be a complex manifold of dimension $n$ and $D$ a normal crossing
divisor on $X$. Let $(E,h)$ be a holomorphic vector bundle on $X\backslash D$
with a smooth hermitian metric $h$. Let ${\bf
a}=(a_{1},\dots,a_{l})\in\mathbb{R}^{l}$ be a tuple of real numbers. For
$\bm{b}=(b_{1},\dots,b_{l})\in\mathbb{R}^{l}$, we denote $\bm{b}<\bm{a}$ if
$b_{i}<a_{i}$ for every $i=1,\dots,l$.

######  Definition 4.1.

Let $X$ be a complex manifold and $D=\cup_{i=1}^{l}D_{i}$ a simple normal
crossing divisor on $X$. A parabolic higgs bundle is a triple
$(\\{E_{\bm{a}}\\},\theta)$ where for each $\bm{a}\in\mathbb{R}^{l}$,
$E_{\bm{a}}$ is a locally free coherent sheaf such that the following hold.

  * •

$E_{\bm{a}+\bm{\epsilon}}=E{{}_{\bm{a}}}$ for any vector
$\bm{\epsilon}=(\epsilon_{1},\cdots,\epsilon_{l})$ with $0<\epsilon_{i}\ll
1,\forall 1\leq i\leq l$.

  * •

$E_{\bm{a}-{\bf 1}_{i}}={E_{\bm{a}}}\otimes\mathscr{O}(-D_{i})$ for every
$1\leq i\leq l$. Here ${\bf 1}_{i}$ denotes $(0,\dots,0,1,0,\dots,0)$ with 1
in the $i$-th component.

  * •

The set of $\bm{a}$ such that ${\rm Gr}_{\bm{a}}E\neq 0$ is discrete in
$\mathbb{R}^{l}$. Such $\bf{a}$ are called the weights.

  * •

The Higgs field $\theta$ has at most logarithmic poles on $E_{\bm{a}}$, that
is, $\theta$ can be extended to

(4.1) $\displaystyle E_{\bm{a}}\to E_{\bm{a}}\otimes\Omega_{X}(\log D)$

for every $\bm{a}\in\mathbb{R}^{l}$.

######  Definition 4.2 (Prolongation).

(Mochizuki[Mochizuki2002], Definition 4.2) Let $U$ be an open subset of $X$
admissible to $D$. For any section $s\in\Gamma(U\backslash D,E)$, let
$|s|_{h}$ denote the norm function of $s$ with respect to the metric $h$. We
describe $|s|_{h}=O(\prod_{i=1}^{l}|z_{i}|^{-a_{i}})$ if there exists a
positive number $C$ such that

$|s|_{h}\leq C\cdot\prod_{i=1}^{l}|z_{i}|^{-a_{i}}.$

We call $-\rm{ord}(s)\leq\bf{a}$ if
$|s|_{h}=O(\prod_{i=1}^{l}|z_{i}|^{-a_{i}-\epsilon})$ for any positive number
$\epsilon$.

The $\mathscr{O}_{X}$-module ${}_{\bm{a}}E$ is defined as follows: For any
open subset $U\subset X$,

$\Gamma(U,{{}_{\bm{a}}E}):=\\{s\in\Gamma(U\backslash
D,E)|-\textrm{ord}(s)\leq{\bm{a}}\\}.$

The sheaf ${}_{\bf a}E$ is called the prolongment of $E$ by an increasing
order $\bm{a}$. Denote

(4.2) $\displaystyle{\rm
Gr}_{\bm{a}}E:={{}_{\bm{a}}}E/\cup_{\bm{b}<\bm{a}}{{}_{\bm{b}}}E.$

######  Theorem 4.3 ([Mochizuki2009], Proposition 2.53).

Let $X$ be a complex manifold and $D=\cup_{i=1}^{l}D_{i}$ a simple normal
crossing divisor on $X$. Let $(E,\theta,h)$ be a tame harmonic bundle on
$X\backslash D$. Then $(\\{_{\bm{a}}E\\},\theta)$ is a parabolic higgs bundle.
The same conclusions hold for the flat bundle $(\mathcal{V},\nabla,h)$
associated with $(E,\theta,h)$ via Simpson’s correspondence.

The following norm estimate for meromorphic sections is crucial in the proof
of our main theorem.

######  Theorem 4.4 ([Mochizuki20072], Part 3, Chapter 13).

Let $(\mathcal{V},\nabla,h)$ be a tame harmonic bundle on
$X^{\ast}=(\Delta^{\ast})^{l}\times\Delta^{n-l}$. Let

$p:\mathbb{H}^{l}\times\Delta^{n-l}\to(\Delta^{\ast})^{l}\times\Delta^{n-l},$
$(z^{\prime}_{1},\dots,z^{\prime}_{l},w_{1},\dots,w_{n-l})\mapsto(e^{2\pi\sqrt{-1}z^{\prime}_{1}},\dots,e^{2\pi\sqrt{-1}z^{\prime}_{l}},w_{1},\dots,w_{n-l})$

be the universal covering. Let ${\bf a}=(a_{1},\dots,a_{l})\in\mathbb{R}^{l}$.
Let $W^{(1)}=W(N_{1}),\dots,W^{(n)}=W(N_{1}+\cdots+N_{n})$ be the residue
weight filtrations on
$V:=\Gamma(\mathbb{H}^{n},p^{\ast}\mathcal{V})^{p^{\ast}\nabla}$. Then for any
$v\in V$ such that

$0\neq[v]\in{\rm Gr}_{l_{n}}^{W^{(n)}}\cdots{\rm
Gr}_{l_{1}}^{W^{(1)}}V\cap{\rm Gr}_{\bf a}\mathcal{V},$

one has

(4.3)
$\displaystyle|v|_{h_{\mathbb{V}}}\sim|s_{1}|^{-a_{1}}\cdots|s_{l}|^{-a_{l}}\left(\frac{\log|s_{1}|}{\log|s_{2}|}\right)^{l_{1}}\cdots\left(-\log|s_{l}|\right)^{l_{n}}$

over any region of the form

$\left\\{(s_{1},\dots,s_{l},w_{1},\dots,w_{n-l})\in(\Delta^{\ast})^{l}\times\Delta^{n-l}\bigg{|}\frac{\log|s_{1}|}{\log|s_{2}|}>\epsilon,\dots,-\log|s_{l}|>\epsilon,(w_{1},\dots,w_{n-l})\in
K\right\\}$

for any $\epsilon>0$ and an arbitrary compact subset $K\subset\Delta^{n-l}$.
The same conclusion holds for the holomorphic sections in the Higgs bundle
$(E,\theta,h)$ associated with $(\mathcal{V},\nabla,h)$ via Simpson’s
correspondence.

###  4.2. $L^{2}$ representation

Let $X$ be a pre-compact open subset of a hermitian manifold $(M,\omega_{M})$
and let $D=\cup_{i=1}^{l}D_{i}$ be a simple normal crossing divisor on $M$.
Let $\sigma_{i}\in H^{0}(M,\mathscr{O}_{M}(D_{i}))$ be the defining section of
$D_{i}$ and $h_{i}$ an arbitrary hermitian metric on $\mathscr{O}_{M}(D_{i})$.
We denote $\varphi_{i}=|\sigma_{i}|_{h_{i}}$. Let $\phi_{P}\in
C^{\infty}(X\backslash D)$ such that
$\omega_{P}:=\sqrt{-1}\partial\bar{\partial}\phi_{P}+\omega_{M}|_{X\backslash
D}$ is a hermitian metric on $X\backslash D$ which has Poincaré type growth
near $D\cap X$. Let $(E,\theta,h)$ be a nilpotent harmonic bundle on
$M\backslash D$ together with $\bar{\partial}_{E}$ its holomorphic structure.
For every indices ${\bf a}=(a_{1},\dots,a_{l})\in\mathbb{R}^{l}$, denote
${}_{\bf a}E$ to be the associated prolongation of $(E,\theta)$. We denote by

${\rm Dol}(_{\bf a}E,\theta):={{}_{\bf a}}E\to{{}_{\bf
a}}E\otimes\Omega_{X}(\log D)\to\cdots$

the associated logarithmic Dolbeault complex.

For $N\in\mathbb{Z}$, denote $h_{N}({\bf
a}):=he^{-N\phi_{P}+\sum_{i=1}^{l}2a_{i}\log\varphi_{i}}$ to be a modified
metric on $E$. Define the operator
$D^{\prime\prime}=\bar{\partial}_{E}+\theta$. Denote by
$\mathscr{D}^{k}_{X,\omega_{P}}(E,D^{\prime\prime},h_{N}({\bf a}))$ the sheaf
of measurable $E$-valued $k$-forms $\alpha$ such that $\alpha$ and
$D^{\prime\prime}\alpha$ are locally square integrable near every point of
$X$. Denote

$\mathscr{D}^{\bullet}_{X,\omega_{P}}(E,D^{\prime\prime},h_{N}({\bf
a})):=\mathscr{D}^{0}_{X,\omega_{P}}(E,D^{\prime\prime},h_{N}({\bf
a}))\stackrel{{\scriptstyle
D^{\prime\prime}}}{{\to}}\mathscr{D}^{1}_{X,\omega_{P}}(E,D^{\prime\prime},h_{N}({\bf
a}))\stackrel{{\scriptstyle D^{\prime\prime}}}{{\to}}\cdots$

to be the associated $L^{2}$-Dolbeault complex. There is a natural inclusion

${\rm Dol}(_{\bf
a}E,\theta)\to\mathscr{D}^{\bullet}_{X,\omega_{P}}(E,D^{\prime\prime},h_{N}({\bf
a}+\bm{\epsilon}))$

for every $\bm{0}<\bm{\epsilon}\in\mathbb{R}^{l}$.

For every ${\bf a}=(a_{1},\dots,a_{l})\in\mathbb{R}^{l}$, define

$\sigma_{E,i}(a_{i})=\min\\{|b-a_{i}||b\in\mathscr{J}_{D_{i}}(E),b\neq
a_{i}\\},\quad\forall 1\leq i\leq l$

and $\bm{\sigma}_{E}({\bf
a})=(\sigma_{E,1}(a_{1}),\dots,\sigma_{E,l}(a_{l}))$. The main result of this
section is the following.

######  Theorem 4.5.

There is a constant $N_{0}$, depending only on $(E,\theta,h)$ and $X$
(independent of $\bm{a}$ and $\bm{\epsilon}$ in the following), such that the
inclusion map

${\rm Dol}(_{{\bf
a}}E,\theta)\to\mathscr{D}^{\bullet}_{X,\omega_{P}}(E,D^{\prime\prime},h_{N}({\bf
a}+\bm{\epsilon}))$

is a quasi-isomorphism for every ${\bf a}\in\mathbb{R}^{l}$, every $N>N_{0}$
and every $\bm{0}<\bm{\epsilon}<\bm{\sigma}_{E}({\bf a})$.

The proof of the theorem is postponed to the end of this section.

###  4.3. Local behavior of the metrics

Since the statement of Theorem 4.5 is local, we assume that $X_{1}=\Delta^{n}$
and $D=\sum_{i=1}^{l}D_{i}$ with $D_{i}=\\{z_{i}=0\\}$ for each $i$. Let
$(E,h)$ be a nilpotent harmonic bundle over $X_{1}^{\ast}:=X_{1}\backslash D$.

