$L^{2}$ -Extension of Adjoint bundles and Kollár’s Conjecture

Junchao Shentu stjc@ustc.edu.cn School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China and Chen Zhao czhao@ustc.edu.cn School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China

Abstract.

We give a new proof of Kollár’s conjecture on the pushforward of the dualizing sheaf twisted by a variation of Hodge structure. This conjecture was settled by M. Saito via mixed Hodge modules and has applications in the investigation of Albanese maps. Our technique is the $L^{2}$ -method and we give a concrete construction and proofs of the conjecture. The $L^{2}$ point of view allows us to generalize Kollár’s conjecture to the context of non-abelian Hodge theory.

1. Introduction

Let $f:X\rightarrow Y$ be a surjective morphism between complex projective varieties. Assume that $X$ is smooth and denote by $\omega_{X}$ its dualizing sheaf. In [Kollar1986_1, Kollar1986_2], J. Kollár proves the following results which is roughly called the Kollár package in this paper.

Torsion Freeness:

$R^{i}f_{\ast}\omega_{X}$ is torsion free for $i\geq 0$ and $R^{i}f_{\ast}\omega_{X}=0$ if $i>\dim X-\dim Y$ .

Vanishing Theorem:

If $L$ is an ample line bundle on $Y$ , then

H^{j}(Y,R^{i}f_{\ast}\omega_{X}\otimes L)=0\textrm{ for }\forall j>0\textrm{ and }\forall i\geq 0.

Decomposition Theorem:

$Rf_{\ast}(\omega_{X})$ splits in $D(Y)$ , i.e.

Rf_{\ast}(\omega_{X})\simeq\bigoplus_{q}R^{q}f_{\ast}(\omega_{X})[-q]\in D(Y).

As a consequence, the spectral sequence

E^{pq}_{2}:=H^{p}(Y,R^{q}f_{\ast}\omega_{X})\Rightarrow H^{p+q}(X,\omega_{X})

degenerates at the $E_{2}$ page.

Motivated by the proofs, Kollár [Kollar1986_2, §5] conjectured that the Kollár package could be put into a more general framework which is closely related to variations of Hodge structure. More precisely, Kollár conjectured that there is a coherent sheaf $S_{X}(\mathbb{V})$ associated to every polarized variation of Hodge structure $\mathbb{V}=(\mathcal{V},\nabla,\mathcal{F}^{\bullet},h_{\mathbb{V}})$ over some dense Zariski open subset of $X_{\rm reg}$ , such that the three results above hold when $\omega_{X}$ is replaced by $S_{X}(\mathbb{V})$ . This conjecture is perfectly settled in [MSaito1991] by M. Saito’s theory of mixed Hodge modules [MSaito1988, MSaito1990] and has applications in the investigation of Albanese maps.

The purpose of this paper has two sides.

(1)
Give a concrete construction of $S_{X}(\mathbb{V})$ by using certain $L^{2}$ holomorphic sections and reprove Kollár’s conjecture without using mixed Hodge modules. There are, in addition to the concrete construction, two other advantages of the $L^{2}$ method:
1. (a)
  
  It allows us to prove Kollár’s conjecture for proper Kähler morphisms. The machinery of mixed Hodge modules does not work for this more general setting because the decomposition theorem of Saito with respect to a proper Kähler morphism is still a conjecture ([MSaito1990(3), Conjecture 0.4]).
2. (b)
  
  It allows us to prove Kollár’s conjecture with a twisted coefficients in a hermitian vector bundle with a Nakano semi-positive curvature. This flexibility is convenient for applications.
(2)

We observe that, rather than the structure of the variation of Hodge structure, the validity of the Kollár package for $S_{X}(\mathbb{V})$ is a consequence of the Nakano semipositivity of the top Hodge bundle $S(\mathbb{V}):=\mathcal{F}^{\max\{k|\mathcal{F}^{k}\neq 0\}}$ (see §4 for the abstract Kollár package). This allows us to further generalize Kollar’s conjecture to the context of non-abelian Hodge theory. For example, we show that the Kollár package holds for $S_{X}(E,h)$ (see below for definition) when $E$ is a subbundle of a tame harmonic bundle $(H,\theta,h)$ such that $\overline{\theta}(E)=0$ and the second fundamental form of $E\subset H$ vanishes. A typical example is a system of Hodge bundles in the sense of C. Simpson [Simpson1988] on some Zariski open subset $X^{o}\subset X_{\rm reg}$ while $E$ is the top nonzero Hodge summand. This is an example that supports the principle that many results for variation of Hodge structure hold for harmonic bundles. Readers may see the hard Lefschetz theorems of Simpson [Simpson1992], C. Sabbah [Sabbah2005] and T. Mochizuki [Mochizuki20072, Mochizuki20071], the vanishing theorems of D. Arapura [Arapura2019] and Y. Deng-F. Hao [Dengya2019] and Mochizuki’s degeneration theory for twistor structures [Mochizuki20072] for other examples supporting this principle. Notice that the Kollár conjecture for complex variations of Hodge structure may be proved in the patten of Saito [MSaito1991] by using the theory of complex Hodge modules (see [Popa-Schnell2017] or the MHM Project by C. Sabbah and C. Schnell on the homepage of Sabbah.).

Let $X$ be a complex space and $X^{o}\subset X_{\rm reg}$ a dense Zariski open subset. Let $(E,h)$ be a hermitian vector bundle on $X^{o}$ . Denote by $j:X^{o}\to X$ the open immersion. The main object of the present paper is the subsheaf $S_{X}(E,h)\subset j_{\ast}(K_{X^{o}}\otimes E)$ consisting of the holomorphic forms which are locally square integrable at every point of $X$ . This class of objects contains Saito’s $S$ -sheaf and shares many properties of the dualizing sheaf. It has the following features:

(1)

$S_{X}(E,h)$ is a torsion free $\mathscr{O}_{X}$ -module. It is coherent when $(E,h)$ is Nakano semi-positive and tame (Definition 2.8).
(2)

$S_{X}(E,h)$ has the functorial property (Proposition 2.5): Let $\pi:X^{\prime}\to X$ be a proper bimeromorphic map such that $\pi|_{\pi^{-1}X^{o}}:\pi^{-1}X^{o}\to X^{o}$ is biholomorphic, then

$\displaystyle S_{X}(E,h)\simeq\pi_{\ast}(S_{X^{\prime}}(\pi^{\ast}E,\pi^{\ast}h)).$

(3)

Let $\mathbb{V}:=(\mathcal{V},\nabla,\mathcal{F}^{\bullet},h_{\mathbb{V}})$ be an $\mathbb{R}$ -polarized variation of Hodge structure where $h_{\mathbb{V}}$ is the Hodge metric. Denote by $S(\mathbb{V}):=\mathcal{F}^{\max\{k|\mathcal{F}^{k}\neq 0\}}$ the top nonzero Hodge piece and by $S(IC_{X}(\mathbb{V}))$ the Saito’s $S$ -sheaf associated to the Hodge module $IC_{X}(\mathbb{V})$ . Then

S_{X}(S(\mathbb{V}),h_{\mathbb{V}})\simeq S(IC_{X}(\mathbb{V}))\quad\textrm{(\cite[cite]{[\@@bibref{}{SC2021}{}{}, Theorem 4.10]})}.

(4)

Consider a tame harmonic bundle $(H,\theta,h)$ and a holomorphic subbundle $E\subset H$ with vanishing second fundamental form. Assume that $\overline{\theta}(E)=0$ . Then the corresponding $S_{X}(E,h)$ is a coherent sheaf (Proposition 5.6) and satisfies the Kollár package (Theorem 1.1). When $(E,\theta,h)$ is the harmonic bundle associated to an $\mathbb{R}$ -polarized variation of Hodge structure $\mathbb{V}:=(\mathcal{V},\nabla,\mathcal{F}^{\bullet},h_{\mathbb{V}})$ and $E=S(\mathbb{V})$ , this is reduced to Saito’s $S(IC_{X}(\mathbb{V}))$ .

All complex spaces are assumed to be separated, reduced, paracompact, countable at infinity and of pure dimension throughout the present paper. We would like to point out that the complex spaces are allowed to be non-irreducible. The main result of the present paper is the following

Theorem 1.1.

Let $f:X\rightarrow Y$ be a proper locally Kähler morphism (Definition 4.1) from a complex space $X$ to an irreducible complex space $Y$ . Assume that every irreducible component of $X$ is mapped onto $Y$ . Let $X^{o}\subset X_{\rm reg}$ be a dense Zariski open subset and $(H,\theta,h)$ a tame harmonic bundle on $X^{o}$ . Let $E\subset H$ be a holomorphic subbundle with vanishing second fundamental form. Let $\overline{\theta}$ be the adjoint of $\theta$ and assume that $\overline{\theta}(E)=0$ . Let $F$ be a Nakano semi-positive vector bundle on $X$ . Then the following statements hold.

Torsion Freeness:

$R^{q}f_{\ast}(S_{X}(E,h)\otimes F)$ is torsion free for every $q\geq 0$ and vanishes if $q>\dim X-\dim Y$ .

Injectivity:

If $L$ is a semi-positive holomorphic line bundle so that $L^{l}$ admits a nonzero holomorphic global section $s$ for some $l>0$ , then the canonical morphism

R^{q}f_{\ast}(\times s):R^{q}f_{\ast}(S_{X}(E,h)\otimes F\otimes L^{\otimes k})\to R^{q}f_{\ast}(S_{X}(E,h)\otimes F\otimes L^{\otimes k+l})

is injective for every $q\geq 0$ and every $k\geq 1$

Vanishing:

If $Y$ is a projective algebraic variety and $L$ is an ample line bundle on $Y$ , then

H^{q}(Y,R^{p}f_{\ast}(S_{X}(E,h)\otimes F)\otimes L)=0,\quad\forall q>0,\forall p\geq 0.

Decomposition:

Assume moreover that $X$ is a compact Kähler space, then $Rf_{\ast}(S_{X}(E,h)\otimes F)$ splits in $D(Y)$ , i.e.

Rf_{\ast}(S_{X}(E,h)\otimes F)\simeq\bigoplus_{q}R^{q}f_{\ast}(S_{X}(E,h)\otimes F)[-q]\in D(Y).

As a consequence, the spectral sequence

E^{pq}_{2}:H^{p}(Y,R^{q}f_{\ast}(S_{X}(E,h)\otimes F))\Rightarrow H^{p+q}(X,S_{X}(E,h)\otimes F)

degenerates at the $E_{2}$ page.

When $(H,\theta,h)$ is a harmonic bundle associated to an $\mathbb{R}$ -polarized variation of Hodge structure $\mathbb{V}:=(\mathcal{V},\nabla,\mathcal{F}^{\bullet},h_{\mathbb{V}})$ , $E=S(\mathbb{V})$ and $F=\mathscr{O}_{X}$ , Theorem 1.1 implies Kollár’s conjecture. In this case, our construction of $S_{X}(\mathbb{V})$ coincides with Saito’s.

The present paper is organized as follows. In Section 2 we introduce the construction of the adjoint $L^{2}$ -extension $S_{X}(E,h)$ of a hermitian bundle $(E,h)$ which generalizes Saito’s $S$ -sheaf. Some fundamental properties of $S_{X}(E,h)$ are proved and its $L^{2}$ -Dolbeault resolution is introduced. Depending on the $L^{2}$ -Dolbeault resolution, we introduce a harmonic representation of the derived pushforwards of $S_{X}(E,h)$ in Section 3 by generalizing K. Takegoshi’s work [Takegoshi1995] to complex spaces. In Section 4 we prove an abstract Kollár package and illustrate the observation that: Kollár’s conjecture is a consequence of the Nakano semi-positivity of the top Hodge piece. We prove the main theorem 1.1 in Section 5 as an application.

Notation: Let $X$ be a complex space. A Zariski closed subset (=closed analytic subset) $Z\subset X$ is a closed subset which is locally defined as the zeros of a set of holomorphic functions. A subset $Y\subset X$ is called Zariski open if $X\backslash Z\subset X$ is Zariski closed.

2. $L^{2}$ -extension and its $L^{2}$ -Dolbeault resolution

2.1. $L^{2}$ -Dolbeault cohomology and $L^{2}$ -Dolbeault complex

A psh (resp. strictly psh) function on a complex space $X$ is a function $\lambda:X\to[-\infty,\infty)$ such that, locally at every point $x\in X$ , there is a neighborhood $x\in U$ , a closed immersion $\iota:U\to\Omega$ into a complex manifold $\Omega$ and a psh (resp. strictly psh) function $\Lambda$ on $\Omega$ such that $\iota^{\ast}\Lambda=\lambda$ . By a $C^{\infty}$ form on $X$ we mean a $C^{\infty}$ form $\alpha$ on $X_{\rm reg}$ so that locally at every point $x\in X$ there is an open neighborhood $x\in U$ , a closed immersion $\iota:U\to\Omega$ into a holomorphic manifold $\Omega$ and $\beta\in C^{\infty}(\Omega)$ such that $\iota^{\ast}\beta=\alpha$ on $U\cap X_{\rm reg}$ .

Let $(Y,ds^{2})$ be a hermitian manifold of dimension $n$ and $(E,h)$ a hermitian vector bundle on $Y$ . Let $L^{p,q}_{(2)}(Y,E;ds^{2},h)$ be the space of measurable $E$ -valued $(p,q)$ -forms on $Y$ which are square integrable 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}_{\rm max}(Y,E;ds^{2},h):=\textrm{Dom}^{p,q}(\bar{\partial}_{\rm max})=\{\phi\in L_{(2)}^{p,q}(Y,E;ds^{2},h)|\bar{\partial}\phi\in L_{(2)}^{p,q+1}(Y,E;ds^{2},h)\}.

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

(2.1)

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

Let $X$ be a complex space and $X^{o}\subset X_{\rm reg}$ a dense Zariski open subset of the regular locus $X_{\rm reg}$ . Let $ds^{2}$ be a hermitian metric on $X^{o}$ and $(E,h)$ a hermitian vector bundle on $X^{o}$ . Let $U\subset X$ be an open subset. Define $L_{X,ds^{2}}^{p,q}(E,h)(U)$ to be the space of measurable $E$ -valued $(p,q)$ -forms $\alpha$ on $U\cap X^{o}$ such that for every point $x\in U$ , there is a neighborhood $x\in V_{x}$ so that

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

For each $0\leq p,q\leq n$ , 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):=\{\phi\in L_{X,ds^{2}}^{p,q}(E,h)(U)|\bar{\partial}_{\rm max}\phi\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

(2.2)

\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.

Definition 2.1.

Let $X$ be a complex space and $ds^{2}$ a hermitian metric on $X_{\rm reg}$ . We say that $ds^{2}$ is a hermitian metric on $X$ if, for every $x\in X$ , there is a neighborhood $U$ and a holomorphic closed immersion $U\subset V$ into a complex manifold $V$ such that $ds^{2}|_{U}\sim ds^{2}_{V}|_{U}$ for some hermitian metric $ds^{2}_{V}$ on $V$ . If $ds^{2}_{0}|_{X_{\rm reg}}$ is moreover a Kähler metric, we say that $ds^{2}_{0}$ is a Kähler hermitian metric.

Lemma 2.2.

