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

Chen Zhao czhao@ustc.edu.cn 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.

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)

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

(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)

(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

{\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)

(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)

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)

(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)

(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)

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)

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]).

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

L2L^{2} representation of Simpson-Mochizuki’s prolongation of Higgs bundles and the Kawamata-Viehweg vanishing theorem for semistable parabolic Higgs bundles

Abstract.

1. Introduction

1.1. Main result

Theorem 1.1.

1.2. Application: vanishing theorems

Theorem 1.2 (=Theorem 5.8).

Remark 1.3.

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

Corollary 1.4.

1.2.2. Saito-type vanishing theorems

Corollary 1.5.

2. Preliminary

2.1. Nilpotent harmonic bundles

Definition 2.1.

Definition 2.2.

Definition 2.3 (Admissible coordinate).

Definition 2.4.

2.2. Boundedness for the Higgs field

Definition 2.5 (Poincaré type metric).

Proposition 2.6 ([CG1975, Zucker1979]).

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

3. L2L^{2} cohomology and L2L^{2} complex

Lemma 3.1 (fineness of the L2L^{2} sheaf).

Proof.

4. L2L^{2} representation of the prolongation of Higgs bundles

4.1. Prolongation and parabolic structure

Definition 4.1.

Definition 4.2 (Prolongation).

Theorem 4.3 ([Mochizuki2009], Proposition 2.53).

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

4.2. L2L^{2} representation

Theorem 4.5.

4.3. Local behavior of the metrics

Remark 4.6.

Proposition 4.7.

Proof.

4.4. Fine bi-complex resolution

Theorem 4.8 ([Demailly1982], Theorem 4.1).

Proposition 4.9.

Proof.

Corollary 4.10.

Proof.

4.5. Proof of Theorem 4.5

Proposition 4.11.

Lemma 4.12.

Proof.

Proposition 4.13.

Proof.

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

Proposition 5.1 ([DF2021], Corollary 2.7).

5.1. Construction of the relavant metrics

Lemma 5.2.

Proof.

Remark 5.3.

Remark 5.4.

Proposition 5.5.

Proof.

5.2. Fine resolution

Proposition 5.6.

Proof.

5.3. Vanishing theorem on weakly pseudoconvex manifolds

Theorem 5.7.

Proof.

Theorem 5.8.

Proof.

Remark 5.9.

5.4. Generalization of Suh’s vanishing theorem

Definition 5.10.

Remark 5.11.

Theorem 5.12.

Theorem 5.13.

References

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

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

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

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

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

4.2. $L^{2}$ representation