For any $N\in\mathbb{Z}_{>0}$ and any
${\bf{a}}=(a_{1},\cdots,a_{l})\in\mathbb{R}^{l}$, we define

(4.4)
$\displaystyle\kappa({\bf{a}},N):=-N(\sum_{j=1}^{l}\log(-\log|z_{j}|^{2})-\sum_{k=l+1}^{n}|z_{k}|^{2})-\sum_{j=1}^{l}a_{j}{\rm{log}}|z_{j}|^{2}.$

Set $h({\bf{a}},N):=h\cdot e^{-\kappa({\bf{a}},N)}$. Then

$R(h({\bf{a}},N))=R(h)+\sqrt{-1}\partial\bar{\partial}\kappa({\bf{a}},N)=R(h)+N\omega_{P}.$

######  Remark 4.6.

For the general settings as in §4.2, the modified hermitian metric
$h_{N}({\bf{a}}):=h\cdot(-\prod_{i=1}^{l}\log|\sigma_{i}|_{h_{i}}^{2})^{N}\cdot\prod_{i=1}^{l}|\sigma_{i}|_{h_{i}}^{2a_{i}}$
is locally quasi-isometric to $h({\bf{a}},N)$.

For every $1\leq i\leq n$, let $p_{i}$ be the projection from
$(\Delta^{\ast})^{l}\times\Delta^{n-l}$ to its $i$-th factor. Note that
$\Omega_{X_{1}^{\ast}}=\oplus_{i=1}^{n}L_{i}$ where $L_{i}$ is the trivial
line bundle defined by $L_{i}:=p_{i}^{\ast}\Omega_{\Delta^{\ast}}$ for
$i=1,...,l$ and $L_{i}=p_{i}^{\ast}\Omega_{\Delta}$ for $i=l+1,...,n$. For any
$p=0,...,n$, set $h_{p}$ to be the hermitian metric on $T_{X_{1}^{\ast}}^{p}$
induced by $\omega_{P}$. Then there is a positive constant $C(p,l)>0$
depending only on $p$ and $l$ so that $|R(h_{p})|_{h_{p},\omega_{P}}\leq
C(p,l)$. Set $C_{0}:=\textrm{sup}_{p=0,...,n;l=1,...,n}C(p,l)$.

######  Proposition 4.7.

Let $(E,\theta,h)$ be a nilpotent harmonic bundle over $X_{1}^{\ast}$. Then
there exists a constant $N_{0}>0$ so that the following property holds after a
possible shrinking of $X_{1}$. For the vector bundle
$\mathcal{E}_{p}:=T_{X_{1}^{\ast}}^{p}\otimes E$ endowed with the metric
$h_{\mathcal{E}_{p}}$ induced by $h({\bf{a}},N)$ and $\omega_{P}$, one has the
following estimate

(4.5) $\displaystyle\sqrt{-1}R(h_{\mathcal{E}_{p}})\geq_{{\rm
Nak}}\omega_{P}\otimes{\rm Id}_{\mathcal{E}_{p}}$

over $X_{1}^{\ast}$ for any $N\geq N_{0}$. Such $N_{0}$ does not depend on the
choice of ${\bf{a}}$.

###### Proof.

Since $(E,\theta,h)$ is a nilpotent harmonic bundle, $|\theta|_{h,\omega_{P}}$
is bounded after a possible shrinking of $X_{1}$ (Theorem 2.7). This, together
with the formula $R(h)=-[\theta,\theta_{h}^{\ast}]$ (2.1), implies that
$|R(h)|_{h,\omega_{P}}\leq C$ for some positive constant $C$. Therefore

$\sqrt{-1}R(h)\geq_{\rm Nak}-C\omega_{P}\otimes{\rm Id}_{E}.$

We also know from $|R(h_{p})|_{h_{p},\omega_{P}}\leq C_{0}$ that
$\sqrt{-1}R(h_{p})\geq_{\rm Nak}-C_{0}\omega_{P}\otimes{\rm
Id}_{T_{X_{1}^{\ast}}^{p}}$.

Hence

$\sqrt{-1}R(h_{p}h)\geq_{\rm Nak}-(C+C_{0})\omega_{P}\otimes{\rm
Id}_{\mathscr{E}_{p}}.$

Modifying the metric $hh_{p}$ to $h({\bf a},N)h_{p}$ on $\mathscr{E}_{p}$, we
obtain that

$\sqrt{-1}R(h_{p}h({\bf a},N))\geq_{\textrm{Nak}}(N-C-C_{0})\omega_{P}\otimes
I_{\mathscr{E}_{p}}.$

Taking $N\geq N_{0}=C+C_{0}+1$, we get the desired result. ∎

###  4.4. Fine bi-complex resolution

Since $|\theta|_{h_{N}({\bf a}),\omega_{P}}=|\theta|_{h,\omega_{P}}$ is
bounded by Theorem 2.7,

$\theta:\mathscr{D}_{X,\omega_{P}}^{p,q}(E,h_{N}({\bf{a}}))\rightarrow\mathscr{D}_{X,\omega_{P}}^{p+1,q}(E,h_{N}({\bf{a}}))$

is bounded for each $0\leq p,q\leq n$.

There is therefore the decomposition

(4.6)
$\displaystyle\mathscr{D}_{X,\omega_{P}}^{m}(E,D^{\prime\prime},h_{N}({\bf{a}}))=\bigoplus_{p+q=m}\mathscr{D}_{X,\omega_{P}}^{p,q}(E,h_{N}({\bf{a}})).$

To construct the $L^{2}$ bi-complex resolution, we recall Demailly’s
formulation of Hormander estimate to solving $\bar{\partial}$-equations on an
incomplete Kähler manifold that admits a complete Kähler metric.

######  Theorem 4.8 ([Demailly1982], Theorem 4.1).

Let $X$ be a complete Kähler manifold with a (possibly) incomplete Kähler
metric $\omega$. Let $(E,h)$ be a smooth hermitian vector bundle on $X$ such
that

$\sqrt{-1}R(h)\geq_{{\rm Nak}}\epsilon\omega\otimes{\rm Id}_{\rm E},$

where $\epsilon>0$ is a positive constant. For every $q\geq 1$, assume that
$g\in L_{(2)}^{n,q}(X,E;\omega,h)$ such that $\bar{\partial}g=0$. Then there
exists $f\in L_{(2)}^{n,q-1}(X,E;\omega,h)$ so that $\bar{\partial}f=g$ and

$\|f\|_{h,\omega}^{2}\leq\frac{1}{\epsilon}\|g\|_{h,\omega}^{2}.$

Now we are ready to prove the following proposition which gives the fine bi-
complex resolution.

######  Proposition 4.9.

Notations as in §4.3. Let $(E,h)$ be a hermitian holomorphic vector bundle on
$X_{1}^{\ast}$. Denote by $\omega_{P}=\sqrt{-1}\partial\bar{\partial}\phi_{P}$
the Poincaré metric on $X_{1}^{\ast}$ as in §2.2. Let $\tilde{h}:=h\cdot
e^{-N\phi_{P}+\sum_{j=1}^{l}a_{j}{\log}|z_{j}|^{2}}$ be a modified metric on
$E$ such that

(4.7)
$\displaystyle\sqrt{-1}R(\mathscr{E}_{p},h_{p}\otimes\tilde{h})\geq\omega_{P}\otimes{\rm
Id}_{\mathscr{E}_{p}},\quad\forall 0\leq p\leq n.$

Here $\mathscr{E}_{p}=T_{X_{1}^{\ast}}^{p}\otimes E$ and $h_{p}$ is the metric
on $T^{p}_{X_{1}^{\ast}}$ induced by $\omega_{P}$. Then the complex

$D_{X_{1},\omega_{P}}^{p,0}(E,\tilde{h})\xrightarrow{\bar{\partial}}D_{X_{1},\omega_{P}}^{p,1}(E,\tilde{h})\xrightarrow{\bar{\partial}}\cdots\xrightarrow{\bar{\partial}}D_{X_{1},\omega_{P}}^{p,n}(E,\tilde{h})$

is exact at each $q\geq 1$ for every $p$.

###### Proof.

First, notice that $X^{\ast}_{1}$ admits a complete Kähler metric by modifying
$\omega_{P}$ to
$\omega_{P}+\sqrt{-1}\partial\bar{\partial}\sum_{i=1}^{n}(1-|z_{i}|^{2})^{-1}$.
Notice that

(4.8) $\displaystyle
L_{(2)}^{n,q}(X_{1}^{\ast},\mathcal{E}_{n-p};\omega_{P},h_{n-p}\otimes\tilde{h})\simeq
L_{(2)}^{p,q}(X_{1}^{\ast},E;\omega_{P},\tilde{h})$

holds for any $0\leq p,q\leq n$. For any $q\geq 1$, any $g\in
L_{(2)}^{n,q}(X_{1}^{\ast},\mathcal{E}_{n-p};\omega_{P},h_{n-p}\otimes\tilde{h})$
with $\bar{\partial}g=0$, by Theorem 4.8 and (4.7) there is $f\in
L_{(2)}^{n,q-1}(X_{1}^{\ast},\mathcal{E}_{n-p};\omega_{P},h_{n-p}\otimes\tilde{h})$
so that $\bar{\partial}f=g$. The proposition therefore follows from (4.8). ∎

The following corollary is a direct consequence of the above proposition.

######  Corollary 4.10.

Notations as in the beginning of §4.2. There is a constant $N_{0}$, depending
only on $X$ and $(E,\theta,h)$, such that complex of sheaves

$\mathscr{D}_{X,\omega_{P}}^{p,0}(E,h_{N}({\bf{a}}))\xrightarrow{\bar{\partial}}\mathscr{D}_{X,\omega_{P}}^{p,1}(E,h_{N}({\bf{a}}))\xrightarrow{\bar{\partial}}\cdots\xrightarrow{\bar{\partial}}\mathscr{D}_{X,\omega_{P}}^{p,n}(E,h_{N}({\bf{a}}))$

is exact at each $q\geq 1$ for every $p$ and every $N>N_{0}$.

###### Proof.

If $x\notin D$, we can take an open neighborhood $U\subset X\backslash D$ of
$x$ which is biholomorphic to a polydisk. Then the corollary follows from the
usual $L^{2}$ Dolbeault lemma. It is therefore sufficient to consider an
arbitrary point $x\in D$. Let $(U;z_{1},\dots,z_{n})$ be an admissible
coordinate neighborhood which is biholomorphic to $\Delta_{r}^{n}$ for some
$r\leq 1$ such that $U\backslash
D\simeq(\Delta_{r}^{\ast})^{l}\times\Delta_{r}^{n-l}$ for some positive
integer $l$.