Let $X$ be a complex space and $X^{o}\subset X_{\rm reg}$ a dense Zariski open subset. Let $ds^{2}$ be a hermitian metric on $X^{o}$ and $(E,h)$ a hermitian vector bundle on $X^{o}$ . Suppose that for every point $x\in X\backslash X^{o}$ there is a neighborhood $x\in U_{x}$ and a hermitian metric $ds^{2}_{0}$ on $U_{x}$ such that $ds^{2}_{0}|_{X^{o}\cap U_{x}}\lesssim ds^{2}|_{X^{o}\cap U_{x}}$ . Then $\mathscr{D}^{p,q}_{X,ds^{2}}(E,h)$ is a fine sheaf for every $p$ and $q$ .

Proof.

If suffices to show that for every $W\subset\overline{W}\subset U\subset X$ where $W$ and $U$ are small open subsets, there is a positive continuous function $f$ on $U$ such that

•

${\rm supp}(f)\subset\overline{W}$ ,
•

$f$ is $C^{\infty}$ on $U\cap X^{o}$
•

$\bar{\partial}f$ has bounded fiberwise norm, with respect to the metric $ds^{2}$ .

Choose a closed embedding $U\subset M$ where $M$ is a smooth complex manifold $M$ . Let $V\subset\overline{V}\subset M$ where $V$ is an open subset such that $V\cap U=W$ . Let $ds^{2}_{M}$ be a hermitian metric on $M$ so that $ds^{2}_{0}|_{U\cap X^{o}}\sim ds^{2}_{M}|_{U\cap X^{o}}$ . Let $g$ be a positive smooth function on $M$ whose support lies in $\overline{V}$ . Denote $f=g|_{U}$ , then apparently ${\rm supp}(f)\subset\overline{W}$ and $f$ is $C^{\infty}$ on $U\cap X^{o}$ . It suffices to show the boundedness of the fiberwise norm of $\bar{\partial}f$ . Since $U\cap X^{o}\subset M$ is a submanifold, one has the orthogonal decomposition

(2.3)

\displaystyle T_{M,x}=T_{U\cap X^{o},x}\oplus T_{U\cap X^{o},x}^{\bot},\quad\forall x\in U\cap X^{o}.

Therefore $|\bar{\partial}f|_{ds^{2}}\lesssim|\bar{\partial}f|_{ds^{2}_{0}}\leq|\bar{\partial}g|_{ds^{2}_{M}}<\infty$ . The lemma is proved. ∎

2.2. $S_{X}(E,h)$ and its basic properties

Let $X$ be a complex space of dimension $n$ and $X^{o}\subset X_{\rm reg}$ a dense Zariski open subset. Let $ds^{2}$ be a hermitian metric on $X^{o}$ and $(E,h)$ a hermitian vector bundle on $X^{o}$ .

Definition 2.3.

Define

(2.4)

\displaystyle S_{X}(E,h):={\rm Ker}\left(\mathscr{D}_{X,ds^{2}}^{n,0}(E,h)\stackrel{{\scriptstyle\bar{\partial}}}{{\to}}\mathscr{D}_{X,ds^{2}}^{n,1}(E,h)\right).

The following proposition shows that $S_{X}(E,h)$ is independent of $ds^{2}$ . Hence $ds^{2}$ is omitted in the notation $S_{X}(E,h)$ .

Proposition 2.4.

$S_{X}(E,h)$ is independent of $ds^{2}$ .

Proof.

Let $\pi:\tilde{X}\to X$ be a desingularization of $X\backslash X^{o}$ so that $\widetilde{X}$ is smooth and $\pi$ is biholomorphic over $X^{o}$ . Let $ds^{2}_{\tilde{X}}$ be a hermitian metric on $\tilde{X}$ . Since $\pi$ is a proper map, a section of $K_{X^{o}}\otimes E$ is locally square integrable at $x$ if and only if it is locally square integrable near $\pi^{-1}\{x\}$ . Thus it suffices to show that

(2.5)

\displaystyle{\rm Ker}\left(\mathscr{D}_{\tilde{X},\pi^{\ast}ds^{2}}^{n,0}(E,h)\stackrel{{\scriptstyle\bar{\partial}}}{{\to}}\mathscr{D}_{\tilde{X},\pi^{\ast}ds^{2}}^{n,1}(E,h)\right)={\rm Ker}\left(\mathscr{D}_{\tilde{X},ds^{2}_{\tilde{X}}}^{n,0}(E,h)\stackrel{{\scriptstyle\bar{\partial}}}{{\to}}\mathscr{D}_{\tilde{X},ds^{2}_{\tilde{X}}}^{n,1}(E,h)\right).

Since the problem is local, we assume that there is an orthogonal frame of cotangent fields $\delta_{1},\dots,\delta_{n}$ such that

(2.6)

\displaystyle\pi^{\ast}ds^{2}\sim\lambda_{1}\delta_{1}\overline{\delta_{1}}+\cdots+\lambda_{n}\delta_{n}\overline{\delta_{n}}

and

(2.7)

\displaystyle ds^{2}_{\tilde{X}}\sim\delta_{1}\overline{\delta_{1}}+\cdots+\delta_{n}\overline{\delta_{n}}.

Let $s=\delta_{1}\wedge\cdots\wedge\delta_{n}\otimes\xi$ . By (2.6) and (2.7) we obtain

(2.8)	$\displaystyle\\|s\\|^{2}_{\pi^{\ast}ds^{2},h}$	$\displaystyle=\int\|\delta_{1}\wedge\cdots\wedge\delta_{n}\otimes\xi\|^{2}_{\pi^{\ast}ds^{2},h}\prod_{i=1}^{n}\lambda_{i}\delta_{i}\wedge\overline{\delta_{i}}$
	$\displaystyle=\int\|\xi\|^{2}_{h}\prod_{i=1}^{n}\delta_{i}\wedge\overline{\delta_{i}}$
	$\displaystyle=\\|s\\|^{2}_{ds^{2}_{\tilde{X}},h}.$

Therefore $\|s\|^{2}_{\pi^{\ast}ds^{2},h}$ is locally finite if and only if $\|s\|^{2}_{ds^{2}_{\tilde{X}},h}$ is locally finite. This proves (2.5). ∎

Proposition 2.5 (Functoriality).

Let $\pi:X^{\prime}\to X$ be a proper holomorphic map between complex spaces which is biholomorphic over $X^{o}$ . Then

\pi_{\ast}S_{X^{\prime}}(\pi^{\ast}E,\pi^{\ast}h)=S_{X}(E,h).

Proof.

It follows from Proposition 2.4 that

S_{X^{\prime}}(\pi^{\ast}E,\pi^{\ast}h)={\rm Ker}\left(\mathscr{D}_{X^{\prime},\pi^{\ast}ds^{2}}^{n,0}(\pi^{\ast}E,\pi^{\ast}h)\stackrel{{\scriptstyle\bar{\partial}}}{{\to}}\mathscr{D}_{X^{\prime},\pi^{\ast}ds^{2}}^{n,1}(\pi^{\ast}E,\pi^{\ast}h)\right)

and

S_{X}(E,h)={\rm Ker}\left(\mathscr{D}_{X,ds^{2}}^{n,0}(E,h)\stackrel{{\scriptstyle\bar{\partial}}}{{\to}}\mathscr{D}_{X,ds^{2}}^{n,1}(E,h)\right).

Since $\pi$ is a proper map, a section of $E$ is locally square integrable at $x$ if and only if it is locally square integrable near $\pi^{-1}\{x\}$ . This proves the lemma. ∎

The following simple lemma is important for applications. Roughly speaking, it says that one can shrink the domain of $(E,h)$ without changing $S_{X}(E,h)$ . This phenomenon also appears in Saito’s $S(M)$ .

Lemma 2.6.

Let $U\subset X^{o}$ be a dense Zariski open subset. Then

S_{X}(E,h)\simeq S_{X}(E|_{U},h|_{U}).

Proof.

This is a consequence of the fact that if a locally $L^{2}$ function on a hermitian manifold is $\bar{\partial}$ -closed away from a Zariski open subset, then it is $\bar{\partial}$ -closed over the whole manifold ([Berndtsson2010, Lemma 1.3]). ∎

Lemma 2.7.

Let $(F,h_{F})$ be a hermitian vector bundle on $X$ , then

S_{X}(E,h)\otimes F\simeq S_{X}(E\otimes F,h\otimes h_{F}).

Proof.

Let $x\in X$ be a point and let $U$ be an open neighborhood of $x$ so that $F|_{U}\simeq\mathscr{O}_{U}^{\oplus r}$ and $h_{F}$ is quasi-isometric to the trivial metric, i.e.

|\sum_{i=1}^{r}a_{i}e_{i}|^{2}_{h_{F}}\sim\sum_{i=1}^{r}\|a_{i}\|^{2}

where $\{e_{i}\}$ is the standard frame of $\mathscr{O}_{U}^{\oplus r}$ and $a_{i}$ s are measurable functions on $U\cap X^{o}$ . Let $ds^{2}$ be an arbitrary hermitian metric on $X^{o}$ and let $\alpha=\sum_{i=1}^{r}\alpha_{i}\otimes e_{i}$ be a measurable section of $(K_{X^{o}}\otimes E\otimes F)|_{U\cap X^{o}}$ . Then

\|\alpha\|^{2}_{ds^{2},h\otimes h_{F}}\sim\sum_{i=1}^{r}\|\alpha_{i}\|^{2}_{ds^{2},h}

is finite if and only if $\|\alpha_{i}\|^{2}_{ds^{2},h}$ is finite for each $i=1,\dots,r$ . This proves the lemma. ∎

At the end of this section, we show that $S_{X}(E,h)$ is a coherent sheaf if the dual metric $h^{\ast}$ has at most polynomially growth at the boundary $X\backslash X^{o}$ and $(E,h)$ is Nakano semi-positive.

Let $Y$ be a complex manifold and $E$ a holomorphic vector bundle on $Y$ . Let $\Theta\in A^{1,1}(Y,{\rm End}(E))$ . Denote $\Theta\geq 0$ if $\Theta$ is Nakano semi-positive, i.e.

(2.9)

\displaystyle\Theta(v,v)\geq 0,\quad\forall v\in\Gamma(T_{Y}\otimes E).

Let $\Theta_{1},\Theta_{2}\in A^{1,1}(Y,{\rm End}(E))$ , then $\Theta_{1}\geq\Theta_{2}$ stands for $\Theta_{1}-\Theta_{2}\geq 0$ . A hermitian vector bundle $(E,h)$ is Nakano semi-positive if $\sqrt{-1}\Theta_{h}(E)\geq 0$ .

Definition 2.8.

$(E,h)$ has tame singularity on $X$ if, for every point $x\in X$ , there is an open neighborhood $U$ , a proper bimeromorphic morphism $\pi:\widetilde{U}\to U$ which is biholomorphic over $U\cap X^{o}$ , and a hermitian vector bundle $(Q,h_{Q})$ on $\widetilde{U}$ such that

(1)

$\pi^{\ast}E|_{\pi^{-1}(X^{o}\cap U)}\subset Q|_{\pi^{-1}(X^{o}\cap U)}$ as a subsheaf.
(2)

There is a hermitian metric $h^{\prime}_{Q}$ on $Q|_{\pi^{-1}(X^{o}\cap U)}$ so that $h^{\prime}_{Q}|_{\pi^{\ast}E}\sim\pi^{\ast}h$ on $\pi^{-1}(X^{o}\cap U)$ and

(2.10) $\displaystyle(\sum_{i=1}^{r}\|\pi^{\ast}f_{i}\|^{2})^{c}h_{Q}\lesssim h^{\prime}_{Q}$

for some $c\in\mathbb{R}$ . Here $(f_{1},\dots,f_{r})$ is an arbitrary local generator of the ideal sheaf defining $\widetilde{U}\backslash\pi^{-1}X^{o}\subset\widetilde{U}$ .

The tameness condition (2.10) is independent of the choice of the local generator. In the present paper, a tame hermitian vector bundle $(E,h)$ is constructed as a subsheaf of a tame harmonic bundle (see §5.2, especially Proposition 5.6) in the sense of Simpson [Simpson1990] and T. Mochizuki [Mochizuki20072, Mochizuki20071]. In this case, condition (2.10) comes from the tameness condition of the harmonic bundle. This is the origin of the name ”tame hermitian vector bundle”.

Proposition 2.9.

$S_{X}(E,h)$ is a coherent sheaf if $(E,h)$ is Nakano semi-positive and tame on $X$ .

Proof.

Since the problem is local, we assume that $X$ is a germ of complex space. Let $\pi:\widetilde{X}\to X$ be a desingularization so that $\pi$ is biholomorphic over $X^{o}$ and $D:=\pi^{-1}(X\backslash X^{o})$ is a simple normal crossing divisor. By abuse of notations we regard $X^{o}\subset\widetilde{X}$ as a subset. Since $(E,h)$ is tame, we assume that there is a hermitian vector bundle $(Q,h_{Q})$ on $\widetilde{X}$ such that $E$ is a subsheaf of $Q|_{X^{o}}$ and there exists $m\in\mathbb{N}$ such that

(2.11)

\displaystyle\|z_{1}\cdots z_{r}\|^{2m}h_{Q}|_{E}\lesssim\pi^{\ast}h

where $(z_{1},\cdots,z_{n})$ is a local coordinate of $\widetilde{X}$ so that $D=\{z_{1}\cdots z_{r}=0\}$ . There is moreover a hermitian metric $h^{\prime}_{Q}$ on $Q|_{X^{o}}$ so that $h^{\prime}_{Q}|_{E}\sim h$ . By Proposition 2.5 there is an isomorphism

(2.12)

\displaystyle S_{X}(E,h)\simeq\pi_{\ast}\left(S_{\widetilde{X}}(E,h)\right).

Since $\pi$ is a proper map, it suffices to show that $S_{\widetilde{X}}(E,h)$ is a coherent sheaf on $\widetilde{X}$ . Since the problem is local and $\widetilde{X}$ is smooth, we may assume that $\widetilde{X}\subset\mathbb{C}^{n}$ is the unit ball so that $D=\{z_{1}\cdots z_{r}=0\}$ . Without loss of generality we assume that $Q$ admits a global holomorphic frame $e_{1},\dots,e_{l}$ which is orthonormal with respect to $h_{Q}$ , i.e.

(2.13)

\displaystyle\langle e_{i},e_{j}\rangle_{h_{Q}}=\begin{cases}1,&i=j\\ 0,&i\neq j\end{cases}.

There is a complete Kähler metric on $X^{o}$ by Lemma 2.13. Since $Q$ is coherent¹¹1We do not distinguish holomorphic vector bundle and its sheaf of holomorphic sections., the space $\Gamma(\widetilde{X},S_{\widetilde{X}}(E,h))$ generates a coherent subsheaf $\mathscr{J}$ of $Q$ by strong Noetherian property of coherent sheaves. We have the inclusion $\mathscr{J}\subset S_{\widetilde{X}}(E,h)$ by the construction. It remains to prove the converse. By the Krull’s theorem ([Atiyah1969, Corollary 10.19]), it suffices to show that

(2.14)

\displaystyle\mathscr{J}_{x}+S_{\widetilde{X}}(E,h)_{x}\cap m_{\widetilde{X},x}^{k+1}Q=S_{\widetilde{X}}(E,h)_{x},\quad\forall k\geq 0,\forall x\in\widetilde{X}.