Note that $h_{N}({\bf a})$ is locally quasi-isometric to $\tilde{h}$ in
Proposition 4.9. Choosing $N$ large enough that satisfies the conditions in
Proposition 4.7, the exactness is obtained from Proposition 4.9. Notice that
the compactness of $\overline{X}$ ensures the existence of the uniformed bound
$N_{0}$. ∎

###  4.5. Proof of Theorem 4.5

Choosing $h_{N}({\bf{a}})$ that assures the validity of Corollary 4.10, we
denote the sheaf

$\displaystyle(\Omega^{m}\otimes E)_{(2),N,{\bf a}}:={\rm
Ker}\left(\mathscr{D}_{X,\omega_{P}}^{m,0}(E,h_{N}({\bf{a}}))\xrightarrow{\bar{\partial}}\mathscr{D}_{X,\omega_{P}}^{m,1}(E,h_{N}({\bf{a}}))\right),\quad\forall
0\leq m\leq n$

and a complex of sheaves

$Dol(E)^{\bullet}_{(2),N,{\bf a}}:=(\Omega^{0}\otimes E)_{(2),N,{\bf
a}}\stackrel{{\scriptstyle{\theta}}}{{\to}}(\Omega^{1}\otimes E)_{(2),N,{\bf
a}}\stackrel{{\scriptstyle{\theta}}}{{\to}}\cdots\stackrel{{\scriptstyle{\theta}}}{{\to}}(\Omega^{n}\otimes
E)_{(2),N,{\bf a}}.$

That is, $(\Omega^{m}\otimes E)_{(2),N,{\bf a}}$ is the sheaf of germs of
holomorphic sections $\sigma\in\Omega^{m}\otimes E$ which are locally square
integrable with respect to $h_{N}({\bf{a}})$ and $\omega_{P}$.

Then there exists a quasi-isomorphism

(4.17)

By taking the total complex in (4.6), one gets

######  Proposition 4.11.

The canonical morphism

(4.18) $\displaystyle Dol(E)^{\bullet}_{(2),N,{\bf
a}}\to\mathscr{D}^{\bullet}_{X,\omega_{P}}(E,D^{\prime\prime},h_{N}({\bf{a}}))$

is a quasi-isomorphism.

Before calculating $Dol(E)^{\bullet}_{(2),N,{\bf a}}$, we would like to state
a simple lemma that will be used.

######  Lemma 4.12.

When $N>1$,

$\int_{\Delta^{\ast}_{\frac{1}{2}}}|z|^{r}({\rm{log}}|z|)^{N}{\rm
vol}_{\omega_{P}}<+\infty$

if and only if $r>0$.

###### Proof.

Let $z=\rho e^{i\theta}$. Then

(4.19)
$\displaystyle\int_{\Delta^{\ast}_{\frac{1}{2}}}|z|^{r}({\rm{log}}|z|)^{N}{\rm
vol}_{\omega_{P}}=\pi\int_{0}^{\frac{1}{2}}\rho^{r-1}({\rm{log}}\rho)^{N-2}d\rho.$

The right hand side of (4.19) is finite if and only if $r>0$. ∎

The proof of Theorem 4.5 therefore follows from the following proposition.

######  Proposition 4.13.

There is a constant $N_{0}$, depending only on $(E,\theta,h)$, such that

$Dol(E)^{\bullet}_{(2),N,{\bf a}+\bm{\epsilon}}={\rm Dol}({{}_{{\bf
a}}}E,\theta)$

for every ${\bf 0}<{\bm{\epsilon}}<\bm{\sigma}_{E}({\bf a})$ and $N>N_{0}$.

###### Proof.

Notations as in Theorem 4.4. For every $0\leq m\leq n$, let
$\tilde{v}:=e_{i_{1}}\wedge\cdots\wedge e_{i_{m}}\otimes v$ be a holomorphic
section of $\Omega^{m}\otimes E$ such that

$0\neq[v]\in{\rm Gr}_{l_{n}}^{W^{(n)}}\cdots{\rm
Gr}_{l_{1}}^{W^{(1)}}E\cap{\rm Gr}_{\bf b}E$

for some $l_{1},\dots,l_{n}\in\mathbb{Z}$ and some ${\bf
b}=(b_{1},\dots,b_{l})\in\mathbb{R}^{l}$. Here

(4.20) $\displaystyle e_{j}=\begin{cases}\frac{dz_{j}}{z_{j}},&j=1,\dots,l\\\
dz_{j},&j=l+1,\dots,m\end{cases}.$

Then one has

(4.21)
$\displaystyle|\tilde{v}|_{h_{N}(\bm{a}+\bm{\epsilon}),\omega_{P}}\sim|s_{1}|^{a_{1}+\epsilon_{1}-b_{1}}\cdots|s_{l}|^{a_{l}+\epsilon_{l}-b_{l}}\big{|}\log|s_{1}|\big{|}^{\frac{N}{2}+l_{1}+\delta(1)}\big{|}\log|s_{2}|\big{|}^{\frac{N}{2}-l_{1}+l_{2}+\delta(2)}\cdots\big{|}\log|s_{l}|\big{|}^{\frac{N}{2}+l_{n}+\delta(n)}.$

Here

(4.22)
$\displaystyle\delta(i)=\begin{cases}1,&i\in\\{i_{1},\dots,i_{m}\\}\cap\\{1,\dots,l\\}\\\
0,&{\textrm{otherwise}}\end{cases}.$

Taking $N_{0}=2\max\\{|l_{i}|+2,|l_{i}-l_{i+1}|+2\\}$. By Lemma 4.12,

$\int|\tilde{v}|_{h_{N}(\bm{a}+\bm{\epsilon}),\omega_{P}}^{2}{\rm
vol}_{\omega_{P}}<\infty$

if and only if

(4.23) $\displaystyle a_{i}+\epsilon_{i}-b_{i}>0,\quad i=1,\dots,l.$

This is equivalent to that $v\in{{}_{{\bf a}}}E$ for any ${\bf
0}<{\bm{\epsilon}}<\bm{\sigma}_{E}({\bf a})$. This shows the proposition. ∎

##  5\. Application: Kawamata-Viehweg type vanishing theorem for Higgs bundles

Let $X$ be an pre-compact open subset of an $n$-dimensional projective Kähler
manifold $(M,\omega)$ such that $X$ is a weakly pseudoconvex manifold and
$\psi$ is a smooth exhausted plurisubharmonic function on $X$. For every
$c\in\mathbb{R}$, denote $X_{c}:=\\{x\in X|\psi(x)<c\\}$. Let
$D:=\cup_{i=1}^{l}D_{i}$ be a simple normal crossing divisor on $M$ and
$(E,\theta,h)$ a nilpotent harmonic bundle on $M\backslash D$. Let
$\sigma_{i}\in H^{0}(M,\mathscr{O}_{M}(D_{i}))$ be the defining section of
$D_{i}$. For each $1\leq i\leq l$, fix some smooth hermitian metric $h_{i}$
for the line bundle $\mathscr{O}_{M}(D_{i})$ so that $|\sigma_{i}|_{h_{i}}<1$.
Since $\overline{X}$ is compact, the associated curvature tensor $R(h_{i})$ is
bounded for every $i$. We denote $\varphi_{i}=|\sigma_{i}|_{h_{i}}$. Let $L$
be a holomorphic line bundle on $M$ so that some positive multiple
$mL=A+F_{1}+F_{2}$ where $A$ is an ample line bundle, $F_{1}$ and $F_{2}$ are
effective divisors supported in $D$ such that $A+F_{1}$ is a nef holomorphic
line bundle. Suppose that $F_{1}=\sum_{i=1}^{l}\alpha_{i}D_{i}$ and
$F_{2}=\sum_{i=1}^{l}r_{i}D_{i}$ where $\alpha_{i},r_{i}\in\mathbb{Z}_{\geq
0},\forall i$. Denote
${\bm{\alpha}}:=(\alpha_{1},\dots,\alpha_{l}),{\bf{r}}:=(r_{1},\dots,r_{l})$.

The main technique that we use to prove Theorem 1.2 is Deng-Hao’s $L^{2}$
estimate for $D^{\prime\prime}$ operators on Higgs bundles following the
pattern of Hormander’s original $L^{2}$ estimate for $\bar{\partial}$
operators.

######  Proposition 5.1 ([DF2021], Corollary 2.7).

Let $(X,\omega)$ be a complete Kähler manifold and $(E,\theta,h)$ any harmonic
bundle on $X$. Let $L$ be a line bundle on $X$ equipped with a hermitian
metric $h_{L}$. Assume that for some $m>0$, one has

(5.1)
$\displaystyle\langle[\sqrt{-1}R(h_{L}),\Lambda_{\omega}]f,f\rangle_{\omega}\geq\epsilon|f|_{\omega}^{2}$

for any $x\in X$, any $f\in\wedge^{p,q}T_{X,x}^{\ast}$ and any $p+q=m$. Set
$(\widetilde{E},\widetilde{\theta},\widetilde{h}):=(E\otimes
L,\theta\otimes{\rm{Id}}_{L},h\otimes h_{L})$ and
$\widetilde{D^{\prime\prime}}=\bar{\partial}_{E\otimes L}+\widetilde{\theta}$.
Then for any $v\in L_{(2)}^{m}(X,\widetilde{E};\omega,\widetilde{h})$ such
that $\widetilde{D^{\prime\prime}}v=0$, there exists $u\in
L_{(2)}^{m-1}(X,\widetilde{E};\omega,\widetilde{h})$ so that
$\widetilde{D^{\prime\prime}}u=v$ and

$\|u\|^{2}\leq\frac{\|v\|^{2}}{\epsilon}.$

###  5.1. Construction of the relavant metrics

Since $A$ is ample, there exists a positive metric $h_{A}$ with weight
$\varphi_{A}$ such that
$\sqrt{-1}\partial\bar{\partial}\varphi_{A}\geq\epsilon_{0}\omega$ for a
certain positive constant $\epsilon_{0}$. Since $A+F_{1}$ is nef, for every
positive constant $\epsilon$ there exists a metric $h_{\epsilon}$ on $A+F_{1}$
with weight $\varphi_{\epsilon}$ such that
$\sqrt{-1}\partial\bar{\partial}\varphi_{\epsilon}\geq-\epsilon\omega$. Let
$\varphi_{F_{1}}=\log|\sigma_{i}|_{h_{i}}^{2\alpha_{i}}$ (resp.
$\varphi_{F_{2}}=\log|\sigma_{i}|_{h_{i}}^{2r_{i}}$) be the weight of a
singular metric on $\mathscr{O}(F_{1})$ (resp. $\mathscr{O}(F_{2})$).

We define a singular metric $h_{\epsilon,\delta}$ on $L$ by the weight

$\varphi_{L}=\frac{1}{m}\left((1-\delta)\varphi_{\epsilon}+\delta(\varphi_{A}+\varphi_{F_{1}})+\varphi_{F_{2}}\right)$

with $\epsilon\ll\delta\ll 1$, $\delta$ rational. Then $\varphi_{L}$ has
singularities along $F_{1}$ and $F_{2}$, and

(5.2) $\displaystyle\sqrt{-1}\partial\bar{\partial}\varphi_{L}=$
$\displaystyle\frac{\sqrt{-1}}{m}\left((1-\delta)\partial\bar{\partial}\varphi_{\epsilon}+\delta\partial\bar{\partial}\varphi_{A}+\delta\partial\bar{\partial}\varphi_{F_{1}}+\partial\bar{\partial}\varphi_{F_{2}}\right)$
$\displaystyle\geq$
$\displaystyle\frac{1}{m}\left(-(1-\delta)\epsilon\omega+\delta(\epsilon_{0}\omega+[F_{1}])+[F_{2}]\right)\geq\frac{\delta}{m}\epsilon\omega$

if we choose $\epsilon\leq\delta\epsilon_{0}$. This is a singular metric which
is smooth and positively curved outside $D$.

Let $\psi_{c}:=\psi+\frac{1}{c-\psi}$ be a smooth exhausted plurisubharmonic
function on $X_{c}$ and $\chi$ a convex increasing function. Choose a positive
constant $N$. For any ${\bf{a}}=(a_{1},\dots,a_{l})\in\mathbb{R}^{l}$, we
modify the metric $h_{\epsilon,\delta}$ to

(5.3) $\displaystyle
h_{\epsilon,\delta}({\bf{a}}):=h_{\epsilon,\delta}\cdot\prod_{i=1}^{l}|\sigma_{i}|_{h_{i}}^{2a_{i}}\cdot(-\prod_{i=1}^{l}\log|\sigma_{i}|_{h_{i}}^{2})^{N}$

and

(5.4)
$\displaystyle{h}_{\epsilon,\delta,c}({\bf{a}}):={h}_{\epsilon,\delta}({\bf{a}})\cdot
e^{-\chi(\psi_{c}^{2})}.$

The associated curvatures are

(5.5) $\displaystyle\sqrt{-1}R({h}_{\epsilon,\delta}({\bf{a}}))=$
$\displaystyle\sqrt{-1}R(h_{\epsilon,\delta})+N\sqrt{-1}\sum_{i=1}^{l}\frac{\partial\log|\sigma_{i}|_{h_{i}}^{2}\wedge\bar{\partial}\log|\sigma_{i}|_{h_{i}}^{2}}{(\log|\sigma_{i}|_{h_{i}}^{2})^{2}}$
$\displaystyle+N\sqrt{-1}\sum_{i=1}^{l}\frac{R(h_{i})}{\log|\sigma_{i}|_{h_{i}}^{2}}+\sqrt{-1}\sum_{i=1}^{l}a_{i}R(h_{i})$

and

(5.6)
$\displaystyle\sqrt{-1}R({h}_{{\epsilon,\delta},c}({\bf{a}}))=\sqrt{-1}R({h}_{\epsilon,\delta}({\bf{a}}))+\sqrt{-1}\partial\bar{\partial}\chi(\psi_{c}^{2}).$

Let $0<\gamma_{1}(x)\leq\cdot\cdot\cdot\leq\gamma_{n}(x)$ be the eigenvalues
of $\sqrt{-1}R(h_{\epsilon,\delta})$ with respect to $\omega$ on $X\backslash
D$. By the assumptions on $h_{L}$, $\gamma_{i}\geq\frac{\delta}{m}\epsilon$
holds on $X\backslash D$ for each $i$.

######  Lemma 5.2.

Fixing $N$, we can pick ${\bf{a}}=(a_{1},\dots,a_{l})\in\mathbb{R}^{l}$ such
that $|{\bf a}|:=\sum_{i=1}^{l}|a_{i}|$ is small enough and rescale each
$h_{i}$ by multiplying a small constant so that

  1. (1)

(5.7)
$\displaystyle\sqrt{-1}R({h}_{\epsilon,\delta}({\bf{a}}))\geq\sqrt{-1}R(h_{\epsilon,\delta})-\frac{\delta\epsilon}{2nm}\omega\geq(1-\frac{1}{2n})\frac{\delta\epsilon}{m}\omega$

on $X^{\ast}:=X\backslash D$.

  2. (2)

The metric

(5.8)
$\displaystyle\omega_{N}({\bf{a}}):=\frac{\delta\epsilon}{2mn}\omega+\sqrt{-1}R({h}_{\epsilon,\delta}({\bf{a}}))$

is a Kähler metric on $X^{\ast}$.

  3. (3)

(5.9)
$\displaystyle\sqrt{-1}R({h}_{{\epsilon,\delta},c}({\bf{a}}))\geq\sqrt{-1}R(h_{\epsilon,\delta})+\sqrt{-1}\partial\bar{\partial}\chi(\psi_{c}^{2})-\frac{\delta\epsilon}{2mn}\omega\geq(1-\frac{1}{2n})\frac{\delta\epsilon}{m}\omega$

on $X_{c}^{\ast}:=X_{c}\backslash D$.

  4. (4)

(5.10)
$\displaystyle\omega_{N,c}({\bf{a}}):=\omega_{N}({\bf{a}})+\sqrt{-1}\partial\bar{\partial}\chi(\psi_{c}^{2})$

is a Kähler metric on $X_{c}^{\ast}$.

###### Proof.

Once (1) is proved, (2) follows immediately. Taking the plurisubarmonicity of
$\chi(\psi_{c}^{2})$ into account, (3) and (4) can also be obtained. Therefore
it suffices to prove (1).

By (5.5), the possible negative terms should appear in
$N\sqrt{-1}\sum_{i=1}^{l}\frac{R(h_{i})}{\log|\sigma_{i}|_{h_{i}}^{2}}+\sqrt{-1}\sum_{i=1}^{l}a_{i}R(h_{i})$.
Since $\overline{X}$ is compact and $N$ is fixed, we can pick $a_{i}$ small
enough and perturb $h_{i}$ to $\epsilon_{0}\cdot h_{i}$ where $\epsilon_{0}$
is a small constant so that

$N\sqrt{-1}\sum_{i=1}^{l}\frac{R(h_{i})}{\log|\sigma_{i}|_{h_{i}}^{2}}+\sqrt{-1}\sum_{i=1}^{l}a_{i}R(h_{i})\geq-\frac{\delta\epsilon}{2mn}\omega.$

Thus the lemma is proved. ∎

######  Remark 5.3.

$\omega_{N}({\bf{a}})$ is a complete Kähler metric on $X^{\ast}$ and
$\omega_{N,c}({\bf{a}})$ is a complete Kähler metric on $X_{c}^{\ast}$. They
are both locally quasi-isometric the Poincaré type metric.

######  Remark 5.4.

For a harmonic bundle $(E,\theta,h)$ on $X^{\ast}$, the modified hermitian
metric $h_{N,c}({\bf{a}}):=h_{N}({\bf{a}})\cdot e^{-\chi(\psi_{c}^{2})}$ on
$E$ is locally quasi-isometric to $h({\bf{a}},N)$ in §4.3.

######  Proposition 5.5.

With the above notations, for every $c\in\mathbb{R}$ one has

(5.11) $\displaystyle\langle[\sqrt{-1}R({h}_{{\epsilon,\delta},c}({\bf
a})),\Lambda_{\omega_{N,c}({\bf a})}]f,f\rangle_{\omega_{N,c}({\bf
a})}\geq\frac{1}{2}|f|_{\omega_{N,c}({\bf{a}})}^{2}$

for any $x\in X_{c}^{\ast}$, any $p+q>n$ and any
$f\in\wedge^{p,q}T_{X_{c}^{\ast},x}^{\ast}$.

###### Proof.

For any point $x\in X_{c}^{\ast}$, one can choose local coordinates
$(z_{1},\dots,z_{n})$ around $x$ so that at $x$,
$\omega=\sqrt{-1}\sum_{i=1}^{n}dz_{i}\wedge d\bar{z}_{i}$ and
$\sqrt{-1}R(h_{\epsilon,\delta,c}({\bf{a}}))=\sqrt{-1}\sum_{i=1}^{n}\tilde{\gamma}_{i}(x)dz_{i}\wedge
d\bar{z}_{i}$ where
$(1-\frac{1}{2n})\frac{\delta\epsilon}{m}\leq\tilde{\gamma}_{1}(x)\leq\tilde{\gamma}_{2}(x)\leq\cdots\leq\tilde{\gamma}_{n}(x)$
are the eigenvalues of $\sqrt{-1}R(h_{\epsilon,\delta,c}({\bf{a}}))$ with
respect to $\omega$. Also
$\sqrt{-1}\partial\bar{\partial}\chi(\psi_{c}^{2})=\sqrt{-1}\sum_{i=1}^{n}{\nu}_{i}(x)dz_{i}\wedge
d\bar{z}_{i}$ with
$0\leq{\nu}_{1}(x)\leq{\nu}_{2}(x)\leq\cdots\leq{\nu}_{n}(x)$ the associated
eigenvalues with respect to $\omega$.

By (5.9),
$\tilde{\gamma}_{i}(x)\geq\gamma_{i}(x)+{\nu}_{i}(x)-\frac{\delta\epsilon}{2mn}$
for each $i$. Let $\lambda_{1}\leq\cdots\leq\lambda_{n}$ be the eigenvalues of
$\sqrt{-1}R(h_{\epsilon,\delta,c}({\bf{a}}))$ with respect to
$\omega_{N,c}({\bf{a}})$. Then
$\lambda_{i}=\frac{\tilde{\gamma}_{i}}{\frac{\delta\epsilon}{2mn}+\tilde{\gamma}_{i}}$
by (3) and (4) in Lemma 5.2. Thus $\frac{2n-1}{2n}\leq\lambda_{i}<1$ for each
$i=1,...,n$.

We assume that $p\geq q$ without loss of generality. Then

(5.12)
$\displaystyle\langle[\sqrt{-1}R(h_{\epsilon,\delta,c}({\bf{a}})),\Lambda_{\omega_{N,c}({\bf{a}})}]f,f\rangle_{\omega_{N,c}({\bf{a}})}\geq$
$\displaystyle(\sum_{i=1}^{p}\lambda_{i}+\sum_{j=1}^{q}\lambda_{j}-\lambda_{1}-\cdot\cdot\cdot-\lambda_{n})|f|_{\omega_{N,c({\bf{a}})}}^{2}$
$\displaystyle\geq$
$\displaystyle\left(p(\frac{2n-1}{2n})-(n-q)\right)|f|^{2}_{\omega_{N,c}({\bf{a}})}$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}|f|^{2}_{\omega_{N,c}({\bf{a}})}.$

∎

###  5.2. Fine resolution

Denote $\widetilde{D^{\prime\prime}}:=\bar{\partial}_{E\otimes
L}+\theta\otimes{\rm Id}_{L}$ to be the operator on $E\otimes L$. Denote
$\mathscr{D}_{X,\omega_{N}({\bf{a}})}^{\bullet}(E\otimes
L,\widetilde{D^{\prime\prime}},h_{N}({\bf{a}})\otimes h_{\epsilon,\delta})$ to
be the complex

$\mathscr{D}^{0}_{X,\omega_{N}({\bf{a}})}(E\otimes
L,\widetilde{D^{\prime\prime}},h_{N}({\bf{a}})\otimes
h_{\epsilon,\delta})\xrightarrow{\widetilde{D^{\prime\prime}}}\mathscr{D}^{1}_{X,\omega_{N}({\bf{a}})}(E\otimes
L,\widetilde{D^{\prime\prime}},h_{N}({\bf{a}})\otimes
h_{\epsilon,\delta})\xrightarrow{\widetilde{D^{\prime\prime}}}\cdots.$

######  Proposition 5.6.

Assume that $a_{i}-\frac{r_{i}}{m}\notin\mathscr{J}_{D_{i}}(E)$ for every
$i=1,\dots,l$. Then the canonical morphism

(5.13) $\displaystyle{\rm Dol}(_{-\frac{\bf r}{m}+{\bf a}}E,\theta)\otimes
L\to\mathscr{D}^{\bullet}_{X,\omega_{N}({\bf{a}})}(E\otimes
L,\widetilde{D^{\prime\prime}},h_{N}({\bf{a}})\otimes h_{\epsilon,\delta})$

is a quasi-isomorphism whenever $\delta>0$ is small enough.