Let $\alpha\in S_{\widetilde{X}}(E,h)_{x}$ be defined in a precompact neighborhood $V$ of $x$ . Choose a $C^{\infty}$ cut-off function $\lambda$ so that $\lambda\equiv 1$ near $x$ and ${\rm supp}\lambda\subset V$ . Denote $\|z\|^{2}:=\sum_{i=1}^{n}\|z_{i}\|^{2}$ . Let

(2.15)

\displaystyle\psi_{k}(z):=2(n+k+rm)\log|z-x|+\|z\|^{2}

and $h_{\psi_{k}}=e^{-\psi_{k}}h$ . Denote $\omega_{0}:=\sqrt{-1}\partial\bar{\partial}\|z\|^{2}$ , then

(2.16)

\displaystyle\sqrt{-1}\Theta_{h_{\psi_{k}}}(E)=\sqrt{-1}\partial\bar{\partial}\psi_{k}+\sqrt{-1}\Theta_{h}(E)\geq\omega_{0}.

Since ${\rm supp}\lambda\alpha\subset V$ and $\bar{\partial}(\lambda\alpha)=0$ near $x$ , we know that

(2.17)

\displaystyle\|\bar{\partial}(\lambda\alpha)\|^{2}_{\omega_{0},h_{\psi_{k}}}\sim\|\bar{\partial}(\lambda\alpha)\|^{2}_{\omega_{0},h}\leq\|\bar{\partial}\lambda\|^{2}_{L^{\infty}}\|\alpha\|^{2}_{\omega_{0},h}+\|\lambda\|^{2}\|\bar{\partial}\alpha\|^{2}_{\omega_{0},h}<\infty

Hence Theorem 2.10 gives a solution of the equation $\bar{\partial}\beta=\bar{\partial}(\lambda\alpha)$ so that

(2.18)

\displaystyle\|\beta\|^{2}_{\omega_{0},h}\lesssim\int_{X^{o}}|\beta|^{2}_{\omega_{0},h}|z-x|^{-2(n+k+rm)}{\rm vol}_{\omega_{0}}\lesssim\|\bar{\partial}(\lambda\alpha)\|^{2}_{\omega_{0},h_{\psi_{k}}}<\infty.

Thus $\gamma=\beta-\lambda\alpha$ is holomorphic and $\gamma\in\Gamma(\widetilde{X},S_{\widetilde{X}}(E,h))$ .

Assume that

\beta=\sum_{i=1}^{l}f_{i}e_{i}dz_{1}\wedge\cdots\wedge dz_{n}

for some holormorphic functions $f_{1},\dots,f_{l}\in\mathscr{O}_{\widetilde{X}}(X^{o})$ . By (2.11) we obtain that

(2.19)			$\displaystyle\sum_{i=1}^{l}\int_{X^{o}}\|f_{i}\|^{2}\\|z_{1}\cdots z_{r}\\|^{2m}\|z-x\|^{-2(n+k+rm)}{\rm vol}_{\omega_{0}}$
	$\displaystyle=$	$\displaystyle\int_{X^{o}}\|\beta\|^{2}_{\omega_{0},h_{Q}}\\|z_{1}\cdots z_{r}\\|^{2m}\|z-x\|^{-2(n+k+rm)}{\rm vol}_{\omega_{0}}$
	$\displaystyle\lesssim$	$\displaystyle\int_{X^{o}}\|\beta\|^{2}_{\omega_{0},h}\|z-x\|^{-2(n+k+rm)}{\rm vol}_{\omega_{0}}<\infty.$

This implies that $z_{1}^{m}\cdots z_{r}^{m}f_{i}\in m_{\widetilde{X},x}^{k+1+rm}$ for every $i=1,\dots,l$ ([Demailly2012, Lemma 5.6]). Therefore $\beta_{x}\in m_{\widetilde{X},x}^{k+1}Q$ and we prove (2.14). ∎

2.3. $L^{2}$ -Dolbeault resolution of $S_{X}(E,h)$

In this subsection we introduce an $L^{2}$ -Dolbeault resolution of $S_{X}(E,h)$ .

First we recall the following useful estimate.

Theorem 2.10.

[Demailly1982, Theorem 5.1] Let $Y$ be a complex manifold of dimension $n$ which admits a complete Kähler metric. Let $(E,h)$ be a hermitian vector bundle such that

\sqrt{-1}\Theta_{h}(E)\geq\omega\otimes{\rm Id}_{E}

for some (not necessarily complete) Kähler form $\omega$ on $Y$ . Then for every $q>0$ and every $\alpha\in L^{n,q}_{(2)}(Y,E;\omega,h)$ such that $\bar{\partial}\alpha=0$ , there is $\beta\in L^{n,q-1}_{(2)}(Y,E;\omega,h)$ such that $\bar{\partial}\beta=\alpha$ and $\|\beta\|^{2}_{\omega,h}\leq q^{-1}\|\alpha\|^{2}_{\omega,h}$ .

The main theorem of this section is the following

Theorem 2.11.

Let $X$ be a complex space of dimension $n$ and $ds^{2}$ a hermitian metric on a dense Zariski open subset $X^{o}\subset X_{\rm reg}$ with $\omega$ its fundamental form. Let $(E,h)$ be a Nakano semi-positive hermitian vector bundle. Assume that, locally at every point $x\in X$ , there is a neighborhood $x\in U$ , a strictly psh function $\lambda\in C^{2}(U)$ and a bounded psh function $\Phi\in C^{2}(U\cap X^{o})$ such that

\sqrt{-1}\partial\bar{\partial}\lambda|_{U\cap X^{o}}\lesssim\omega|_{U\cap X^{o}}\lesssim\sqrt{-1}\partial\bar{\partial}\Phi

Then the canonical map

(2.20)

\displaystyle S_{X}(E,h)\to\mathscr{D}^{n,\bullet}_{X,ds^{2}}(E,h)

is a quasi-isomorphism. As a consequence, there is a canonical isomorphism

H^{q}(X,S_{X}(E,h))\simeq H^{n,q}_{(2)}(X^{o},E;ds^{2},h),\quad\forall q\geq 0

if $X$ is compact.

Proof.

It suffices to show that (2.20) is exact at $\mathscr{D}^{n,q}_{X,ds^{2}}(E,h)$ for $q>0$ . Since the problem is local, we consider a point $x\in X$ and an open neighborhood $x\in U$ which is small enough so that there is $C>0$ and a bounded psh function $\Phi\in C^{2}(U\cap X^{o})$ such that $C\sqrt{-1}\partial\bar{\partial}\Phi\geq\omega|_{U\cap X^{o}}$ . Let $h^{\prime}=e^{-C\Phi}h$ . Since $\Phi$ is bounded and $(E,h)$ is Nakano semi-positive, we have $h^{\prime}\sim h$ and

\sqrt{-1}\Theta_{h^{\prime}}(E|_{U\cap X^{o}})=C\sqrt{-1}\partial\bar{\partial}\Phi\otimes{\rm Id}_{E}|_{U\cap X^{o}}+\sqrt{-1}\Theta_{h}(E|_{U\cap X^{o}})\geq\omega\otimes{\rm Id}_{E}|_{U\cap X^{o}}.

By Lemma 2.13, we may assume that $U\cap X^{o}$ admits a complete Kähler metric. By Theorem 2.10, we therefore have

H^{n,q}_{(2)}(U\cap X^{o},E|_{U\cap X^{o}};ds^{2},h)=H^{n,q}_{(2)}(U\cap X^{o},E|_{U\cap X^{o}};ds^{2},h^{\prime})=0,\quad\forall q>0.

This proves the exactness of (2.20) at $\mathscr{D}^{n,q}_{X,ds^{2}}(E,h)$ , $q>0$ . Since $\omega$ is locally bounded from below by a hermitian metric, $\mathscr{D}^{n,q}_{X,ds^{2}}(E,h)$ is a fine sheaf for every $q$ by Lemma 2.2. The second claim therefore follows from the compactness of the space $X$ . ∎

In order to apply Theorem 2.11, we introduce a type of hermitian metric that is crucial for the present paper.

Definition 2.12.

Let $X$ be a complex space and $X^{o}\subset X_{\rm reg}$ a dense Zariski open subset. Let $ds^{2}$ be a hermitian metric on $X^{o}$ .

(1)

$ds^{2}$ admits a $(\infty,1)$ bounded potential locally on $X$ if, for every point $x\in X$ , there is a neighborhood $U$ of $x$ , a function $\Phi\in C^{\infty}(U)$ such that $|\Phi|+|d\Phi|_{ds^{2}}<\infty$ and $ds^{2}|_{U}\sim\sqrt{-1}\partial\bar{\partial}\Phi$ .
(2)

$ds^{2}$ is called locally complete on $X$ if, for every point $x\in X$ , there is a neighborhood $U$ of $x$ such that $(\overline{U}\cap X^{o},ds^{2})$ is complete.
(3)

$ds^{2}$ is locally bounded from below by a hermitian metric if, for every point $x\in X$ , there is a neighborhood $U$ of $x$ and a hermitian metric $ds^{2}_{0}$ on $U$ such that $ds^{2}_{0}|_{U}\lesssim ds^{2}|_{U}$ .

Lemma 2.13.

Let $X$ be a weakly pseudoconvex Kähler space with $\psi$ a smooth exhausted psh function on $X$ . Denote $X_{c}:=\{x\in X|\psi(x)<c\}$ . Let $X^{o}\subset X$ be a dense Zariski open subset. Then, for every $c\in\mathbb{R}$ , there exists a complete Kähler metric $ds^{2}$ on $X_{c}\cap X^{o}$ such that

(1)

$ds^{2}_{0}\lesssim ds^{2}$ for some hermitian metric $ds^{2}_{0}$ on $X_{c}$ ;
(2)

$ds^{2}$ admits a $(\infty,1)$ bounded potential locally on $X_{c}$ .

Proof.

Let $U$ be a neighborhood of a point $x\in X\backslash X^{o}$ . Assume that $U\backslash X^{o}\subset U$ is defined by $f_{1},\dots,f_{r}\in\mathscr{O}_{U}(U)$ . Let

\varphi_{U}:=\frac{1}{\log(-\log\sum_{i=1}^{r}\|f_{i}\|^{2})}+\phi_{U},

where $\phi_{U}$ is a strictly $C^{\infty}$ psh function on $U$ so that $\varphi_{U}$ is strictly psh. Then the quasi-isometric class of $\sqrt{-1}\partial\bar{\partial}\varphi_{U}$ is independent of the choice of $\{f_{1},\dots,f_{r}\}$ and $\phi_{U}$ . By partition of unity the potential functions $\varphi_{U}$ can be glued to a global $\varphi$ on $X$ so that

(2.21)

\displaystyle\sqrt{-1}\partial\bar{\partial}\varphi|_{U}\sim\sqrt{-1}\partial\bar{\partial}\varphi_{U}

near every point $x\in X\backslash X^{o}$ and $\varphi\equiv 0$ away from a neighborhood $V$ of $X\backslash X^{o}$ .

Denoting $u=-\log\sum_{i=1}^{r}\|f_{i}\|^{2}$ , we assume that $u>e$ on $V$ by a possible shrinking of $V$ . Then

(2.22)		$\displaystyle\sqrt{-1}\partial\bar{\partial}\varphi\|_{U}$	$\displaystyle\sim\sqrt{-1}\frac{2+\log u}{u^{2}\log^{3}u}\partial u\wedge\bar{\partial}u+\sqrt{-1}\frac{\partial\bar{\partial}u}{u\log^{2}u}+\sqrt{-1}\partial\bar{\partial}\phi_{U}$
		$\displaystyle\sim\sqrt{-1}\frac{\partial u\wedge\bar{\partial}u}{u^{2}\log^{2}u}+\sqrt{-1}\frac{\partial\bar{\partial}u}{u\log^{2}u}+\sqrt{-1}\partial\bar{\partial}\phi_{U}.$

Hence

|\varphi|+|d\varphi|_{\sqrt{-1}\partial\bar{\partial}\varphi}\lesssim\frac{1}{\log u}<1.

Since $\log\log u$ is a smooth exhausted function such that

|d\log\log u|_{\sqrt{-1}\partial\bar{\partial}\varphi}\leq 2.

By the Hopf-Rinow theorem, $\sqrt{-1}\partial\bar{\partial}\varphi$ is locally complete near $X\backslash X^{o}$ .

Let $c\in\mathbb{R}$ and let $\omega_{0}$ be a Kähler hermitian metric on $X$ . By adding a constant to $\psi$ , we assume $\psi\geq 0$ . Then $\psi_{c}:=\psi+\frac{1}{c-\psi}$ is a smooth exhausted psh function on $X_{c}=\{x\in X|\psi(x)<c\}$ . Hence $\omega_{0}+\sqrt{-1}\partial\bar{\partial}\psi^{2}_{c}$ is a complete Kähler hermitian metric on $X_{c}$ ([Demailly1982, Theorem 1.3]).

Since $\overline{X_{c}}$ is compact,

\sqrt{-1}\partial\bar{\partial}\varphi+K(\omega_{0}+\sqrt{-1}\partial\bar{\partial}\psi^{2}_{c}),\quad K\gg 0

is positive definite and it gives a desired complete Kähler metric on $X_{c}\cap X^{o}$ which we need. ∎

3. Harmonic Representation of the derived pushforwards

Let $f:X\rightarrow Y$ be a proper locally Kähler morphism between complex spaces and $X^{o}\subset X_{\rm reg}$ a dense Zariski open subset. Let $(E,h)$ be a hermitian vector bundle on $X^{o}$ with a Nakano semi-positive curvature. The aim of this section is to introduce a harmonic representation of $R^{q}f_{\ast}S_{X}(E,h)$ . The results of this section are generalizations of Section 4 of [Takegoshi1995] to $L^{2}$ -Dolbeault complexes on complex spaces.

3.1. Harmonic forms on weakly 1-complete spaces

Let $Z$ be a complex space of pure dimension $n$ and $Z^{o}\subset Z_{\rm reg}$ a dense Zariski open subset. Let $ds^{2}$ be Kähler metric on $Z^{o}$ which is locally complete and admits a $(\infty,1)$ bounded potential locally on $X$ (Definition 2.12). Denote $\omega$ to be the associated Kähler form. Let $(E,h)$ be a hermitian vector bundle on $Z^{o}$ . Let $A^{p,q}(Z^{o},E)$ (resp. $A^{k}(Z^{o},E)$ ) be the space of $C^{\infty}$ $E$ -valued $(p,q)$ (resp. $k$ ) forms on $Z^{o}$ and let $A^{p,q}_{\rm cpt}(Z^{o},E)\subset A^{p,q}(Z^{o},E)$ be the subspace of forms with compact support in $Z^{o}$ . Let $\ast:A^{p,q}(Z^{o},E)\to A^{n-q,n-p}(Z^{o},E)$ be the Hodge star operator relative to $ds^{2}$ and let $\sharp_{E}:A^{p,q}(Z^{o},E)\to A^{q,p}(Z^{o},E^{\ast})$ be the anti-isomorphism defined by $h$ . Denote by $\langle-,-\rangle_{h}$ the pointwise inner product on $A^{p,q}(Z^{o},E)$ . These operators are related by

(3.1)

\displaystyle\langle\alpha,\beta\rangle_{h}{\rm vol}_{ds^{2}}=\alpha\wedge\ast\sharp_{E}\beta.

Denote

(3.2)

\displaystyle(\alpha,\beta)_{h}:=\int_{Z^{o}}\langle\alpha,\beta\rangle_{h}{\rm vol}_{ds^{2}}

and $\|\alpha\|_{h}:=\sqrt{(\alpha,\alpha)_{h}}$ . Let $\nabla=D^{\prime}+\bar{\partial}$ be the Chern connection associated to $h$ . Let $\bar{\partial}^{\ast}_{h}=-\ast D^{\prime}\ast$ and $D^{\prime\ast}_{h}=-\ast\bar{\partial}\ast$ be the formal adjoints of $\bar{\partial}$ and $D^{\prime}$ respectively.

For two operators $S$ and $T$ acting on $A^{r}(Z^{0},E)$ with degree $a$ and $b$ respectively, we define the graded Lie bracket $[S,T]:=S\circ T-(-1)^{ab}T\circ S$ .

Denote by $L$ the Lefschetz operator with respect to $ds^{2}$ and by $\Lambda$ the formal adjoint of $L$ . Then we have the following Kähler identities ([Wells1980], Chapter V):

(3.3)

\displaystyle D_{h}^{{}^{\prime}\ast}=-\sqrt{-1}[\bar{\partial},\Lambda],

(3.4)

\displaystyle\bar{\partial}_{h}^{\ast}=\sqrt{-1}[D^{\prime},\Lambda]\quad and

(3.5)

\displaystyle e(\theta)^{\ast}=\sqrt{-1}[e(\bar{\theta}),\Lambda]

for $\theta\in A^{1,0}(Z^{o},E)$ .

Denote $\Delta_{\bar{\partial}}=\bar{\partial}\bar{\partial}^{\ast}_{h}+\bar{\partial}^{\ast}_{h}\bar{\partial}$ and $\Delta_{D^{\prime}}=D^{\prime}D^{\prime\ast}_{h}+D^{\prime\ast}_{h}D^{\prime}$ . Since $e(\Theta_{h})=[\bar{\partial},D^{\prime}]$ , by (3.3), (3.4) and Jacobi’s identity

[[S,T],U]+(-1)^{a(b+c)}[[T,U],S]+(-1)^{c(a+b)}[[U,S],T]\equiv 0

where $a=\deg S$ , $b=\deg T$ and $c=\deg U$ respectively, we obtain that the formula

(3.6)

\displaystyle\Delta_{\bar{\partial}}=\Delta_{D^{\prime}}+\sqrt{-1}[e(\Theta_{h}),\Lambda]

holds on $Z^{o}$ where $e(\alpha)(\beta):=\alpha\wedge\beta$ .

Let $\varphi:Z^{o}\to\mathbb{R}$ be a $C^{\infty}$ function and let $h_{\varphi}:=e^{-\varphi}h$ . Since $D^{\prime}_{h_{\varphi}}=D^{\prime}-e(\partial\varphi)$ , we obtain the formulae:

(3.7)

\displaystyle\bar{\partial}_{h_{\varphi}}^{\ast}=\bar{\partial}_{h}^{\ast}+e(\bar{\partial}\varphi)^{\ast}

and

(3.8)

\displaystyle\Delta_{\bar{\partial}_{h_{\varphi}}}=\Delta_{D^{\prime}_{h_{\varphi}}}+\sqrt{-1}[e(\Theta(h)+\partial\bar{\partial}\varphi),\Lambda]

where $\Delta_{\bar{\partial}_{h_{\varphi}}}:=[\bar{\partial},\bar{\partial}_{h_{\varphi}}^{\ast}]$ and $\Delta_{D^{\prime}_{h_{\varphi}}}:=[D^{\prime}_{h_{\varphi}},D^{\prime\ast}_{h_{\varphi}}]$ .

By (3.3), (3.4), (3.5) and Jacobi’s identity, the Donnelly and Xavier’s formula ([Takegoshi1995, (1.9), (1.10)]) can be stated as follows.

(3.9)

\displaystyle[\bar{\partial},e(\bar{\partial}\varphi)^{\ast}]+[D^{\prime\ast}_{h},e(\partial\varphi)]=\sqrt{-1}[e(\partial\bar{\partial}\varphi),\Lambda]

and

(3.10)

\displaystyle[D^{\prime},e(\partial\varphi)^{\ast}]+[\bar{\partial}^{\ast}_{h},e(\bar{\partial}\varphi)]=\sqrt{-1}[e(\partial\bar{\partial}\varphi),\Lambda)].

Fix a Whitney stratification of $Z$ so that $Z^{o}$ is the union of open strata. By Sard’s theorem there is a subset $\Sigma\subset\mathbb{R}$ of measure zero so that for every $c\in\mathbb{R}\backslash\Sigma$ and every stratum $S$ of $Z$ , $c$ is a regular value of $\varphi|_{S}$ . Any $c\in\mathbb{R}\backslash\Sigma$ is called a regular value of $\varphi$ . For any regular value $c$ of $\varphi$ , $\{\varphi=c\}$ is a piecewise smooth submanifold of $Z$ . In particular, $\{\varphi\leq c\}\cap Z^{o}$ is a submanifold of $Z^{o}$ with a smooth boundary $\{\varphi=c\}\cap Z^{o}$ .

Denote

(\alpha,\beta)_{c,h}=\int_{\{\varphi\leq c\}\cap Z^{o}}\langle\alpha,\beta\rangle_{h}{\rm vol}_{ds^{2}},\quad[\alpha,\beta]_{c,h}:=\int_{\{\varphi=c\}\cap Z^{o}}\langle\alpha,\beta\rangle_{h}{\rm vol}_{\{\varphi=c\}}

and $\|\alpha\|_{c,h}:=\sqrt{(\alpha,\alpha)_{c,h}}$ for any regular value $c$ of $\varphi$ . By Stokes’s theorem we acquire that ([Takegoshi1995, (1.1)])

(3.11)

\displaystyle(\bar{\partial}\alpha,\beta)_{c,h}=(\alpha,\bar{\partial}^{\ast}_{h}\beta)_{c,h}+[\alpha,e(\bar{\partial}\varphi)^{\ast}\beta]_{c,h},

and

(3.12)

\displaystyle(D^{\prime}\alpha,\beta)_{c,h}=(\alpha,D^{\prime\ast}_{h}\beta)_{c,h}+[\alpha,e(\partial\varphi)^{\ast}\beta]_{c,h}

hold for every $C^{\infty}$ forms $\alpha$ and $\beta$ on $\{\varphi\leq c\}\cap Z^{o}$ such that either $\alpha$ or $\beta$ has compact support in $\{\varphi\leq c\}\cap Z^{o}$ .

Proposition 3.1.

Let $(M,\omega_{M})$ be a complete Kähler manifold of dimension $n$ and let $(E,h)$ be a Nakano semi-positive holomorphic vector bundle on $M$ . If $q\geq 1$ and $\alpha\in\mathscr{D}^{n,q}(M,E;\omega_{M},h)\cap{\rm Ker}\Delta_{\bar{\partial}}$ , then $\alpha$ satisfies the following equations:

(1)

$\bar{\partial}\alpha=0$ , $\bar{\partial}^{\ast}_{h}\alpha=0$ , $D^{\prime\ast}_{h}\alpha=0$ and $\langle e(\Theta_{h})\Lambda\alpha,\alpha\rangle_{h}=0$ on $M$ . In particular $\bar{\partial}(\ast\alpha)=0$ , i.e. $\ast\alpha\in\Gamma(M,\Omega^{n-q}_{M}(E))$ .
(2)

$e(\bar{\partial}\varphi)^{\ast}\alpha=0$ and $\langle e(\partial\bar{\partial}\varphi)\Lambda\alpha,\alpha\rangle_{h}=0$ on $M$ for any $C^{\infty}$ psh function $\varphi$ on $M$ with

${\rm sup}_{x\in M}\{|\varphi(x)|+|d\varphi(x)|_{\omega_{M}}\}<\infty.$

Proof.

See [Takegoshi1995, Theorem 3.4]. ∎

Denote by $\mathscr{P}_{e}(Z)$ the set of $C^{\infty}$ psh functions $\varphi:Z\to(-\infty,c_{\ast})$ for some $c_{\ast}\in(-\infty,\infty]$ such that $X_{c}:=\{z\in Z|\varphi(z)<c\}$ is precompact in $Z$ for every $c<c_{\ast}$ . $Z$ is called a weakly 1-complete space if $\mathscr{P}_{e}(Z)\neq\emptyset$ . For every $\varphi\in\mathscr{P}_{e}(Z)$ , denote

(3.13)

\displaystyle\mathscr{H}^{p,q}(Z,E,h,\varphi):=\left\{\alpha\in\mathscr{D}^{p,q}_{Z,ds^{2}}(E,h)(Z)\big{|}\bar{\partial}\alpha=\bar{\partial}^{\ast}_{h}\alpha=0,e(\bar{\partial}\varphi)^{\ast}\alpha=0\right\}.

By the regularity theorem for elliptic operators of second order, every element of $\mathscr{H}^{p,q}(Z,E,h,\varphi)$ is $C^{\infty}$ on $Z^{o}$ .

Proposition 3.2.

Let $Z$ be a weakly 1-complete space of pure dimension $n$ and $Z^{o}\subset Z_{\rm reg}$ a dense Zariski open subset. Let $ds^{2}$ be a Kähler metric on $Z^{o}$ which is locally complete on $Z$ and let $(E,h)$ be a hermitian vector bundle on $Z^{o}$ with a Nakano semi-positive curvature. Then the following assertions hold.

(1)

Let $\varphi\in\mathscr{P}_{e}(Z)$ . Assume that $\alpha\in\mathscr{D}^{n,q}_{Z,ds^{2}}(E,h)(Z)$ satisfies $e(\bar{\partial}\varphi)^{\ast}\alpha=0$ . Then $\bar{\partial}\alpha=\bar{\partial}^{\ast}_{h}\alpha=0$ if and only if $D_{h}^{\prime\ast}\alpha=\langle\sqrt{-1}e(\Theta_{h}+\partial\bar{\partial}\varphi)\Lambda\alpha,\alpha\rangle_{h}=0$ . Here $\langle\sqrt{-1}e(\Theta_{h}+\partial\bar{\partial}\varphi)\Lambda\alpha,\alpha\rangle_{h}=0$ is equivalent to $\langle\sqrt{-1}(\partial\bar{\partial}\varphi)\Lambda\alpha,\alpha\rangle_{h}=\langle\sqrt{-1}(\Theta_{h})\Lambda\alpha,\alpha\rangle_{h}=0$ .
(2)

Let $\varphi,\psi\in\mathscr{P}_{e}(Z)$ . Then $\mathscr{H}^{n,q}(Z,E,h,\varphi)=\mathscr{H}^{n,q}(Z,E,h,\psi)$ for every $q\geq 0$ .

(3)

For every $q\geq 0$ and every $\varphi\in\mathscr{P}_{e}(Z)$ , the Hodge star operator gives a well defined map

(3.14)

\displaystyle\ast:\mathscr{H}^{n,q}(Z,E,h,\varphi)\to{\rm Ker}\bar{\partial}\left(\mathscr{D}^{n-q,0}_{Z,ds^{2}}(E,h)(Z)\to\mathscr{D}^{n-q,1}_{Z,ds^{2}}(E,h)(Z)\right).

Proof.

To show (1), we suppose that $e(\bar{\partial}\varphi)^{\ast}\alpha=0$ and $\bar{\partial}\alpha=\bar{\partial}^{\ast}_{h}\alpha=0$ . Take any regular value $c$ of $\varphi$ . Since $(\{\varphi\leq c\}\cap Z^{o},ds^{2})$ is complete, the Hopf-Rinow theorem implies that there exists an exhaustive sequence $(K_{\nu})$ of compact sets of $Z^{o}\cap\{\varphi\leq c\}$ and functions $\theta_{\nu}\in C^{\infty}(\{\varphi\leq c\}\cap Z^{o},\mathbb{R})$ such that

\theta_{\nu}=1\quad\textrm{on a neighbourhood of $K_{\nu}$,}\quad\textrm{Supp $\theta_{\nu}\subset K^{\circ}_{\nu+1}$},

0\leq\theta_{\nu}\leq\theta_{\nu+1}\leq 1\quad\textrm{and}\quad|d\theta_{\nu}|_{g}\leq 2^{-\nu},\quad\forall\nu.

Then $\theta_{\nu}\alpha$ converges to $\alpha$ under the norm $\|-\|+\|\bar{\partial}-\|+\|D^{\prime}-\|+\|\bar{\partial}^{\ast}_{h}-\|+\|D^{\prime\ast}_{h}-\|$ .

Let $h_{\varphi}=e^{-\varphi}h$ . Since $\bar{\partial}\alpha=\bar{\partial}^{\ast}_{h}\alpha=0$ , by (3.6) we obtain that

(3.15)

\displaystyle(D^{\prime}D^{\prime\ast}_{h_{\varphi}}\alpha,\theta_{\nu}\alpha)_{c,h_{\varphi}}+(\sqrt{-1}e(\Theta_{h_{\varphi}})\Lambda\alpha,\theta_{\nu}\alpha)_{c,h_{\varphi}}=0.

By (3.12), this is equivalent to

(3.16)

\displaystyle(D^{\prime\ast}_{h_{\varphi}}\alpha,D^{\prime\ast}_{h_{\varphi}}(\theta_{\nu}\alpha))_{c,h_{\varphi}}+(\sqrt{-1}e(\Theta_{h_{\varphi}})\Lambda\alpha,\theta_{\nu}\alpha)_{c,h_{\varphi}}+[e(\partial\varphi)D^{\prime\ast}_{h_{\varphi}}\alpha,\theta_{\nu}\alpha]_{c,h_{\varphi}}=0.

By the assumptions and (3.9) we have

(3.17)

\displaystyle e(\partial\varphi)D^{\prime\ast}_{h_{\varphi}}\alpha=\sqrt{-1}e(\partial\bar{\partial}\varphi)\Lambda\alpha.

Substituting (3.17) into (3.16), we get that

(3.18)

\displaystyle(D^{\prime\ast}_{h_{\varphi}}\alpha,D^{\prime\ast}_{h_{\varphi}}(\theta_{\nu}\alpha))_{c,h_{\varphi}}+(\sqrt{-1}e(\Theta_{h_{\varphi}})\Lambda\alpha,\theta_{\nu}\alpha)_{c,h_{\varphi}}+[\sqrt{-1}e(\partial\bar{\partial}\varphi)\Lambda\alpha,\theta_{\nu}\alpha]_{c,h_{\varphi}}=0.

Letting $\nu\to\infty$ , we obtain that

(3.19)

\displaystyle\|D^{\prime\ast}_{h_{\varphi}}\alpha\|^{2}_{c,h_{\varphi}}+(\sqrt{-1}e(\Theta_{h_{\varphi}})\Lambda\alpha,\alpha)_{c,h_{\varphi}}+[\sqrt{-1}e(\partial\bar{\partial}\varphi)\Lambda\alpha,\alpha]_{c,h_{\varphi}}=0.

Since $\varphi$ is a psh function and $\sqrt{-1}\Theta_{h}$ is Nakano semi-positive, $\sqrt{-1}\Theta_{h_{\varphi}}=\sqrt{-1}\Theta_{h}+\sqrt{-1}\partial\bar{\partial}\varphi$ is also Nakano semi-positive. Then all the three terms in (3.19) are semi-positive. Hence they are all zero for every regular value $c$ of $\varphi$ . This proves the necessity of (1).