###### Proof.

Since $a_{i}-\frac{r_{i}}{m}\notin\mathscr{J}_{D_{i}}(E)$, there exists
$\epsilon_{i}\in\mathbb{R}_{>0}$ small enough such that
$a_{i}-\frac{r_{i}}{m}-\epsilon_{i}\notin\mathscr{J}_{D_{i}}(E)$. Denote
$\bm{\epsilon}=(\epsilon_{1},\dots,\epsilon_{l})\in\mathbb{R}_{>0}^{l}$. Thus
${\rm Dol}(_{-\frac{\bf r}{m}+{\bf a}}E,\theta)={\rm Dol}(_{-\frac{\bf
r}{m}+{\bf a}-\bm{\epsilon}}E,\theta)$.

By Theorem 4.5, it suffices to show that

(5.14) $\displaystyle\mathscr{D}^{\bullet}_{X,\omega_{N}({\bf{a}})}(E\otimes
L,\widetilde{D^{\prime\prime}},h_{N}({\bf{a}})\otimes
h_{\epsilon,\delta})\simeq\mathscr{D}^{\bullet}_{X,\omega_{N}({\bf{a}})}(E,D^{\prime\prime},h_{N}(-\frac{{\bf
r}+\delta\bm{\alpha}}{m}+{\bf{a}}))\otimes L$

when $\delta$ is chosen small enough. By constructions one has

$h_{N}({\bf{a}})\otimes h_{\epsilon,\delta}\sim h_{N}(-\frac{{\bf
r}+\delta\bm{\alpha}}{m}+{\bf{a}})\otimes h_{L}$

for some smooth hermitian metric $h_{L}$ on $L$. Let
$\beta=\beta^{\prime}\otimes e_{L}$ be a differential form with value in
$E\otimes L$, where $\beta^{\prime}$ is an $E$-valued form and $e_{L}$ is a
holomorphic local generator of $L$. Since $|e_{L}|_{h_{L}}\sim 1$ and
$|\theta|_{\omega_{N}(\bm{a})}$ is bounded, one obtains that
$\|\beta\|<\infty,\|\widetilde{D^{\prime\prime}}(\beta)\|<\infty$ if and only
if $\|\beta^{\prime}\|<\infty,\|D^{\prime\prime}(\beta^{\prime})\|<\infty$.
This proves (5.14).

∎

###  5.3. Vanishing theorem on weakly pseudoconvex manifolds

Now we are ready to prove the following vanishing theorem.

######  Theorem 5.7.

Let $X$ be a pre-compact open subset of an $n$-dimensional projective manifold
$M$ such that $X$ is a weakly pseudoconvex manifold and $\psi$ is a smooth
exhausted plurisubharmonic function on $X$. Let $D$ be a simple normal
crossing divisor on $M$ and $(E,\theta,h)$ a nilpotent harmonic bundle on
$M\backslash D$. Suppose that some positive multiple $mL=A+F_{1}+F_{2}$ where
$A$ is an ample line bundle, $F_{1}$ and $F_{2}$ are effective divisors
supported in $D$ such that $A+F_{1}$ is a nef holomorphic line bundle. Assume
that $F_{2}=\sum_{i=1}^{l}r_{i}D_{i}$ where $r_{i}\in\mathbb{Z}_{\geq
0},\forall i$. Denote ${\bf{r}}:=(r_{1},\dots,r_{l})$ and denote
$X_{c}:=\\{x\in X|\psi(x)<c\\}$ for every $c\in\mathbb{R}$. Then

$\mathbb{H}^{i}(X_{c},{\rm Dol}(_{-\frac{\bf r}{m}+{\bf a}}E,\theta)\otimes
L)=0$

holds for every $c\in\mathbb{R}$, every $i>n$ and every ${\bf
a}=(a_{1},\dots,a_{l})\in\mathbb{R}^{l}$ such that
$|a_{j}|<\sigma_{E,j}(-\frac{r_{j}}{m})$ for every $j=1,\dots,l$.

###### Proof.

By Proposition 5.6, there is a quasi-isomorphism

${\rm Dol}(_{-\frac{\bf r}{m}+{\bf a}}E,\theta)\otimes L\simeq_{\rm
qis}\mathscr{D}_{X,\omega_{N}({\bf{a}})}^{\bullet}(E\otimes
L,\widetilde{D^{\prime\prime}},h_{N}({\bf{a}})\otimes h_{\epsilon,\delta})$

under the assumption of the theorem. Since
$\mathscr{D}_{X,\omega_{N}({\bf{a}})}^{p,q}(E\otimes
L,\widetilde{D^{\prime\prime}},h_{N}({\bf{a}})\otimes h_{\epsilon,\delta})$ is
a fine sheaf for each $p$ and $q$ (Lemma 3.1), by (4.6) we obtain that
$\mathscr{D}_{X,\omega_{N}({\bf{a}})}^{i}(E\otimes
L,\widetilde{D^{\prime\prime}},h_{N}({\bf{a}})\otimes h_{\epsilon,\delta})$ is
a fine sheaf for each $i$. Notice that $\overline{X_{c}}$ is always compact
since $\psi$ is exausted. For every $i>n$ and every
$\alpha\in\mathscr{D}_{X_{c},\omega_{N}({\bf{a}})}^{i}(E\otimes
L,\widetilde{D^{\prime\prime}},h_{N}({\bf{a}})\otimes
h_{\epsilon,\delta})(X_{c}^{\ast})$ such that
$\widetilde{D^{\prime\prime}}\alpha=0$, it suffices to show that there exists
$\beta\in\mathscr{D}_{X_{c},\omega_{N}({\bf{a}})}^{i-1}(E\otimes
L,\widetilde{D^{\prime\prime}},h_{N}({\bf{a}})\otimes
h_{\epsilon,\delta})(X_{c}^{\ast})$ such that
$\widetilde{D^{\prime\prime}}\beta=\alpha$.

Note that $h_{N}({\bf{a}})\otimes
h_{\epsilon,\delta}=h\otimes{h}_{\epsilon,\delta}({\bf{a}})$,
$h_{N,c}({\bf{a}})\otimes h_{\epsilon,\delta}=h\otimes
h_{\epsilon,\delta,c}({\bf{a}})$ and $\omega_{N}({\bf{a}})$ is incomplete on
$X_{c}^{\ast}$. To apply Proposition 5.1, we modify the metric
$\omega_{N}({\bf{a}})$ to the complete Kähler metric $\omega_{N,c}({\bf{a}})$.
Let $\chi$ be a convex increasing function as in [[Demailly2012], Chapter
VIII, Lemma (5.7)] such that

(5.15)
$\displaystyle\int_{X_{c}^{\ast}}|\alpha|_{\omega_{N,c}({\bf{a}}),h_{N,c}({\bf{a}})\otimes
h_{\epsilon,\delta}}^{2}{\rm vol}_{\omega_{N,c}({\bf a})}$
$\displaystyle\lesssim\int_{X_{c}^{\ast}}|\alpha|_{\omega_{N}({\bf{a}}),h_{N}({\bf{a}})\otimes
h_{\epsilon,\delta}}^{2}e^{-\chi(\psi_{c}^{2})}(1+\chi^{\prime}\circ\psi_{c}^{2}+\chi^{\prime\prime}\circ\psi_{c}^{2})^{n}{\rm
vol}_{\omega_{N}({\bf a})}$
$\displaystyle\lesssim\int_{X_{c}^{\ast}}|\alpha|_{\omega_{N}({\bf{a}}),h_{N}({\bf{a}})\otimes
h_{\epsilon,\delta}}^{2}{\rm vol}_{\omega_{N}({\bf{a}})}<+\infty.$

Since $\omega_{N,c}({\bf{a}})$ is a complete Kähler metric on $X_{c}^{\ast}$,
it follows from Proposition 5.1 and Proposition 5.5 that there exists
$\beta\in L_{(2)}^{i-1}(X_{c}^{\ast},E\otimes
L;\omega_{N,c}({\bf{a}}),h_{N,c}({\bf{a}})\otimes
h_{\epsilon,\delta})(X_{c}^{\ast})$ such that
$\widetilde{D^{\prime\prime}}\beta=\alpha$. Then
$\beta\in\mathscr{D}_{X_{c},\omega_{N}({\bf{a}})}^{i-1}(E\otimes
L,\widetilde{D^{\prime\prime}},h_{N}({\bf{a}})\otimes
h_{\epsilon,\delta})(X_{c}^{\ast})$ since
$\omega_{N}({\bf{a}})\sim\omega_{N,c}({\bf{a}})$ and $h_{N}({\bf{a}})\sim
h_{N,c}({\bf{a})}$ locally on $X_{c}^{\ast}$. Thus the theorem is proved. ∎

A relative version of the vanishing theorem follows from the above theorem.

######  Theorem 5.8.

Let $X$ be a projective manifold of dimension $n$, $Y$ be an analytic space
and $f:X\rightarrow Y$ be a proper surjective smooth map. Let
$D=\sum_{i=1}^{l}D_{i}$ be a normal crossing divisor on $X$. Let
$(E,\theta,h)$ be a nilpotent harmonic bundle on $X\backslash D$. Let $L$ be a
holomorphic line bundle on $X$. Suppose that some positive multiple
$mL=A+F_{1}+F_{2}$ where $A$ is an ample line bundle, $F_{1}$ and $F_{2}$ are
effective divisors supported in $D$ such that $A+F_{1}$ is a nef holomorphic
line bundle. Suppose that $F_{2}=\sum_{i=1}^{l}r_{i}D_{i}$ with
$r_{i}\in\mathbb{Z}_{\geq 0},\forall i$. Denote ${\bf{r}}:=(r_{1},..,r_{l})$.

Then

$R^{i}f_{\ast}({\rm Dol}(_{-\frac{\bf{r}}{m}+{\bf a}}E,\theta)\otimes L)=0$

for any $i>n$ and any ${\bf a}=(a_{1},\dots,a_{l})\in\mathbb{R}^{l}$ such that
$|a_{j}|<\sigma_{E,j}(-\frac{r_{j}}{m})$ for every $j=1,\dots,l$.

###### Proof.

Since the problem is local, we may assume that $Y$ admits a non-negative
smooth exhausted strictly plurisubharmonic function $\psi$ so that
$\psi^{-1}\\{0\\}=\\{y\\}\subset Y$ is a point. To achieve this one may embed
$Y$ into $\mathbb{C}^{N}$ as a closed Stein analytic subspace and take
$\psi=\sum_{i=1}^{N}\|z_{i}\|^{2}$. Since $f$ is proper, $f^{\ast}\psi$ is a
smooth exhausted plurisubharmonic function on $X$.

Fixing $c\in\mathbb{R}_{>0}$, $f^{-1}Y_{c}$ is a weakly pseudoconvex manifold
with $f^{\ast}\psi$ a smooth exhausted plurisubharmonic function on
$f^{-1}Y_{c}$ and $\overline{f^{-1}Y_{c}}$ is compact. It follows from Theorem
5.7 that

(5.16) $\displaystyle H^{i}(f^{-1}Y_{d},\textrm{Dol}(_{-\frac{\bf{r}}{m}+{\bf
a}}E,\theta)\otimes L)=0,\quad\forall i>n\quad\textrm{and}\quad\forall 0<d<c.$

Taking $d\to 0$ we acquire that

(5.17) $\displaystyle R^{i}f_{\ast}(\textrm{Dol}(_{-\frac{\bf{r}}{m}+{\bf
a}}E,\theta)\otimes L)_{y}=0,\quad\forall i>n.$

This proves the theorem. ∎

######  Remark 5.9.

By [Mochizuki2009], a parabolic stable higgs bundle admits a tame harmonic
metric such that the parabolic structure coincide with the Simpson-Mochizuki’s
parabolic structure. Therefore Theorem 5.8 implies Theorem 1.2.