To prove the sufficiency of (1), we assume that $D_{h}^{\prime\ast}\alpha=\sqrt{-1}\langle e(\Theta_{h}+\partial\bar{\partial}\varphi)\Lambda\alpha,\alpha\rangle_{h}=0$ and $e(\bar{\partial}\varphi)^{\ast}\alpha=0$ . By (3.6) we have $\Delta_{\bar{\partial}}\alpha=0$ . By (3.11) we obtain

(3.20)			$\displaystyle(\Delta_{\bar{\partial}}\alpha,\theta_{\nu}\alpha)_{c,h}$
	$\displaystyle=$	$\displaystyle(\bar{\partial}^{\ast}_{h}\alpha,\bar{\partial}^{\ast}_{h}(\theta_{\nu}\alpha))_{c,h}+(\bar{\partial}\alpha,\bar{\partial}(\theta_{\nu}\alpha))_{c,h}+[\bar{\partial}^{\ast}_{h}\alpha,e(\bar{\partial}\varphi)^{\ast}(\theta_{\nu}\alpha)]_{c,h}-[e(\bar{\partial}\varphi)^{\ast}\bar{\partial}\alpha,\theta_{\nu}\alpha]_{c,h}$
	$\displaystyle=$	$\displaystyle 0.$

By (3.9) we acquire that

(3.21)

\displaystyle\langle\sqrt{-1}e(\bar{\partial}\varphi)\bar{\partial}\alpha,\theta_{\nu}\alpha\rangle_{h}=\langle\sqrt{-1}e(\partial\bar{\partial}\varphi)\Lambda\alpha,\theta_{\nu}\alpha\rangle_{h}=\theta_{\nu}\sqrt{-1}\langle e(\partial\bar{\partial}\varphi)\Lambda\alpha,\alpha\rangle_{h}=0.

Combining (3.20), (3.21) and $e(\bar{\partial}\varphi)^{\ast}(\theta_{\nu}\alpha)=\theta_{\nu}e(\bar{\partial}\varphi)^{\ast}\alpha=0$ , we obtain that

(3.22)

\displaystyle(\bar{\partial}^{\ast}_{h}\alpha,\bar{\partial}^{\ast}_{h}(\theta_{\nu}\alpha))_{c,h}+(\bar{\partial}\alpha,\bar{\partial}(\theta_{\nu}\alpha))_{c,h}=0.

Taking its limit we know that

(3.23)

\displaystyle\|\bar{\partial}^{\ast}_{h}\alpha\|^{2}_{c,h}+\|\bar{\partial}\alpha\|^{2}_{c,h}=0

for every regular value $c$ of $\varphi$ . This implies that $\bar{\partial}\alpha=\bar{\partial}^{\ast}_{h}\alpha=0$ .

To prove (2) we set $h_{-\psi}=e^{\psi}h$ . By (3.9) we obtain that

(3.24)

\displaystyle\bar{\partial}e(\bar{\partial}\psi)^{\ast}(\theta_{\nu}\alpha)+e(\bar{\partial}\psi)^{\ast}\bar{\partial}(\theta_{\nu}\alpha)=\sqrt{-1}e(\partial\bar{\partial}\psi)\Lambda(\theta_{\nu}\alpha)

if $\alpha\in\mathscr{H}^{n,q}(Z,E,h,\varphi)$ . By (3.7) and (3.11), we get that

(3.25)	$\displaystyle(\bar{\partial}e(\bar{\partial}\psi)^{\ast}(\theta_{\nu}\alpha),\alpha)_{c,h_{-\psi}}$	$\displaystyle=(e(\bar{\partial}\psi)^{\ast}(\theta_{\nu}\alpha),\bar{\partial}^{\ast}_{h_{-\psi}}\alpha)_{c,h_{-\psi}}+[e(\bar{\partial}\psi)^{\ast}(\theta_{\nu}\alpha),e(\bar{\partial}\varphi)^{\ast}\alpha]_{c,h_{-\psi}}$
	$\displaystyle=(e(\bar{\partial}\psi)^{\ast}(\theta_{\nu}\alpha),(\bar{\partial}^{\ast}_{h}-e(\bar{\partial}\psi)^{\ast})\alpha)_{c,h_{-\psi}}$
	$\displaystyle=-(e(\bar{\partial}\psi)^{\ast}(\theta_{\nu}\alpha),e(\bar{\partial}\psi)^{\ast}\alpha)_{c,h_{-\psi}}.$

Taking $\nu\to\infty$ on (3.24), we know that

(3.26)

\displaystyle(\sqrt{-1}e(\partial\bar{\partial}\psi)\Lambda(\alpha),\alpha)_{c,h_{-\psi}}+\|e(\bar{\partial}\psi)^{\ast}\alpha\|^{2}_{c,h_{-\psi}}=0

for every regular value $c$ of $\varphi$ . Since both terms are semi-positive, we show that $e(\bar{\partial}\psi)^{\ast}\alpha=0$ . Hence $\alpha\in\mathscr{H}^{n,q}(Z,E,h,\psi)$ .

It remains to show (3). Since $\ast$ is a bounded operator, it suffices to show that $\bar{\partial}\ast\alpha=0$ for every $\alpha\in\mathscr{H}^{n,q}(Z,E,h,\varphi)$ . This follows from $-\ast\bar{\partial}\ast\alpha=D^{\prime\ast}_{h}\alpha=0$ which is proved in (1). ∎

By Proposition 3.2-(2), $\mathscr{H}^{n,q}(Z,E,h,\varphi)$ is independent of the choice of $\varphi\in\mathscr{P}_{e}(Z)$ . Hence we simply denote $\mathscr{H}^{n,q}(Z,E,h):=\mathscr{H}^{n,q}(Z,E,h,\varphi)$ for every $Z$ with $\mathscr{P}_{e}(Z)\neq\emptyset$ .

3.2. Harmonic Representation

Let us return to the relative setting. Let $f:X\rightarrow Y$ be a proper morphism from a complex space $X$ to an irreducible complex space $Y$ . Denote $n:={\rm dim}X$ and $m:={\rm dim}Y$ respectively. Assume that every irreducible component of $X$ is mapped onto $Y$ . Let $X^{o}\subset X_{\rm reg}$ be a dense Zariski open subset and $(E,h)$ a hermitian vector bundle on $X^{o}$ with a Nakano semi-positive curvature. Assume that there is a Kähler metric $ds^{2}$ on $X^{o}$ such that

(1)

$ds^{2}$ admits a $(\infty,1)$ bounded potential locally on $X$ ;
(2)

$ds^{2}$ is locally complete on $X$ and is locally bounded from below by a hermitian metric.

By Lemma 2.13, such kind of metric exists locally near every fiber of $f$ and globally on $X^{o}$ when $X$ is a compact Kähler space. By Theorem 2.11 and Lemma 2.2, there is a resolution by fine sheaves

(3.27)

\displaystyle S_{X}(E,h)\to\mathscr{D}^{n,\bullet}_{X,ds^{2}}(E,h).

Denote $X(T):=f^{-1}T$ for every subset $T\subset Y$ . Denote by $L$ the Lefschetz operator with respect to $ds^{2}$ and denote by $\Lambda$ the formal adjoint of $L$ .

Proposition 3.3.

Notations as above. Let $S\subset Y$ be a Stein open subset. Let $S^{o}\subset S_{\rm reg}$ and $X(S)^{o}\subset X(S)_{\rm reg}\cap X(S^{o})$ be dense Zariski open subsets so that $f:X(S)^{o}\to S^{o}$ is a submersion. Then the Hodge star operator

(3.28)

\displaystyle\ast:\mathscr{H}^{n,q}(X(S),E,h)\to\left\{\alpha\in\mathscr{D}^{n-q,0}_{X(S),ds^{2}}(E,h)(X(S))\bigg{|}\bar{\partial}\alpha=0,\alpha|_{X(S)^{o}}\in\Gamma(X(S)^{o},\Omega_{X^{o}}^{n-m-q}\otimes f^{\ast}\Omega^{m}_{S^{o}})\right\}

is well defined and injective for every $q$ . As a consequence,

(1)

$\mathscr{H}^{n,q}(X(S),E,h)=0$ for every $q>n-m$ ;
(2)

For every open subset $S^{\prime}\subset S$ such that $\mathscr{P}_{e}(S^{\prime})\neq\emptyset$ , the restriction map

(3.29) $\displaystyle\mathscr{H}^{n,q}(X(S),E,h)\to\mathscr{H}^{n,q}(X(S^{\prime}),E,h)$

is well defined.

Proof.

Let $\alpha\in\mathscr{H}^{n,q}(X(S),E,h)$ . By Proposition 3.2-(1), $D^{\prime\ast}_{h}\alpha=0$ , i.e. $\bar{\partial}\ast\alpha=0$ . It remains to show that $\ast\alpha|_{X(S)^{o}}\in\Gamma(X(S)^{o},\Omega_{X^{o}}^{n-m-q}\otimes f^{\ast}\Omega^{m}_{S^{o}})$ .

Fix a closed immersion $S\subset\mathbb{C}^{N}$ where $z_{1},\dots,z_{N}$ is the coordinate of $\mathbb{C}^{N}$ . Let $\varphi=\sum_{i=1}^{N}\|z_{i}\|^{2}\in\mathscr{P}_{e}(S)$ . By Proposition 3.2-(1) and (3.5), one has

(3.30)		$\displaystyle\sum_{i=1}^{N}\|e(\overline{\partial f^{\ast}(z_{i})})^{\ast}\alpha\|^{2}_{h}=$	$\displaystyle\sum_{i=1}^{N}\langle e(\overline{\partial f^{\ast}(z_{i})})^{\ast}\alpha,\sqrt{-1}e(\partial f^{\ast}(z_{i}))\Lambda\alpha\rangle\quad(\ref{align_theta})$
	$\displaystyle=$	$\displaystyle\sum_{i=1}^{N}\langle\sqrt{-1}e(\overline{\partial f^{\ast}(z_{i})})e(\partial f^{\ast}(z_{i}))\Lambda\alpha,\alpha\rangle$
	$\displaystyle=$	$\displaystyle\langle\sqrt{-1}e(\partial\bar{\partial}f^{\ast}\varphi)\Lambda\alpha,\alpha\rangle_{h}=0.$

Hence $f^{\ast}(dz_{i})\wedge\ast\alpha=0$ , $\forall i=1,\dots,N$ . This implies that $\alpha|_{X(S)^{o}}=0$ if $q>n-m$ and $\ast\alpha|_{X(V)\cap X(S)^{o}}$ can be divided by $f^{\ast}\theta$ for any open subset $V\subset S^{o}$ which admits a non-vanishing holomorphic $m$ -form $\theta$ . Therefore

\ast\alpha|_{X(S)^{o}}\in\Gamma(X(S)^{o},\Omega_{X^{o}}^{n-m-q}\otimes f^{\ast}\Omega^{m}_{S^{o}}).

This proves that (3.28) is well defined and $\mathscr{H}^{n,q}(X(S),E,h)=0$ for every $q>n-m$ . (3.28) is injective because

(3.31)

\displaystyle L^{q}\circ\ast=c(n,q){\rm Id},\quad c(n,q)=\sqrt{-1}^{(n-q)(n-q+3)}q!

holds on $(n,q)$ -forms ([Wells1980, Theorem 3.16]).

It remains to show that $\alpha|_{X(S^{\prime})}\in\mathscr{H}^{n,q}(X(S^{\prime}),E,h)$ , i.e. $e(\bar{\partial}f^{\ast}\psi)^{\ast}\alpha|_{X(S^{\prime})}=0$ for some $\psi\in\mathscr{P}_{e}(S^{\prime})$ . By (3.31), $\alpha=L^{q}\beta$ for $\beta=c(n,q)^{-1}\ast\alpha$ . Choose an arbitrary $\psi\in\mathscr{P}_{e}(S^{\prime})$ . Since $\beta|_{X(S)^{o}}\in\Gamma(X(S)^{o},\Omega_{X^{o}}^{n-m-q}\otimes f^{\ast}\Omega^{m}_{S^{o}})$ , $e(\partial(f^{\ast}\psi))L^{k}\beta|_{X(S^{\prime})\cap X(S)^{o}}=0$ for every $k=0,\dots,q-1$ . Because

(3.32)

\displaystyle\sqrt{-1}[e(\bar{\partial}(f^{\ast}\psi))^{\ast},L]=e(\partial(f^{\ast}\psi)),\textrm{\quad(\cite[cite]{[\@@bibref{}{Wells1980}{}{}, Chapter V, (3.22)]})},

we obtain that

(3.33)

\displaystyle e(\bar{\partial}(f^{\ast}\psi))^{\ast}L^{q}\beta|_{X(S^{\prime})\cap X(S)^{o}}=L^{q}e(\bar{\partial}(f^{\ast}\psi))^{\ast}\beta|_{X(S^{\prime})\cap X(S)^{o}}=0.

Thus $e(\bar{\partial}(f^{\ast}\psi))^{\ast}\alpha|_{X(S^{\prime})}=0$ by continuity. This proves the proposition. ∎

By Proposition 3.3, the restriction map

\mathscr{H}^{n,q}(X(V),E,h)\to\mathscr{H}^{n,q}(X(U),E,h)

is well defined for any pair of Stein open subsets $U\subset V\subset Y$ . Hence the data

\displaystyle U\mapsto\mathscr{H}^{n,q}(X(U),E,h),\quad U\subset Y\textrm{ is a Stein open subset}

determines a sheaf $\mathscr{H}^{n,q}_{f}(E,h)$ on $Y$ (after a sheafification).

By (3.27) and Lemma 2.2, there is a natural morphism

(3.34)

\displaystyle\mathscr{H}^{n,\bullet}_{f}(E,h)\to f_{\ast}(\mathscr{D}_{X,ds^{2}}^{n,\bullet}(E,h))\simeq Rf_{\ast}(S_{X}(E,h)).

This induces a canonical morphism

\mathscr{H}^{n,q}_{f}(E,h)\to R^{q}f_{\ast}S_{X}(E,h)

for every $0\leq q\leq n$ . The main result of this section is

Theorem 3.4.

$\mathscr{H}^{n,q}_{f}(E,h)$ is a sheaf of $\mathscr{O}_{Y}$ -modules for every $q\geq 0$ . Assume that $S_{X}(E,h)$ is a coherent sheaf, then the canonical morphism

\mathscr{H}^{n,q}_{f}(E,h)\to R^{q}f_{\ast}S_{X}(E,h)

is an isomorphism of $\mathscr{O}_{Y}$ -modules for every $q\geq 0$ . Moreover,

\mathscr{H}^{n,q}_{f}(E,h)(U)=\mathscr{H}^{n,q}(f^{-1}(U),E,h)

for every Stein open subset $U\subset Y$ .

Proof.

Let $S\subset Y$ be a Stein open subset and $\varphi\in\mathscr{P}_{e}(X(S))$ . For every $\alpha\in\mathscr{H}^{n,q}(X(S),E,h)$ and every $g\in\mathscr{O}_{Y}(S)$ , denote $g^{\prime}:=f^{\ast}g$ . Then Proposition 3.2-(1) implies that

D^{\prime\ast}_{h}(g^{\prime}\alpha)=-\ast\bar{\partial}\ast(g^{\prime}\alpha)=g^{\prime}D^{\prime\ast}_{h}\alpha=0

and

\langle\sqrt{-1}e(\Theta_{h}+\partial\bar{\partial}\varphi)\Lambda(g^{\prime}\alpha),g^{\prime}\alpha\rangle_{h}=\|g^{\prime}\|^{2}\langle\sqrt{-1}e(\Theta_{h}+\partial\bar{\partial}\varphi)\Lambda(\alpha),\alpha\rangle_{h}=0.

Hence $g^{\prime}\alpha\in\mathscr{H}^{n,q}(X(S),E,h,\varphi)$ by Proposition 3.2-(1). This shows that $\mathscr{H}^{n,q}_{f}(E,h)$ is a sheaf of $\mathscr{O}_{Y}$ -modules for every $0\leq q\leq n$ .

Since $S_{X}(E,h)$ is a coherent sheaf, to prove the remaining claims, it suffices to show that the natural morphism

(3.35)

\displaystyle\tau^{q}_{U}:\mathscr{H}^{n,q}(X(U),E,h)\to H^{q}\Gamma(X(U),\mathscr{D}^{n,\bullet}_{X,ds^{2}}(E,h))

is an isomorphism for every Stein open subset $U\subset Y$ and every $q\geq 0$ . Fix a $C^{\infty}$ exhausted strictly psh function $\varphi_{U}$ on $U$ . Denote $U_{c}:=\{\varphi_{U}<c\}$ and $\varphi:=f^{\ast}\varphi_{U}$ .

Claim 1: $\tau^{q}_{U}$ is injective. Assume that $\alpha\in\mathscr{H}^{n,q}(X(U),E,h)$ and $\alpha=\bar{\partial}\beta$ for some $\beta\in\mathscr{D}^{n,q-1}_{X,ds^{2}}(E,h)(X(U))$ . By (3.11), we obtain that

(\alpha,\alpha)_{c,h}=(\beta,\bar{\partial}^{\ast}_{h}\alpha)_{c,h}=0

for every regular value $c$ of $\varphi$ . Hence $\alpha=0$ .

Claim 2: $\tau^{q}_{U}$ is surjective. Let $\alpha\in\Gamma(X(U),\mathscr{D}^{n,q}_{X,ds^{2}}(E,h))$ be $\bar{\partial}$ -closed.

Step 1: In this step we show that for every $c\in\mathbb{R}$ , $\alpha|_{X(U_{c})}=u_{c}+\bar{\partial}\beta_{c}$ for some $u_{c}\in\mathscr{H}^{n,q}(X(U_{c}),E,h)$ and $\beta_{c}\in\Gamma(X(U_{c}),\mathscr{D}^{n,q-1}_{X,ds^{2}}(E,h))$ .

Fix $0<c_{0}<c_{1}$ . Denote $\omega_{ds^{2}}$ to be the Kähler form associated to $ds^{2}$ and let $\omega_{\lambda}:=\omega_{ds^{2}}+\sqrt{-1}\partial\bar{\partial}\lambda(\varphi-c)$ for some smooth convex function $\lambda:\mathbb{R}\to\mathbb{R}_{\geq 0}$ such that $\lambda(t)=0$ if $t\leq c_{0}$ , $\lambda(t)>0$ , $\lambda^{\prime}(t)>0$ , $\lambda^{\prime\prime}(t)>0$ if $t>c_{0}$ and $\int_{0}^{c_{1}}\sqrt{\lambda^{\prime\prime}(t)}dt=+\infty$ . Let $h_{\lambda}:=e^{-\lambda(\varphi)}h$ . Then $\omega_{\lambda}\geq\omega_{ds^{2}}$ is a complete Kähler metric on $X(U_{c+c_{1}})$ and $(E,h_{\lambda})$ is Nakano semi-positive. By choosing $\lambda$ sufficiently large we assume that $\alpha\in L^{n,q}_{(2)}(X(U_{c+c_{1}}),E;\omega_{\lambda},h_{\lambda})$ . Noting that $\bar{\partial}\alpha=0$ , there is a unique decomposition $\alpha=v_{c}+\gamma_{c}$ so that $v_{c}$ is a harmonic form on $U_{c+c_{1}}$ with respect to $\omega_{\lambda}$ and $h_{\lambda}$ while $\gamma_{c}$ lies in the closure of the range of $\bar{\partial}$ in the Hilbert space $L^{n,q}_{(2)}(X(U_{c+c_{1}}),E;\omega_{\lambda},h_{\lambda})$ . Since $\omega_{\lambda}|_{U_{c}}=\omega_{ds^{2}}|_{U_{c}}$ and $h_{\lambda}|_{U_{c}}=h|_{U_{c}}$ , by Proposition 3.1 and Proposition 3.5 below, we see that $v_{c}|_{U_{c}}\in\mathscr{H}^{n,q}(X(U_{c}),E,h)$ and $\gamma_{c}|_{U_{c}}=\bar{\partial}\beta_{c}$ for some $\beta_{c}\in\Gamma(X(U_{c}),\mathscr{D}^{n,q-1}_{X,ds^{2}}(E,h))$ .

Step 2: By step 1, there is a decomposition

\alpha|_{X(U_{k})}=u_{k}+\bar{\partial}\beta_{k}

where $u_{k}\in\mathscr{H}^{n,q}(X(U_{k}),E,h)$ and $\beta_{k}\in\Gamma(X(U_{k}),\mathscr{D}^{n,q-1}_{X,ds^{2}}(E,h))$ for every $k\in\mathbb{N}$ . Since $u_{k+1}$ and $u_{k}$ are cohomologous on $X(U_{k})$ , we have $u_{k+1}|_{X(U_{k})}=u_{k}$ by Claim 1. Hence there is a unique $u\in\mathscr{H}^{n,q}(X(U),E,h)$ such that $u|_{X(U_{k})}=u_{k}$ . Hence the cohomology classes $[\alpha]$ and $[u]$ are equal over every $X(U_{k})$ , $k\in\mathbb{N}$ . Since $U_{k}$ is a Stein space for each $k\in\mathbb{N}\cup\{+\infty\}$ and $S_{X}(E,h)$ is a coherent sheaf, there is a canonical isomorphism

\Gamma(X(U_{k}),\mathscr{D}^{n,q}_{X,ds^{2}}(E,h))\simeq H^{q}(X(U_{k}),S_{X}(E,h))\simeq\Gamma(U_{k},R^{q}f_{\ast}(S_{X}(E,h)))

for every $k\in\mathbb{N}\cup\{+\infty\}$ . By regarding $[\alpha]$ and $[u]$ as sections of $R^{q}f_{\ast}(S_{X}(E,h))$ (which are equal over every $U_{k}$ , $k\in\mathbb{N}$ ) we obtain that $[\alpha]=[u]$ . This proves the surjectivity of $\tau^{q}_{U}$ . ∎

To complete the proof of Theorem 3.4, we go on proving the following proposition.

Proposition 3.5.

Assume that $S_{X}(E,h)$ is a coherent sheaf on $X$ . Let $V\subset X$ be an open subset and denote $V^{o}:=V\cap X^{o}$ . Let

\bar{\partial}:L^{n,q-1}_{(2)}(V^{o},E;ds^{2},h)\to L^{n,q}_{(2)}(V^{o},E;ds^{2},h)

be the unbounded operator in the sense of distribution where $q\geq 1$ . Suppose that $\alpha\in\overline{{\rm Im}\bar{\partial}}$ . Then there is an $(n,q-1)$ -form $\beta\in L^{n,q-1}_{V,ds^{2}}(E,h)(V)$ such that $\alpha=\bar{\partial}\beta$ .

Proof.

To prove the proposition, we need the following lemma which is a modified version of Theorem 2.10 and we leave its proof to the end of this section.

Lemma 3.6.

Let $Y$ be a complex manifold of dimension $n$ which admits a complete Kähler metric. Let $\omega=\sqrt{-1}\partial\bar{\partial}\varphi$ be a Kähler metric of $Y$ with $\sup|\varphi|<\infty$ . Let $(E,h)$ be a Nakano semi-positive hermitian vector bundle on $Y$ . Then for every $q>0$ and every $\alpha\in L^{n,q}_{(2)}(Y,E;\omega,h)$ such that $\bar{\partial}\alpha=0$ , there is $\beta\in L^{n,q-1}_{(2)}(Y,E;\omega,h)$ such that $\bar{\partial}\beta=\alpha$ and $\|\beta\|^{2}_{\omega,h}\leq C\|\alpha\|^{2}_{\omega,h}$ . Here $C$ is a constant depending on $q$ and $\varphi$ , but not depending on $\alpha$ .

Proof.

Let $h^{\prime}:=he^{-\varphi}$ , then $h^{\prime}\sim h$ . Since $(E,h)$ is Nakano semi-positive, we know that

\sqrt{-1}\Theta_{h^{\prime}}(E)=\sqrt{-1}\partial\bar{\partial}\varphi\otimes{\rm Id}_{E}+\sqrt{-1}\Theta_{h}(E)\geq\omega\otimes{\rm Id}_{E}.

By Theorem 2.10, there exists $\beta\in L_{(2)}^{n,q-1}(Y,E;\omega,he^{-\varphi})$ such that $\bar{\partial}\beta=\alpha$ and $\|\beta\|_{\omega,he^{-\varphi}}^{2}\leq\frac{1}{q}\|\alpha\|_{\omega,h^{-\varphi}}$ . Since $\varphi$ is a bounded function, we have $\|\beta\|^{2}_{\omega,h}\leq C\|\alpha\|^{2}_{\omega,h}$ where $C$ is a constant depending on $q$ and $\varphi$ and not depending on $\alpha$ . The lemma is proved. ∎

Let us return to the proof of the proposition. First we take a locally finite Stein open cover $\{V_{j}\}$ of $V$ . Denote $V_{j_{0},j_{1},\dots,j_{p}}:=V_{j_{1}}\cap\cdots\cap V_{j_{p}}$ . By Lemma 2.13, after a possible refinement we assume that there is a complete Kähler metric on $V_{j_{0},j_{1},\dots,j_{p}}^{o}:=V_{j_{0},j_{1},\dots,j_{p}}\cap X^{o}$ which has a bounded potential for every $j_{0},j_{1},\dots,j_{p}$ . Since $\alpha\in\overline{{\rm Im}\bar{\partial}}\subset L^{n,q}_{(2)}(V^{o},E;ds^{2},h)$ and $\rm Ker\bar{\partial}$ is closed in $L^{n,q}_{(2)}(V^{o},E;ds^{2},h)$ , we know that $\bar{\partial}\alpha=0$ .

Since $S_{X}(E,h)$ is a coherent sheaf on $X$ , by Theorem 2.11 and Lemma 2.2 there are isomorphisms of cohomologies

(3.36)

\displaystyle H^{k}(\{V_{j}\},S_{X}(E,h))\simeq H^{k}(V,S_{X}(E,h))\simeq H^{k}(\Gamma(V,\mathscr{D}^{\bullet}_{X,E;ds^{2},h})),

where $H^{k}(\{V_{j}\},S_{X}(E,h))$ is the Čech cohomology with respect to the covering $\{V_{j}\}$ .

Let us recall the corresponding Čech cocycle of $\alpha$ . Let $\alpha_{j}=\alpha|_{V_{j}^{o}}$ . By Lemma 3.6, there exists $b_{j}\in L_{(2)}^{n,q-1}(V_{j}^{o},E;ds^{2},h)$ such that $\alpha_{j}=\bar{\partial}b_{j}$ on $V_{j}^{o}$ for each $j$ .

Suppose that $\{\alpha_{j_{0},\dots,j_{r}}\}$ and $\{b_{j_{0},\dots,j_{r}}\}$ are determined in the way as:

(3.37)			$\displaystyle b_{j_{0},\dots,j_{r}}\in L_{(2)}^{n,q-r-1}(V_{j_{0},j_{1},\dots,j_{p}}^{o},E;ds^{2},h),\quad\alpha_{j_{0},\dots,j_{r}}\in L_{(2)}^{n,q-r}(V_{j_{0},j_{1},\dots,j_{p}}^{o},E;ds^{2},h),$
		$\displaystyle\alpha_{j_{0},\dots,j_{r}}=\bar{\partial}b_{j_{0},\dots,j_{r}}\quad{\rm on}\quad V_{j_{0},\dots,j_{r}}^{o}\quad{\rm and}\quad(\delta\alpha)_{j_{0},\dots,j_{r+1}}=0.$

Set

\alpha_{j_{0},\dots,j_{r+1}}:=(\delta b)_{j_{0},\dots,j_{r+1}},

which is a $\bar{\partial}$ -closed form. It follows from Lemma 3.6 that the same statements in (3.37) also hold for $\alpha_{j_{0},\dots,j_{r+1}}$ . Repeating the above steps, we can obtain a $q$ -cocycle $\{\alpha_{j_{0},\dots,j_{q}}\}\in Z^{q}(\{V_{j}\},S_{X}(E,h))$ which corresponds to $\alpha$ by (3.36).

By the hypothesis there exits a sequence $\{\gamma_{m}\}\in L_{(2)}^{n,q-1}(V^{o},E;ds^{2},h)$ such that $\|\alpha-\bar{\partial}\gamma_{m}\|\rightarrow 0$ as $m\rightarrow\infty$ . Let $\nu_{m}:=\alpha-\bar{\partial}\gamma_{m}$ . Since $\bar{\partial}\nu_{m}=0$ , by Lemma 3.6 there exists $\{\mu_{j,m}\}\in L_{(2)}^{n,q-1}(V_{j}^{o},E;ds^{2},h)$ such that $\nu_{m}=\bar{\partial}\mu_{j,m}$ and $\|\mu_{j,m}\|\leq C_{j}\|\nu_{m}\|$ for $C_{j}$ independent of $m$ but depending on $j$ . Set $\rho_{j,m}:=b_{j}-\mu_{j,m}-\gamma_{m}|_{V_{j}^{o}}$ . Then we have $\bar{\partial}\rho_{j,m}=0$ , $\bar{\partial}(\alpha_{ij}-\delta\{\rho_{j,m}\})=0$ and $\|\alpha_{ij}-\delta\{\rho_{j,m}\}\|\leq C_{ij}\|\nu_{m}\|$ .

Suppose that $\{\rho_{j_{0},\dots,j_{r},m}\}$ are already determined as:

(3.38)			$\displaystyle\rho_{j_{0},\dots,j_{r},m}\in L_{(2)}^{n,q-r-1}(V_{j_{0},\dots,j_{r}}^{o},E;ds^{2},h),\bar{\partial}(\alpha_{j_{0},\dots,j_{r+1}}-(\delta\rho)_{j_{0},\dots.,j_{r+1},m})=0,$
		$\displaystyle\bar{\partial}\rho_{j_{0},\dots,j_{r},m}=0,\\|\alpha_{j_{0},\dots,j_{r+1}}-(\delta\rho)_{j_{0},\dots.,j_{r+1},m}\\|\leq C_{j_{0},\dots,j_{r+1}}\\|\nu_{m}\\|.$

We construct $\{\rho_{j_{0},\dots,j_{r+1},m}\}$ as follows. (3.38) and Lemma 3.6 imply that there exists $\gamma_{j_{0},\dots,j_{r},m}\in L_{(2)}^{n,q-r-2}(V_{j_{0},\dots,j_{r}}^{o},E;ds^{2},h)$ and $\mu_{j_{0},\dots,j_{r+1},m}\in L_{(2)}^{n,q-r-2}(V_{j_{0},\dots,j_{r+1}}^{o},E;ds^{2},h)$ such that $\rho_{j_{0},\dots,j_{r},m}=\bar{\partial}\gamma_{j_{0},\dots,j_{r},m}$ , $\alpha_{j_{0},\dots,j_{r+1}}-(\delta\rho_{m})_{j_{0},\dots,j_{r+1}}=\bar{\partial}\mu_{j_{0},\dots,j_{r+1},m}$ and $\|\mu_{j_{0},\dots,j_{r+1},m}\|\leq C_{j_{0},\dots,j_{r+1}}\|\nu_{m}\|$ . Then we set

\rho_{j_{0},\dots,j_{r+1},m}=b_{j_{0},\dots,j_{r+1}}-\mu_{j_{0},\dots,j_{r+1},m}-(\delta\gamma)_{j_{0},\dots,j_{r+1},m},

which satisfies the statements in (3.38). Repeating the above steps, we obtain a $q-1$ cochain $\rho_{m}=\{\rho_{j_{0},\dots,j_{q-1},m}\}\in C^{q-1}(\{V_{j}\},S_{X}(E,h))$ such that $\|\alpha-\delta\rho_{m}\|\rightarrow 0$ as $m\rightarrow\infty$ . By [Hormander1990, Theorem 2.2.3], $\delta\rho_{m}$ tends to $\alpha$ as $m\rightarrow\infty$ uniformly on compact subsets of $X$ . Since $S_{X}(E,h)$ is coherent, $\delta C^{q-1}(\{V_{j}\},S_{X}(E,h))$ is a closed subspace of $C^{q}(\{V_{j}\},S_{X}(E,h))$ with respect to the topology of uniform convergence on compact subsets. Hence there exists $\rho\in C^{q-1}(\{V_{j}\},S_{X}(E,h))$ such that $\delta\rho=\alpha$ . By (3.36), $[\alpha]=0\in H^{q}(\Gamma(V,\mathscr{D}^{\bullet}_{X,E;ds^{2},h}))$ . Thus we prove the proposition. ∎

4. An abstract Kollár package

Definition 4.1.