###  5.4. Generalization of Suh’s vanishing theorem

In this subsection we generalize Suh’s Kawamata-Viehweg type vanishing theorem
with coefficients for the Deligne extension of a variation of Hodge structure
([Suh2018, Theorem 1]). This generalization has two sides: one is that we
treat polarized complex variations of Hodge structure rather than
$\mathbb{Q}$-polarized or $\mathbb{R}$-polarized variations of Hodge
structure; the other is that we treat other prolongations so that we obtain
the vanishing theorems for line bundles which may not be nef and big.

######  Definition 5.10.

Let $M$ be a complex manifold. A polarized complex variation of Hodge
structure on $M$ of weight $k$ is a harmonic bundle $(\mathcal{V},\nabla,h)$
on $M$ which consists of a holomorphic flat bundle $(\mathcal{V},\nabla)$ and
a harmonic metric, together with a decomposition
$\mathcal{V}\otimes_{\mathscr{O}_{M}}\mathscr{A}^{0}_{M}=\bigoplus_{p+q=k}\mathcal{V}^{p,q}$
of $C^{\infty}$ bundles such that

  1. (1)

The Griffiths transversality condition

(5.18)
$\displaystyle\nabla(\mathcal{V}^{p,q})\subset\mathscr{A}^{0,1}_{M}(\mathcal{V}^{p+1,q-1})\oplus\mathscr{A}^{0,1}_{M}(\mathcal{V}^{p,q})\oplus\mathscr{A}^{1,0}_{M}(\mathcal{V}^{p,q})\oplus\mathscr{A}^{1,0}_{M}(V^{p-1,q+1})$

holds for every $p$ and $q$. Here $\mathscr{A}^{i,j}_{M}(\mathcal{V}^{p,q})$
(resp. $\mathscr{A}^{k}_{M}(\mathcal{V}^{p,q})$) denotes the sheaf of
$C^{\infty}$ $(i,j)$-forms (resp. $k$-forms) with values in
$\mathcal{V}^{p,q}$.

  2. (2)

The hermitian form $Q$ which equals $(-1)^{p}h$ on $\mathcal{V}^{p,q}$ is
parallel with respect to $\nabla$.

We have a decomposition

(5.19) $\displaystyle\nabla=\overline{\theta}+\bar{\partial}+\partial+\theta$

according to (5.18).

Let us fix some notations. For every $p$, $F^{p}:=\oplus_{i\geq
p}\mathcal{V}^{i,k-i}\cap{\rm Ker}(\bar{\partial}+\overline{\theta})$ is a
holomorphic subbundle which satisfies the Griffiths transversality condition
that $\nabla(F^{p})\subset F^{p-1}\otimes\Omega_{M}$ for every $p$. In
literatures one usually regards $(\mathcal{V},\nabla,\\{F^{p}\\},h)$ as a
polarized $\mathbb{C}$-variation of Hodge structure.

Due to (5.18), the graded quotient $({\rm Gr}_{F}\mathcal{V},{\rm
Gr}_{F}\nabla)$ is a Higgs bundle in the sense of Simpson [Simpson1988, §8].
Denote $H:={\rm
Ker}(\bar{\partial}:\mathcal{V}\to\mathscr{A}^{1}_{M}(\mathcal{V}))$ and
$H^{p,q}:=H\cap\mathcal{V}^{p,q}$. Then
$(H\simeq\oplus_{p+q=k}H^{p,q},\theta)$ is a Higgs bundle which is canonically
isomorphic to $({\rm Gr}_{F}\mathcal{V},{\rm Gr}_{F}\nabla)$. Hence $({\rm
Gr}_{F}\mathcal{V},{\rm Gr}_{F}\nabla)$ is related to $(\mathcal{V},\nabla)$
under Simpson’s correspondence between flat connections and semistable Higgs
bundles with vanishing Chern classes ([Simpson1992, §4]).

######  Remark 5.11.

The Higgs bundle associated to a polarized complex variation of Hodge
structure is always nilpotent.

Let $X$ be a complex manifold and $D=\cup_{i=1}^{l}D_{i}$ a simple normal
crossing divisor on $X$. Let
$(\mathcal{V},\nabla,\\{F^{p}\\}_{p\in\mathbb{Z}})$ be a polarizable variation
of Hodge structure of weight $k$. For every
${\bm{a}}=(a_{1},\dots,a_{l})\in\mathbb{R}^{l}$, let
$\mathcal{V}_{\geq\bm{a}}$ be the unique locally free $\mathscr{O}_{X}$-module
extending $\mathcal{V}$ such that $\nabla$ induces a connection with
logarithmic singularities
$\nabla:\mathcal{V}_{\geq{\bm{a}}}\to\mathcal{V}_{\geq{\bm{a}}}\otimes\Omega_{X}(\log
D)$ whose real parts of the eigenvalues of the residue of $\nabla$ along
$D_{i}$ belong to $[{{a}}_{i},{{a}}_{i}+1)$. Let $j:X\backslash D\to X$ be the
open immersion. Denote
$F^{p}_{\geq{\bm{a}}}:=j_{\ast}F^{p}\cap\mathcal{V}_{\geq{\bm{a}}}$. The
recent work of Deng [Deng2022] shows the following theorem, which generalizes
the nilpotent orbit theorem of Schmid [Schmid1973] and Cattani-Kaplan-Schmid
[Cattani_Kaplan_Schmid1986].

######  Theorem 5.12.

For every ${\bm{a}}\in\mathbb{R}^{l}$ and every $p\in\mathbb{Z}$,
$F_{\geq{\bm{a}}}^{p}$ is a subbundle of $\mathcal{V}_{\geq{\bm{a}}}$ and

$F_{\geq{\bm{a}}}^{p}/F_{\geq{\bm{a}}}^{p+1}\simeq{{}_{{\bm{a}}}}H^{p,k-p},$

where ${{}_{{\bm{a}}}}H^{p,k-p}$ is Mochizuki’s prolongation (a locally free
$\mathscr{O}_{X}$-modules) of the Hodge bundle $H^{p,k-p}:=F^{p}/F^{p+1}$. The
connection $\nabla$ admits logarithmic singularities on
$F_{\geq{\bm{a}}}^{p}$, i.e., $\nabla:F^{p}\to
F^{p-1}\otimes\Omega_{X\backslash D}$ extends to

$\nabla:F_{\geq{\bm{a}}}^{p}\to F_{\geq{\bm{a}}}^{p-1}\otimes\Omega_{X}(\log
D).$

Thus one has a logarithmic de Rham complex

$\mathcal{V}_{\geq{\bm{a}}}\stackrel{{\scriptstyle\nabla}}{{\to}}\mathcal{V}_{\geq{\bm{a}}}\otimes\Omega_{X}(\log
D)\stackrel{{\scriptstyle\nabla}}{{\to}}\mathcal{V}_{\geq{\bm{a}}}\otimes\Omega^{2}_{X}(\log
D)\stackrel{{\scriptstyle\nabla}}{{\to}}\cdots$

and subcomplexes

$F^{p}_{\geq{\bm{a}}}\stackrel{{\scriptstyle\nabla}}{{\to}}F^{p-1}_{\geq{\bm{a}}}\otimes\Omega_{X}(\log
D)\stackrel{{\scriptstyle\nabla}}{{\to}}F^{p-2}_{\geq{\bm{a}}}\otimes\Omega^{2}_{X}(\log
D)\stackrel{{\scriptstyle\nabla}}{{\to}}\cdots,\quad\forall p.$

Denote the graded quotient complex as

${}_{\bf a}{\rm
DR}_{(X,D)}(\mathcal{V},F):=\bigoplus_{p}F_{\geq{\bm{a}}}^{p}/F_{\geq{\bm{a}}}^{p+1}\stackrel{{\scriptstyle{\rm
Gr}\nabla}}{{\to}}\bigoplus_{p}F_{\geq{\bm{a}}}^{p-1}/F_{\geq{\bm{a}}}^{p}\otimes\Omega_{X}(\log
D)\stackrel{{\scriptstyle{\rm Gr}\nabla}}{{\to}}\cdots.$

By Theorem 5.12 we have

(5.20) ${}_{\bf a}{\rm DR}_{(X,D)}(\mathcal{V},F)\simeq{\rm
Dol}({{}_{\bm{a}}}H,\theta)$

where $(H,\theta,h)$ is the Higgs bundle associated to
$(\mathcal{V},\nabla,h)$. Applying Theorem 1.2 we obtain the following
vanishing result.

######  Theorem 5.13.

Let $X$ be a projective manifold of dimension $n$, $Y$ be an analytic space
and $f:X\rightarrow Y$ be a proper surjective smooth map. Let
$D=\sum_{i=1}^{l}D_{i}$ be a normal crossing divisor on $X$. Let
$(\mathcal{V},\nabla,\\{\mathcal{V}^{p,q}\\}_{p+q=k},h)$ be a polarizable
variation of Hodge structure of weight $k$ on $X\backslash D$. Let $L$ be a
holomorphic line bundle on $X$. Suppose that some positive multiple
$mL=A+F_{1}+F_{2}$ where $A$ is an ample line bundle, $F_{1}$ and $F_{2}$ are
effective divisors supported in $D$ such that $A+F_{1}$ is a nef holomorphic
line bundle. Suppose that $F_{2}=\sum_{i=1}^{l}r_{i}D_{i}$ where
$r_{i}\in\mathbb{Z}_{\geq 0},\forall i$. Denote
${\bf{r}}:=(r_{1},\dots,r_{l})$.

Then

$R^{i}f_{\ast}\left({}_{-\frac{\bf{r}}{m}+{\bf a}}{\rm
DR}_{(X,D)}(\mathcal{V},F)\otimes L\right)=0$

for any $i>n$ and any ${\bf a}=(a_{1},\dots,a_{l})\in\mathbb{R}^{l}$ such that
$|a_{j}|<\sigma_{E,j}(-\frac{r_{j}}{m})$ for every $j=1,\dots,l$.

To consider the canonical extension of $\mathcal{V}$, let us take
${\bm{a}}=(0,\dots,0)$. In this case ${}_{\bf{0}}{\rm
DR}_{(X,D)}(\mathcal{V},F)$ is the logarithmic de Rham complex associated with
the canonical extension of $(\mathcal{V},F)$ considered in [Suh2018]. Taking
$Y={\rm pt}$, $F_{2}=0$ and ${\bm{a}}=(0,\dots,0)$, Theorem 5.13 implies Suh’s
vanishing theorem ([Suh2018, Theorem 1]).

## References


15(0.5,0.3) Please cite: Hussein S. Al-Olimat, Valerie L. Shalin,
Krishnaprasad Thirunarayan, and Joy Prakash Sain. 2019. Towards Geocoding
Spatial Expressions (Vision Paper). In Proceedings of the 27th ACM SIGSPATIAL
International Conference on Advances in Geographic Information Systems
(SIGSPATIAL ’19). Association for Computing Machinery, New York, NY, USA,
75-78. DOI:https://doi.org/10.1145/3347146.3359356

# Towards Geocoding Spatial Expressions

Hussein S. Al-Olimat, Valerie L. Shalin, Krishnaprasad Thirunarayan and Joy
Prakash Sain<EMAIL_ADDRESS>Kno.e.sis Center,
Wright State University, Dayton, OH.