[Takegoshi1995, Definition 6.1] A morphism $f:X\rightarrow Y$ between complex spaces is locally Kähler if $f^{-1}U$ is a Kähler space for any relatively compact open subset $U\subset Y$ .

Throughout this section, $f:X\rightarrow Y$ is a proper locally Kähler morphism from a complex space $X$ to an irreducible complex space $Y$ . Assume that every irreducible component of $X$ is mapped onto $Y$ . $X^{o}\subset X_{\rm reg}$ is a dense Zariski open subset and $(E,h)$ is a hermitian vector bundle on $X^{o}$ with a Nakano semi-positive curvature. In this section we establish the Kollár package of $S_{X}(E,h)$ under the coherence assumptions.

Theorem 4.2 (Torsion Freeness).

Assume that $S_{X}(E,h)$ is a coherent sheaf on $X$ , then $R^{q}f_{\ast}S_{X}(E,h)$ is torsion free for every $q\geq 0$ and vanishes if $q>\dim X-\dim Y$ .

Proof.

Since the problem is local, we assume that $Y$ is Stein and there is a Kähler metric on $X^{o}$ which is locally complete and locally bounded from below by a hermitian metric and which admits a $(\infty,1)$ potential locally on $X$ (Lemma 2.13). Define the sheaf $\Omega^{p}_{X,ds^{2},(2)}(E,h)$ as

\Omega^{p}_{X,ds^{2},(2)}(E,h)(U)=\left\{\alpha\in\mathscr{D}^{p,0}_{X,ds^{2}}(E,h)(U)\bigg{|}\bar{\partial}\alpha=0\right\}

for every open subset $U\subset X$ .

By Proposition 3.2-(3), the Hodge star operator induces a well defined map

\ast:\mathscr{H}^{n,q}_{f}(E,h)\to f_{\ast}\Omega^{n-q}_{X,ds^{2},(2)}(E,h).

Since the Lefschetz operator $L$ with respect to $ds^{2}$ is bounded and $[L,\bar{\partial}]=0$ , taking the finesss of $\mathscr{D}^{n,\bullet}_{X,ds^{2}}(E,h)$ (Lemma 2.2) into account we get a well defined map

L^{q}:f_{\ast}\Omega^{n-q}_{X,ds^{2},(2)}(E,h)\to R^{q}f_{\ast}(\mathscr{D}^{n,\bullet}_{X,ds^{2}}(E,h)).

By Theorem 2.11 and Theorem 3.4, we get the morphisms

R^{q}f_{\ast}S_{X}(E,h)\stackrel{{\scriptstyle\ast}}{{\to}}f_{\ast}\Omega^{n-q}_{X,ds^{2},(2)}(E,h)\stackrel{{\scriptstyle L^{q}}}{{\to}}R^{q}f_{\ast}S_{X}(E,h)

such that

(4.1)

\displaystyle L^{q}\circ\ast=c(n,q){\rm Id},\quad c(n,q)=\sqrt{-1}^{(n-q)(n-q+3)}q!\quad(\ref{align_Last=1}).

This proves the first claim since $f_{\ast}\Omega^{n-q}_{X,ds^{2},(2)}(E,h)$ is torsion free for every $q\geq 0$ . The vanishing result follows from Proposition 3.3-(1). ∎

Theorem 4.3 (Injectivity Theorem).

Assume that $S_{X}(E,h)$ is a coherent sheaf on $X$ . If $L$ is a semi-positive holomorphic line bundle on $X$ so that $L^{l}$ admits a nonzero holomorphic global section $s$ for some $l>0$ , then the canonical morphism

R^{q}f_{\ast}(\otimes s):R^{q}f_{\ast}(S_{X}(E,h)\otimes L^{\otimes k})\to R^{q}f_{\ast}(S_{X}(E,h)\otimes L^{\otimes k+l})

is injective for every $q\geq 0$ and every $k\geq 1$ .

Proof.

Since the problem is local, we assume that $Y$ is Stein and there is a Kähler metric on $X^{o}$ which is locally complete and locally bounded from below by a hermitian metric and which admits a $(\infty,1)$ potential locally on $X$ (Lemma 2.13). Let $h_{L}$ be the hermitian metric on $L$ with a semi-positive curvature, then we have

S_{X}(E,h)\otimes L^{\otimes k}\simeq S_{X}(E\otimes L^{\otimes k},h\otimes h_{L}^{k}),\quad k\geq 1

by Lemma 2.7. By Theorem 3.4, it is therefore sufficient to show that the canonical map

(4.2)

\displaystyle\otimes s:\mathscr{H}^{n,q}(X(U),E\otimes L^{\otimes k},h\otimes h_{L}^{k})\to\mathscr{H}^{n,q}(X(U),E\otimes L^{\otimes k+l},h\otimes h_{L}^{k+l})

is well defined for every Stein open subset $U$ of $Y$ . Let $\alpha\in\mathscr{H}^{n,q}(X,E\otimes L^{\otimes k},h\otimes h_{L}^{k},\varphi)$ for some $\varphi\in\mathscr{P}_{e}(X)$ ( $\mathscr{P}_{e}(X)\neq\emptyset$ since $Y$ is Stein and $f$ is proper). It follows from Proposition 3.2-(1) that

D^{\prime\ast}_{h}(\alpha\otimes s)=-\ast\bar{\partial}\ast(\alpha\otimes s)=D^{\prime\ast}_{h}\alpha\otimes s=0

and

	$\displaystyle 0$	$\displaystyle\leq\langle\sqrt{-1}e(\Theta_{h\otimes h_{L}^{k+l}}(E\otimes L^{k+l})+\partial\bar{\partial}\varphi)\Lambda(\alpha\otimes s),\alpha\otimes s\rangle_{h\otimes h_{L}^{k+l}}$
		$\displaystyle\leq\frac{k+l}{k}\|s\|_{h_{L}^{l}}^{2}\langle\sqrt{-1}e(\Theta_{h\otimes h_{L}^{k}}(E\otimes L^{k})+\partial\bar{\partial}\varphi)\Lambda(\alpha),\alpha\rangle_{h\otimes h_{L}^{k}}=0.$

Hence $\alpha\otimes s\in\mathscr{H}^{n,q}(X,E\otimes L^{\otimes k+l},h\otimes h_{L}^{k+l},\varphi)$ by Proposition 3.2-(1). Hence $\otimes s$ is well defined and injective. The proof is finished. ∎

Theorem 4.4 (Decomposition).

Assume that there exists a Kähler metric $ds^{2}$ on $X^{o}$ such that

(1)

$ds^{2}$ admits a $(\infty,1)$ bounded potential locally on $X$ ;
(2)

$ds^{2}$ is locally complete on $X$ and is locally bounded from below by a hermitian metric.

Assume that $S_{X}(E,h)$ is a coherent sheaf on $X$ . Then $Rf_{\ast}S_{X}(E,h)$ splits in $D(Y)$ , i.e.

Rf_{\ast}S_{X}(E,h)\simeq\bigoplus_{q}R^{q}f_{\ast}S_{X}(E,h)[-q]\in D(Y).

As a consequence, the spectral sequence

(4.3)

\displaystyle E^{pq}_{2}:=H^{p}(Y,R^{q}f_{\ast}S_{X}(E,h))\Rightarrow H^{p+q}(X,S_{X}(E,h))

degenerates at the $E_{2}$ page if $Y$ is compact.

Remark 4.5.

By Lemma 2.13, such kind of metric exists if $X$ is compact.

Proof.

Let

(4.4)

\displaystyle\alpha:\bigoplus_{q}\mathscr{H}^{n,q}_{f}(E,h)[-q]\to f_{\ast}\mathscr{D}^{\bullet}_{X,E;ds^{2},h}\simeq Rf_{\ast}\mathscr{D}^{\bullet}_{X,E;ds^{2},h}

be the inclusion map. By Theorem 3.4, $\alpha$ is a quasi-isomorphism under the hypothesis that $S_{X}(E,h)$ is coherent. Hence by Theorem 2.11 and Theorem 3.4 we obtain that

Rf_{\ast}S_{X}(E,h)\simeq Rf_{\ast}\mathscr{D}^{\bullet}_{X,E;ds^{2},h}\simeq\bigoplus_{q}\mathscr{H}^{n,q}_{f}(E,h)[-q]\simeq\bigoplus_{q}R^{q}f_{\ast}S_{X}(E,h)[-q]

in $D(Y)$ . The degeneration of the spectral sequence follows from standard arguments. ∎

Theorem 4.6 (Vanishing Theorem).

Assume that $S_{X}(E,h)$ is a coherent sheaf on $X$ . If $Y$ is a projective algebraic variety and $L$ is an ample line bundle on $Y$ , then

H^{q}(Y,R^{p}f_{\ast}S_{X}(E,h)\otimes L)=0,\quad\forall q>0,\forall p\geq 0.

Proof.

Since $S_{X}(E,h)$ is coherent, so is $R^{p}f_{\ast}S_{X}(E,h)$ . Then there is $k$ large enough so that $H^{0}(Y,L^{\otimes k})\neq 0$ and

(4.5)

\displaystyle H^{q}(Y,R^{p}f_{\ast}S_{X}(E,h)\otimes L^{k+1})=0,\quad\forall q>0,\forall p\geq 0.

By Lemma 2.7, we acquire that

(4.6)

\displaystyle S_{X}(E,h)\otimes f^{\ast}L^{\otimes l}\simeq S_{X}(E\otimes L^{\otimes l},h\otimes f^{\ast}h_{L}^{l}),\quad\forall l>0

where $h_{L}$ is a hermitian metric on $L$ with positive curvature. By Theorem 4.3, the canonical map

\otimes f^{\ast}s:H^{q}(X,S_{X}(E\otimes L,h\otimes f^{\ast}h_{L}))\to H^{q}(X,S_{X}(E\otimes L^{\otimes k+1},h\otimes f^{\ast}h_{L}^{k+1})),\quad\forall q\geq 0

is injective. By Theorem 4.4, we know that the canonical map

\otimes s:H^{q}(Y,R^{p}f_{\ast}S_{X}(E\otimes L,h\otimes f^{\ast}h_{L}))\to H^{q}(Y,R^{p}f_{\ast}S_{X}(E\otimes L^{\otimes k+1},h\otimes f^{\ast}h_{L}^{k+1})),\quad\forall p,q\geq 0

is injective. Combining this with (4.5) and (4.6), we prove the theorem. ∎

5. Non-abelian Hodge theory and Kollár package

5.1. Harmonic bundle and variation of Hodge structure

The notion of harmonic bundle is used by Simpson [Simpson1992] to establish a correspondence between local systems and semistable higgs bundles with vanishing Chern classes over a compact Kähler manifold. A typical example of harmonic bundles comes from a polarized variation of Hodge structure (loc. cit.). A harmonic bundle produces a $\lambda$ -connection structure which gives a $\mathscr{D}$ -module on $\lambda=1$ and a higgs bundle on $\lambda=0$ . This is the main subject of non-abelian Hodge theory. We only review the necessary knowledge of this topic that is used in the present paper. Readers may consult [Simpson1988, Simpson1990, Simpson1992, Sabbah2005, Mochizuki20072, Mochizuki20071] for more details.

Let $(M,ds^{2})$ be a Kähler manifold and $(\mathcal{V},\nabla)$ a holomorphic vector bundle with a flat connection on $M$ . Let $h$ be a hermitian metric on $\mathcal{V}$ which is not necessarily compatible with $\nabla$ .

Let $\nabla=\nabla^{1,0}+\nabla^{0,1}$ be the bi-degree decomposition. There are unique operators $\delta^{\prime}_{h}$ and $\delta^{\prime\prime}_{h}$ so that $\nabla^{1,0}+\delta^{\prime\prime}_{h}$ and $\nabla^{0,1}+\delta^{\prime}_{h}$ are connections compatible with $h$ . Denote $\nabla^{c}_{h}=\delta^{\prime\prime}_{h}-\delta^{\prime}_{h}$ and $\Theta_{h}(\nabla)=\nabla\nabla^{c}_{h}+\nabla^{c}_{h}\nabla$ .

Definition 5.1.

$(\mathcal{V},\nabla,h)$ is called a harmonic bundle if $\Theta_{h}(\nabla)=0$ . In this case, $h$ is called a harmonic metric.

Let $(\mathcal{V},\nabla,h)$ be a harmonic bundle. Denote $\theta=\frac{1}{2}(\nabla^{1,0}-\delta^{\prime}_{h})$ and $\bar{\partial}=\frac{1}{2}(\nabla^{0,1}+\delta^{\prime\prime}_{h})$ . Then $\bar{\partial}^{2}=0$ . Denote by $\widetilde{\mathcal{V}}$ the underlying complex $C^{\infty}$ -vector bundle of $\mathcal{V}$ , then $H=(\widetilde{\mathcal{V}},\bar{\partial})$ is a holomorphic vector bundle and $\theta$ is a higgs field on $H$ (i.e. $\theta$ is $\mathscr{O}_{M}$ -linear and $\theta^{2}=0$ ).

Let $\nabla_{h}$ be the Chern connection on $H$ with respect to $h$ and let $\Theta_{h}(H)=\nabla_{h}^{2}$ be its curvature form. Then we have the self-dual equation

(5.1)

\displaystyle\Theta_{h}(H)+\theta\wedge\overline{\theta}+\overline{\theta}\wedge\theta=0

where $\overline{\theta}=\frac{1}{2}(\nabla^{0,1}-\delta^{\prime\prime}_{h})$ is the adjoint of $\theta$ with respect to the metric $h$ .

Conversely, let $(H,\theta,h)$ be a hermitian higgs bundle . Let $\overline{\theta}$ be the adjoint of $\theta$ and let $\partial$ be the unique $(1,0)$ -connection such that $\partial+\bar{\partial}$ is compatible with $h$ . $h$ is called harmonic if $\Theta_{h}(\theta):=(\partial+\bar{\partial}+\theta+\overline{\theta})^{2}=0$ . There is a $1:1$ correspondence (c.f. [Simpson1988, Simpson1992])

(5.2)

\displaystyle\left\{(\mathcal{V},\nabla,h):\Theta_{h}(\nabla)=0\right\}\stackrel{{\scriptstyle 1:1}}{{\leftrightarrow}}\left\{(H,\theta,h):\Theta_{h}(\theta)=0\right\}.