(2019)

###### Abstract.

Imprecise _composite_ location references formed using _ad hoc spatial
expressions_ in English text makes the geocoding task challenging for both
inference and evaluation. Typically such spatial expressions fill in
unestablished areas with new toponyms for finer spatial referents. For
example, the spatial extent of the ad hoc spatial expression ”north of” or ”50
minutes away from” in relation to the toponym ”Dayton, OH” refers to an
ambiguous, imprecise area, requiring translation from this qualitative
representation to a quantitative one with precise semantics using systems such
as WGS84. Here we highlight the challenges of geocoding such referents and
propose a formal representation that employs background knowledge, semantic
approximations and rules, and fuzzy linguistic variables. We also discuss an
appropriate evaluation technique for the task that is based on human
contextualized and subjective judgment.

††copyright: rightsretained††conference: Pre-print; June 2019; Dayton, OH,
USA††journalyear: 2019††article: 4††price: 15.00††ccs: Information systems
Evaluation of retrieval results††ccs: Information systems Geographic
information systems

##  1\. Introduction

A geographic coordinate system (GCS), such as the World Geodetic System 1984
(WGS84), quantitatively describes the exact area of locations. However,
natural language refers to a spatial area qualitatively posing a challenge to
precision in the face of the complexity of human thought process. Human
qualitative spatial referents appear as established toponyms (e.g., ” _Dayton_
, Ohio”) or _ad hoc_ spatial expressions (e.g., ” _North of_ Dayton, Ohio”)
linked to established/atomic toponyms with relations. In English, these often
employ prepositions. The mismatch between quantitative coordinate-based
referents and qualitative spatial expressions prohibits their straightforward
alignment. Moreover, human spatial referents are not precise or formally
defined, making them ambiguous, and laden with idiosyncrasies. According to
Landau and Jackendoff’s (Landau and Jackendoff, 1993) spatial cognition theory
”Whatever we can talk about we can also represent”. Two properties of spatial
expressions challenge their representation for computing systems: they are
ambiguous (i.e., requiring context) and they employ fuzzy location referents
(i.e., requiring approximation). Therefore, we seek a tolerant formal
representation, to serve as an _interlingua_ between quantitative and
qualitative spatial referents acknowledging Zadeh’s incompatibility principle
(Zadeh, 1973) (i.e., ”the ineffectiveness of computers in dealing with human
systems”).

World map projects, such as OpenStreetMap, apparently solve the problem for
established toponyms. Toponyms with quantitative spatial extents are
represented as polygons in GeoJSON shapefiles. E.g., the city of ”Seattle, WA”
is associated with a polygon that covers the whole
city111https://www.openstreetmap.org/relation/237385 as defined and agreed on
by the OpenStreetMap community. The existence of alternative quantitative
representations, including lat/long point representations, anticipate
potential disagreement in practice, where humans may not employ officially
established boundaries. Moreover, only established, coarse-grained toponyms
(aka. atomic toponyms) link to agreed-upon quantitative representations.
However, such resources are inevitably incomplete. Humans use spatial
expressions to fill in the unestablished areas with new toponyms for finer
spatial referents. For example, ”downtown Seattle” or ”northern Seattle” refer
to an unestablished, _ad hoc_ portion of the larger, but _atomically specified
toponym_ (i.e., ”Seattle”) with imprecise and fuzzy boundaries. The number of
possible mappings between this _ad hoc_ referent and its quantitative extent,
and the lack of precise and agreed upon interpretation defies both enumeration
and evaluation and therein lies the complexity.

Dealing with ad hoc spatial expressions as linguistic variables (Zadeh, 1975)
captures the inherent fuzziness of the referents reflecting day to day usage.
While a lat/long point on a map, as in WGS84 has precise and distinctive
semantics, adding linguistic expressions such as ”north of” or ”not far from”
to that point makes it difficult for spatial systems to infer the exact
spatial extent of the referent because the area is not precisely defined. This
requires translation from a qualitative representation to an (imprecise)
quantitative one that is adequate for the intended application and context.

In this paper, we identify and then operate on polygons to represent
qualitative spatial referents. A linguistic spatial expression contrasts with
polygons referenced using the WGS84 system in that such linguistic expressions
come with fuzzy boundaries. This representation requires us to design
heuristics to draw polygons and scale them automatically _based on the context
of referents_ (e.g., the size of the referenced toponyms (Clementini et al.,
1997)). The use of an ad hoc spatial expressions serves as an _approximate
characterization_ for a descriptive phrase of a location referent.

Standard evaluation methods that rely on precise boundaries are inherently
incompatible with these new kinds of unstandardized spatial referents.
Precision/Recall metrics do not readily characterize the degree of error
associated with referents specified with fuzzy boundaries and are missing from
gazetteers. Therefore, we suggest evaluating proposed representations using a
measurement scale of a higher resolution relative to a binary decision (the
Likert scale), to collect human judgment and better capture the intuition
behind the use of _ad hoc_ spatial expressions.

##  2\. Related Work

Vernacular Geography (VG) concerns the vagueness of location referents (aka.
imprecise regions) that are not part of a gazetteer but are part of the
regional culture (e.g., a ”city center”, ”the downtown”, or ”the old
downtown”) (Hollenstein and Purves, 2010). These referents can be as big as
the Midwest United States or as small as ”downtown Dayton”. Disagreement,
between official boundaries of the locations and what people tend to consider
as the boundaries, reflects contextual factors such as population density,
housing systems, land use policy, and other social, cultural, and physical
artifacts (Hollenstein and Purves, 2010). However, VG techniques largely have
not developed formal linguistic rules and methods to deal with the semantics
of location referents. Instead, these attempts are mostly deductive methods
drawing on conclusions from patterns in geotagged data and textual
descriptions.

Some toponym resolution techniques that map location names to their
unambiguous spatial footprint rely on spatially annotated documents to learn
the spatial distributions of document words (DeLozier et al., 2015). Usually,
such supervised techniques deduce the spatial footprint of words to
georeference documents based on lexical choice (e.g., ”Y’all” suggests origins
in the”southern US”). Woodruff and Plaunt (Woodruff and Plaunt, 1994)
performed syntactic operations to retain relevant portions of phrases and omit
others (e.g., retaining ”Delta” from the phrase ”literature on the Delta”).
They use a relevance weighting method to assign weights to generated polygons
for referents. They also support lexical transformation to improve coverage,
for example, transforming ”Valleys” into ”Valley” that can then be found in a
gazetteer. They strip tokens from toponyms, such as ”County” from ”Kern
County”, to create what is called ”evidonyms” to retrieve locations like ”Kern
Water Bank” which is not part of the gazetteer. The technique considers the
highest stacked polygons of all georeferenced locations as the focus area of a
document to deduce the semantics of spatial expressions such as ”south of the
Delta”. Lieberman and Samet (Lieberman and Samet, 2012) use geographical
distances and background knowledge from gazetteers to disambiguate location
references based on their relative proximity in texts. However, none of these
techniques deal with isolated fine-grained and _ad hoc_ spatial referents in
expressions that include relations such as ”near” or ”between”.

Clementini et al. (Clementini et al., 1997) analyzed qualitative distances and
defined context using a three-component frame of reference (i.e., distance
system, scale, and type). Crucially, the analysis shows that distance is
dependent on scale. For example, the meaning of ”close” in ” _x_ is close to
_y_ ” depends on the relative position of both locations and their respective
sizes. This is very useful in practice for developing heuristics to deal with
spatial expressions, especially when inferring the meaning of ad hoc spatial
expressions, such as ”near”, that relies on the contextual toponyms in the
same expression (e.g., ”I live in a country near Mexico” vs. ”I live in a city
near Columbus, OH”).

Finally, in Geographic Information Retrieval (GIR) (Purves et al., 2018), the
understanding of spatial relationships such as ”near” in the example ”beaches
near Dayton” requires an understanding of the theme ”beaches” and the spatial
extent of the toponym ”Dayton”. Returning all beaches within a predefined
distance, or returning the closest few beaches around the atomic toponym
”Dayton”, regardless of distance, are alternative ways of satisfying the
query. Such approaches are limited by the availability of an entry for
”beaches” in the geographic coordinate system.

##  3\. Referencing Spatial Expressions

Reference objects (aka. anchors), such as established atomic toponyms or any
geocoded spatial extents (e.g., the northern side of a city), provide an
initial frame of reference for the _ad hoc_ spatial referents (Liu et al.,
2009). We acknowledge that ambiguity may persist in understanding the spatial
extent of atomic toponyms. However, here, we assume that atomic toponyms are
correctly geocoded and disambiguated. Additionally, we assume that all atomic
and ad hoc spatial referents are correctly delimited in text using techniques
such as locative expression extraction (Liu et al., 2014) and the spatial
relationships are correctly annotated using techniques such as spatial role
labeling (Kordjamshidi et al., 2010).

The focus of this paper is to infer and geocode relative/partial spatial
referents with respect to one or more identified spatial extents of anchor
objects. Thus, an atomic toponym, for example, is not the head of the
expression, but rather a modifier on the head, e.g., ”banks” or ”gas stations”
in ”Dayton” is more about the ”banks” and ”gas stations” that appear in
”Dayton”, than ”Dayton” _per se_. The task then is to specify how the
_”Dayton”_ toponym enables the interpretation of _”in”_ and the scope of the
referents for ”banks” and ”gas stations”. Landau and Jackendoff (Landau and
Jackendoff, 1993) suggest that representing space requires the understanding
of toponyms and routes (aka. places and paths) and that prepositions extend
the spatial area of a toponym to related paths and regions. The spatial
relation expressed in prepositional phrases would, therefore, play the
defining role of spatial extents222In this work, we will not talk about the
spatial extents of routes. For example, we are not yet concerned about how to
represent ”via” and ”through”.. In general, there are two types of spatial
relations expressed in prepositional phrases:

  1. (1)

Binary spatial relations: These are the spatial referents expressed in
reference to a single anchor object (e.g., ”near $x$”, ”away_from $x$”,
”beside $x$”, or ”in $x$”).

  2. (2)

N-ary spatial relations: These are the spatial referents expressed in
reference to two or more anchor objects (e.g., ”near $x$ and $y$”, ”between
$x$ and $y$”, or ”among $x$, $y$, and $z$”).

There are three main frames of reference that affect the interpretation of
spatial extents and influence the semantics of spatial relations (Purves et
al., 2018):

  1. (1)

Relative: These are the ones that are in direct relationship with the anchor
objects (e.g., ”in”, ”left_of”, or ”near”).

  2. (2)

Absolute: This uses an external frame of reference, such as the earth’s
cardinal directions (e.g., ”north_of” or ”south_of”)

  3. (3)

Intrinsic: Different from the Relative type, this adheres to the inherent
properties of toponyms. For example, in ”The clock tower in front of the
palace”, the orientation here is not important because there is only one front
of the palace regardless of the relative binary relationship between the
palace and the clock tower (Levinson and Levinson, 2003).

Other linguistic facts and cognitive constraints of language inform the
understanding of spatial relations and prepositional phrases. In English, the
encoding of subjects and objects of spatial prepositions reflects the size of
objects (Landau and Jackendoff, 1993). The larger sized objects/locations
always appear as the object in the case of ”inside”, ”in”, ”outside”, etc
(e.g., for ”Our office is in Ohio”, the area of the ”office” is much smaller
than the area of ”Ohio”). Similarly for adjacency relationships where bigger
sized spatial extents, usually of toponyms that are familiar or culturally
significant, are used as a reference to smaller spatial extents, as in the
example of ”behind $x$”. The use of ”among” is interpreted as n-ary in
contrast to the binary relation ”between”. Additionally, distances can be
combined with direction markers to yield finer distinctions of spatial
referents (e.g., ”$5km$ away from $x$”).

##  4\. Spatial referents representation

Given a sentence $W=\langle w_{1},w_{2},\dots,w_{n}\rangle$, let $W$ contain
one or more atomic toponyms in addition to _ad hoc_ spatial expressions
defining the resolution of the referents and the extent of the focal area with
spatial relations (see Section 3). While the semantics of the spatial
footprint of an atomic toponym is clear (i.e., available in mapping systems
and gazetteers), the spatial semantics and extent of an _ad hoc_ expression is
not clear or formally defined. Inspired by Zadeh’s work on linguistic
variables and values (Zadeh, 1975), we define spatial linguistic terms in ad
hoc spatial expressions, such as ”between” or ”north_of”, by defining _fuzzy_
restrictions on their spatial extent.

Figure 1. Examples of abstract spatial representations of anchor objects
($z_{(\cdot)}$) and the ad hoc referents ”northern $z_{1}$”
($\mathbf{a_{1}}$), ”north_of $z_{1}$” ($\mathbf{a_{2}}$), ”$t$ time/distance
away_from $z_{1}$” ($\mathbf{a_{3}}$), ”between $z_{2}$ and $z_{3}$”
($\mathbf{a_{4}}$), ”along/on $z_{4}$” ($\mathbf{a_{5}}$), ”$z5$ with $z_{6}$”
($\mathbf{a_{6}}$), ”outside/out_of $z_{7}$” ($\mathbf{a_{7}}$), and
”in/inside $z_{8}$” ($\mathbf{a_{8}}$).

Ad hoc expressions on a linguistic value (e.g., ”north_of”) such as ”$t$
minutes” or ”$m$ miles”, modify (or make precise) the semantics induced by
predefined syntactic rules. To represent qualitative spatial expressions, we
expand the formulation of Zadeh’s linguistic variable. We characterize a
linguistic variable by a sextuple
$(\mathcal{V},\mathcal{I},\mathcal{C},\mathcal{P},\mathcal{G},\mathcal{S})$,
where $\mathcal{V}$ is a linguistic variable (e.g., ”distance”), $\mathcal{I}$
is the set of linguistic terms for a fuzzy value or functional relation (e.g.,
”away_from” and ”minutes_from”), $\mathcal{C}$ is the set of all anchor
objects, $\mathcal{P}$ is the set of spatial extents of the objects in
$\mathcal{C}$, $\mathcal{G}$ is the set of spatial expression syntactic rules
that generates the terms in $\mathcal{I}$, and $\mathcal{S}$ is the set of
semantic rules restricting the values of each term in $\mathcal{I}$ referenced
using $\mathcal{C}$.

We employ the cognitive constraints of the language from Section 3 while
constructing semantic rules to further constrain the spatial extent of
referents. For example, in the case of ”between”, the sides of the fuzzy
polygon should not exceed the edge of the shortest side of the two facing
polygons as we discuss next.

Figure 1 shows abstractions of different spatial referents and the
corresponding spatial expressions that can be formulated using the predefined
sextuple. Below is an example formulation of $II$:

$\displaystyle\mathcal{V}=\text{"topological"}$
$\displaystyle\mathcal{I}=\text{\\{ "between"; "betwixt" \\}}$
$\displaystyle\mathcal{C}=\\{c_{1},c_{2}\\}$
$\displaystyle\mathcal{P}=\\{c_{1}\to[lat_{1},long_{1},lat_{2},long_{2}],$
$\displaystyle\ \ \ \ \ \ \ \ \ \
c_{2}\to[lat_{1},long_{1},lat_{2},long_{2}]\mid c_{1},c_{2}\in\mathcal{C}\\}$
$\displaystyle\mathcal{G}=\\{\langle i,c_{1},\text{"and"},c_{2}\rangle\mid
i\in\mathcal{I}\wedge c_{1},c_{2}\in\mathcal{C}\\}$
$\displaystyle\mathcal{S}=[\mathcal{P}(c_{1}).lat_{2}-\alpha,\mathcal{P}(c_{1}).long_{1}-\beta,$
$\displaystyle\ \ \ \ \ \ \ \ \
\mathcal{P}(c_{2}).lat_{1}+\alpha,\mathcal{P}(c_{1}).long_{2}+\beta]$

where $c_{i}$ is the string name of an anchor object (e.g., ”Walmart”),
$\mathcal{P}$ containing the polygon of each $c_{i}$ (e.g., the simple
bounding box [38.40, -84.82, 42.32, -80.51]). Without loss of generality and
for the convenience of illustration and notational simplicity, we narrow down
the spatial representation of each anchor object from complex polygons to
rectangular form (i.e., bounding boxes) with bounds on both sides (i.e., top-
right $[lat_{1},long_{1}]$ and bottom-left $[lat_{2},long_{2}]$ points).
$\mathcal{G}$ contains only one spatial expression which describes how the
spatial relationship in $\mathcal{I}$ must be used (e.g., ”between Walmart and
Sam’s Club”). Finally, the semantics of the lexical item ”between” is defined
as a polygon between these toponyms333For more complex polygons of the two
toponyms, the area inside the convex hull would make the ”between” area
(Billen and Clementini, 2004).. As shown in Figure 1-$II$, the area takes the
shortest facing edge of the two polygons (shown as bounding boxes for ease of
representation) with the height to be the distance between them. Therefore,
the possible area of ”between” can be $\mathcal{S}$, where $lat_{(\cdot)}$ and
$long_{(\cdot)}$ are the latitudes and longitudes of the $i^{th}$ anchor
object, and $\alpha$ and $\beta$ are two offsets defined based on the
semantics of $I$ to create buffer distances (Chen et al., 2018).444The offsets
values would be different for ”between” and ”on” since ”on” is for exterior
referents with contact while ”between” does not mean contact.

Figure 2. Spatial representation of ”6 hours” and ”south of” ad hoc terms in
relation to the atomic toponym ”Ohio”.

As for the directional markers in $\langle d,\text{"of"},z\rangle\mid
d\in\\{\text{"north"; "east"};\\\ \dots\\}$, we create a polygon representing
that area in relation to the anchor object $z$ using a cone-based cardinal
direction model (Liu et al., 2005). For example, for ”south of Ohio”, we
create a polygon $\mathcal{S}_{1}$ that sits on the bottom side of Ohio’s
polygon $\mathcal{P}_{1}$ (see Figure 2). The extent of $\mathcal{S}_{1}$
continues to a maximum of half of the equatorial circumference. Now, the
constraint ”6 hours” on the ”south of Ohio” expression would create a sub-
polygon $\mathcal{S}_{2}$ of the previously induced spatial extent
$\mathcal{P}_{2}$ to only be the red polygon. Notice that since ”6 hours” is
fuzzy, the created polygon would capture that fuzziness by adding buffer
distances to the sides of $\mathcal{S}_{2}$ equal to, for example, 5% of the
modifier value (i.e., $\pm 18$ minutes of the $6$ hours constraint). Our
formulation scales automatically with the addition of more constraints and ad
hoc expressions. For example, if we add ”near Asheville” to the end of ”6
hours south of Ohio”, we would create another polygon as in Figure 2, showing
the constraint on $\mathcal{S}_{2}$, in this case.

Similarly, we represent the examples in Figure 1, without loss of generality,
using a complete polygon (as in
$\mathbf{a_{1}},\mathbf{a_{2}},\mathbf{a_{4}},\mathbf{a_{6}},$ and
$\mathbf{a_{8}}$) or a polygon with a hole in it (as in $\mathbf{a_{3}}$ and
$\mathbf{a_{7}}$). The challenge is not only the determination of the fuzzy
boundaries for spatial extents but also the range of possible interpretations
of the relative position of the polygons. In the case of $\mathbf{a_{3}}$,
there are an infinite number of points that are $t$ minutes/miles away from
$\mathbf{z_{1}}$, which requires more context to draw the fuzzy
polygon555Different distance calculation methods can be used including ”as the
crow flies” distance or route-based distance calculation which requires route
navigation assuming ”cars” as the culturally determined mode of
transportation.. Alternatively, absolute frames of reference using lexical
markers such as ”north_of” or ”south_of” constrain the area of the referent,
but without context or other lexical markers, we would draw a polygon with a
hole in it covering all possible spatial extents around the atomic toponym.
Finally, the alternative intrinsic frame of reference needs a full
understanding of the surrounding area and requires a gazetteer with rich
metadata and toponym semantics. For example, finding the front of a university
campus might require searching for clusters of points of interest to suggest
the orientation of the toponym. We leave further exploration and evaluation of
this as future work.

##  5\. Evaluation

The majority of the techniques in the literature focus on geocoding the
spatial extent of atomic toponyms (i.e., Toponym Resolution) (Kamalloo and
Rafiei, 2018). The default evaluation methods are Precision, Recall, and the
F1 measure as the task is to link a toponym to a gazetteer record or compare
the geocoordinates with annotated toponyms in documents. Other measures
include micro and macro averaging, mean error distance, AUC of geocoding
errors, and Acc@161, which calculates the accuracy of geocoding within a 161km
tolerance distance (Gritta et al., 2018). Clearly, such evaluation methods are
incompatible with the qualitative, imprecise location referents that have no
records in a gazetteer, have fuzzy meaning, and are finer than typical atomic
toponyms with (sometimes) a huge error tolerance of 161km.

As there is no satisfactory approach in the literature or gold data available
for fine-grained referents involving imprecise spatial regions, we devise an
approach using a Likert scale, similar to (Aflaki et al., 2018) but while
adding the negative side of the scale (i.e., including ”Strongly Agree” to
”Strongly Disagree”). Effectively, the individual scores ranging from 1 to -1
to be plugged in the following equation to score all geocodings:

$Score=\frac{1}{mn}\sum_{i=0}^{n}\sum_{j=0}^{m}\sigma_{i,j}$

where $n$ is the number of spatial expressions, $m$ is the number of
annotators, $\sigma$ is the weight (1 to -1 Likert scale) given to a geocoded
expression by the annotator $j$, and $Score$ is the cumulative system score
averaged over all weights capturing the judges agreements and disagreements to
evaluate the geocoder.

##  6\. Conclusions and Future Work

In this paper, we devised a formal representation to geocode imprecise and _ad
hoc_ location referents as opposed to atomic gazetteer toponyms. The potential
disagreement between the different interpretations and the unavailability of
appropriate gold standard data made conventional evaluation methods
incompatible with the task at hand and required from us to develop more
approximate evaluation measures. This paper constitutes an initial step in
this complex task, and we plan to continue this work by expanding on the
formulation and the semantic rules, and evaluating our solution on real-world
data.

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