Therefore by a harmonic bundle we mean an object on either side of (5.2).

For the purpose of the present paper, we are interested in tame harmonic bundles in the sense of Simpson [Simpson1990] and Mochizuki [Mochizuki20072, Mochizuki20071].

Definition 5.2.

Let $X$ be a complex space and $X^{o}\subset X_{\rm reg}$ a dense Zariski open subset. A harmonic bundle $(\mathcal{V},\nabla,h)$ on $X^{o}$ is called a tame harmonic bundle on $(X,X^{o})$ if, locally at every point $x\in X$ , there exists some $c>0$ such that

(5.3)

\displaystyle(\sum_{i=1}^{r}\|f_{i}\|^{2})^{c}\lesssim|v|_{h}\lesssim(\sum_{i=1}^{r}\|f_{i}\|^{2})^{-c}

holds for every (multivalued) flat section $v$ . Here $\{f_{1},\dots,f_{r}\}$ is a local generator of the ideal sheaf defining $X\backslash X^{o}\subset X$ and $c>0$ is a constant independent of $v$ .

The tameness is independent of the choice of the local generator $\{f_{1},\dots,f_{r}\}$ . Let $\pi:\widetilde{X}\to X$ be a desingularization of $X\backslash X^{o}\subset X$ so that the preimage of $X\backslash X^{o}$ is supported on a simple normal crossing divisor $E$ . Then (5.3) is equivalent to the estimate

(5.4)

\displaystyle\|z_{1}\cdots z_{r}\|^{c^{\prime}}\lesssim|v|_{h}\lesssim\|z_{1}\cdots z_{r}\|^{-c^{\prime}},\quad\textrm{ for some }c^{\prime}>0

where $(z_{1},\dots,z_{n})$ is an arbitrary holomorphic local coordinate of $\widetilde{X}$ such that $E=\{z_{1}\cdots z_{r}=0\}$ .

Remark 5.3.

There is an equivalent definition of tameness, using the corresponding higgs bundle. By desingularization, we assume that $X$ is smooth and $X\backslash X^{o}\subset X$ is a normal crossing divisor. Then the higgs field $\theta$ of the higgs bundle $(H,\theta)$ associated to $(\mathcal{V},\nabla,h)$ can be described as:

(5.5)

\displaystyle\theta=\sum_{i=1}^{r}f_{i}\frac{dz_{i}}{z_{i}}+\sum_{j=r+1}^{n}g_{j}dz_{j}

under a holomorphic coordinate $U\subset X$ such that $U\cap(X\backslash X^{o})=\{z_{1}\cdots z_{r}=0\}$ . The harmonic bundle $(\mathcal{V},\nabla,h)$ is tame if and only if the coefficients of the characteristic polynomials $\det(t-f_{i})$ , $i=1,\dots,r$ and $\det(t-g_{j})$ , $j=r+1,\dots,n$ can be extended to the holomorphic functions on $U$ . Readers may see [Mochizuki2002] for more details.

A typical type of tame harmonic bundles is the variation of Hodge structure.

Definition 5.4.

[Simpson1988, §8] Let $X$ be a complex space and $X^{o}\subset X_{\rm reg}$ a dense Zariski open subset. Denote by $\mathscr{A}^{0}_{X^{o}}$ the sheaf of $C^{\infty}$ functions on $X^{o}$ . A polarized complex variation of Hodge structure on $X^{o}$ of weight $k$ is a harmonic bundle $(\mathcal{V},\nabla,h)$ on $X^{o}$ together with a decomposition $\mathcal{V}\otimes_{\mathscr{O}_{X}}\mathscr{A}^{0}_{X^{o}}=\bigoplus_{p+q=k}\mathcal{V}^{p,q}$ of $C^{\infty}$ -bundles such that

(1)

The Griffiths transversality condition

(5.6)

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

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

(2)

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

Denote $S(\mathbb{V}):=\mathcal{F}^{p_{\rm max}}$ where $p_{\rm max}=\max\{p|\mathcal{F}^{p}\neq 0\}$ .

The following proposition is known to experts. We recall the proof in sketch for the convenience of readers.

Proposition 5.5.

If a harmonic bundle $(\mathcal{V},\nabla,h)$ admits a complex variation of Hodge structure, then $(\mathcal{V},\nabla,h)$ is tame.

Proof.

Let $\mathbb{V}:=(\mathcal{V},\nabla,\{\mathcal{V}^{p,q}\},h_{\mathbb{V}})$ be a complex variation of Hodge structure of weight $k$ . Take the decomposition

\nabla=\overline{\theta}+\partial+\bar{\partial}+\theta

according to (5.6). By [Simpson1988, §8], the corresponding higgs bundle of $(\mathcal{V},\nabla,h_{\mathbb{V}})$ is $(H={\rm Ker}\bar{\partial},\theta,h_{\mathbb{V}})$ . There is moreover an orthogonal decomposition of holomorphic subbundles $H=\oplus_{p+q=k}H^{p,q}$ where $H^{p,q}=H\cap\mathcal{V}^{p,q}$ and

\theta(H^{p,q})\subset H^{p-1,q+1}\otimes\Omega_{X^{o}}.

Thus the higgs field is nilpotent. Therefore $f_{i}$ , $i=1,\dots,r$ and $g_{j}$ , $j=r+1,\dots,n$ in the local expression (5.5) are nilpotent matrix. Hence $\det(t-f_{i})=t^{n}$ and $\det(t-g_{j})=t^{n}$ . So the harmonic bundle is tame due to the alternative definition of tameness (Remark 5.3). ∎

5.2. Harmonic bundle and Kollár package

Proposition 5.6.

Let $X$ be a complex space and $X^{o}\subset X_{\rm reg}$ a dense Zariski open subset. Let $(H,\theta,h)$ be a tame harmonic bundle on $X^{o}$ . Let $E\subset H$ be a holomorphic subbundle with vanishing second fundamental form. Assume that $\overline{\theta}(E)=0$ . Then $(E,h)$ is a tame hermitian vector bundle with a Nakano semi-positive curvature. As a consequence, $S_{X}(E,h)$ is a coherent sheaf on $X$ .

Proof.

To prove the tameness (Definition 2.8), we construct $Q$ by Mochizuki’s prolongation construction. Since the problem is local, we assume that there is a desingularization $\pi:\widetilde{X}\to X$ such that $\pi$ is biholomorphic over $X^{o}$ and $D:=\pi^{-1}(X\backslash X^{o})$ is a simple normal crossing divisor. By abuse of notations we identify $X^{o}$ and $\pi^{-1}X^{o}$ . Since $(H,\theta,h)$ is tame, by [Mochizuki20072, Theroem 8.58] there is a logarithmic higgs bundle $(\widetilde{H},\widetilde{\theta})$ :

\displaystyle\widetilde{\theta}:\widetilde{H}\to\Omega_{\widetilde{X}}(\log D)\otimes\widetilde{H},

such that $(\widetilde{H},\widetilde{\theta})|_{X^{o}}$ is holomorphically equivalent to $(H,\theta)$ . Let $D=\cup_{i=1}^{k}D_{i}$ be the irreducible decomposition and let $(z_{1},\dots,z_{n})$ be a local coordinate such that $D_{i}=\{z_{i}=0\}$ , $i=1,\dots,k$ . By [Mochizuki20072, Part 3, Chapter 13], the tameness of $(H,\theta,h)$ forces a norm estimate

(5.7)

\displaystyle|z_{1}\cdots z_{k}|^{2b}\lesssim|s|_{h}

for any local holomorphic section $s$ of $\widetilde{H}$ and a constant $b>0$ which is independent of $s$ .

Let $Q:=\widetilde{H}$ , then $E$ is a holomorphic subbundle of $Q|_{X^{o}}$ . By (5.7), we see that $(E,h)$ is tame.

To see that $(E,h)$ is Nakano semi-positive, we take the decomposition

\nabla=\overline{\theta}+\partial+\bar{\partial}+\theta

according to (5.6). Since $E\oplus E^{\bot}$ is an orthogonal holomorphic direct sum of $H$ and $\overline{\theta}(E)=0$ , we get from (5.1) that

\sqrt{-1}\Theta_{h}(E)=\sqrt{-1}\theta\wedge\overline{\theta}\geq 0.

Hence $(E,h)$ is Nakano semi-positive. By Proposition 2.9, $S_{X}(E,h)$ is a coherent sheaf on $X$ . ∎

Let $\mathbb{V}:=(\mathcal{V},\nabla,\{\mathcal{V}^{p,q}\},h_{\mathbb{V}})$ be a complex variation of Hodge structure of weight $k$ . Let

\nabla=\overline{\theta}+\partial+\bar{\partial}+\theta

be the decomposition according to (5.6). The corresponding higgs bundle of $(\mathcal{V},\nabla,h_{\mathbb{V}})$ is $(H={\rm Ker}\bar{\partial},\theta,h_{\mathbb{V}})$ by [Simpson1988, §8]. There is moreover an orthogonal decomposition of holomorphic subbundles $H=\oplus_{p+q=k}H^{p,q}$ where $H^{p,q}=H\cap\mathcal{V}^{p,q}$ and

\theta(H^{p,q})\subset H^{p-1,q+1}\otimes\Omega_{X^{o}}.

For the reason of degrees, we have $\overline{\theta}(S(\mathbb{V}))=0$ . By Proposition 5.5, we acquire that $(H={\rm Ker}\bar{\partial},\theta,h_{\mathbb{V}})$ and $S(\mathbb{V})$ satisfy the conditions in Proposition 5.6. Hence $S_{X}(S(\mathbb{V}),h_{\mathbb{V}})$ is a coherent sheaf.

The following proposition shows that $S_{X}(S(\mathbb{V}),h_{\mathbb{V}})$ coincides with Saito’s $S$ -sheaf associated to $IC_{X}(\mathbb{V})$ .

Proposition 5.7.

Let $\mathbb{V}:=(\mathbb{V},\nabla,\mathcal{F}^{\bullet},h_{\mathbb{V}})$ be an $\mathbb{R}$ -polarized variation of Hodge structure. Denote by $IC_{X}(\mathbb{V})$ the intermediate extension of $\mathbb{V}$ on $X$ as a pure Hodge module and by $S(IC_{X}(\mathbb{V}))$ the Saito’s $S$ -sheaf associated to $IC_{X}(\mathbb{V})$ ([MSaito1991]). Then

S_{X}(S(\mathbb{V}),h_{\mathbb{V}})\simeq S(IC_{X}(\mathbb{V})).

Proof.

See [Schnell2020] or [SC2021, Theorem 4.10]. ∎

Now we are ready to show the Kollár package with respect to a tame harmonic bundle.

Theorem 5.8.

Let $f:X\rightarrow Y$ be a proper locally Kähler morphism from a complex space $X$ to an irreducible complex space $Y$ . Assume that every irreducible component of $X$ is mapped onto $Y$ . Let $X^{o}\subset X_{\rm reg}$ be a dense Zariski open subset and $(H,\theta,h)$ a tame harmonic bundle on $X^{o}$ . Let $E\subset H$ be a holomorphic subbundle with vanishing second fundamental form. Assume that $\overline{\theta}(E)=0$ . Let $F$ be a Nakano semi-positive vector bundle on $X$ . Then the following statements hold.

Torsion Freeness:

$R^{q}f_{\ast}(S_{X}(E,h)\otimes F)$ is torsion free for every $q\geq 0$ and vanishes if $q>\dim X-\dim Y$ .

Injectivity:

If $L$ is a semi-positive holomorphic line bundle so that $L^{\otimes l}$ admits a nonzero holomorphic global section $s$ for some $l>0$ , then the canonical morphism

R^{q}f_{\ast}(\times s):R^{q}f_{\ast}(S_{X}(E,h)\otimes F\otimes L^{\otimes k})\to R^{q}f_{\ast}(S_{X}(E,h)\otimes F\otimes L^{\otimes k+l})

is injective for every $q\geq 0$ and every $k\geq 1$ .

Vanishing:

If $Y$ is a projective algebraic variety and $L$ is an ample line bundle on $Y$ , then

H^{q}(Y,R^{p}f_{\ast}(S_{X}(E,h)\otimes F)\otimes L)=0,\quad\forall q>0,p\geq 0.

Decomposition:

Assume moreover that $X$ is a compact Kähler space, then $Rf_{\ast}(S_{X}(E,h)\otimes F)$ splits in $D(Y)$ , i.e.

Rf_{\ast}(S_{X}(E,h)\otimes F)\simeq\bigoplus_{q}R^{q}f_{\ast}(S_{X}(E,h)\otimes F)[-q]\in D(Y).

As a consequence, the spectral sequence

E^{pq}_{2}:H^{p}(Y,R^{q}f_{\ast}(S_{X}(E,h)\otimes F))\Rightarrow H^{p+q}(X,S_{X}(E,h)\otimes F)

degenerates at the $E_{2}$ page.

Proof.

By Proposition 5.6, $S_{X}(E,h)$ is a coherent sheaf. Let $h_{F}$ be a hermitian metric on $F$ with Nakano semi-positive curvature. By Lemma 2.7, we see that

S_{X}(E,h)\otimes F\simeq S_{X}(E\otimes F,h\otimes h_{F})

is a coherent sheaf on $X$ . Note that $(E\otimes F,h\otimes h_{F})$ is Nakano semi-positive by Proposition 5.6. It follows from the abstract Kollár package in §4 that the claims of the theorem are valid. ∎

Taking $S_{X}(S(\mathbb{V}),h_{\mathbb{V}})\simeq S(IC_{X}(\mathbb{V}))$ for a variation of Hodge structure $\mathbb{V}$ and $F=\mathscr{O}_{X}$ , we obtain Kollár’s conjecture.

We end this section with remarks on two other packages of Kollár’s conjecture.

Remark 5.9 (Remarks on the Intersection cohomology package).

In [Kollar1986_2, §5.8], Kollár also predicts that $S(IC_{X}(\mathbb{V}))$ is related to the intersection complex $IC_{X}(\mathbb{V})$ when $\mathbb{V}=(\mathcal{V},\nabla,\mathcal{F}^{\bullet},h)$ is a polarized variation of Hodge structure. This involves the $L^{2}$ -representation of the intersection complex.

Theorem 5.10.

Let $X$ be a compact Kähler space of pure dimension $n$ and $X^{o}\subset X_{\rm reg}$ a dense Zariski open subset. Let $\mathbb{V}=(\mathcal{V},\nabla,\mathcal{F}^{\bullet},h)$ be an $\mathbb{R}$ -polarized variation of Hodge structure of weight $r$ on $X^{o}$ . Then $IH^{k}(X,\mathbb{V})$ admits a pure Hodge structure of weight $k$ :

IH^{k}(X,\mathbb{V})=\bigoplus_{\stackrel{{\scriptstyle p,q\geq 0}}{{p+q=k+r}}}IH^{p,q}(X,\mathbb{V}),\quad IH^{p,q}(X,\mathbb{V})=\overline{IH^{q,p}(X,\mathbb{V})}

for every $0\leq k\leq 2\dim X$ . There is moreover a morphism

IC_{X}(\mathbb{V})\to S_{X}(\mathbb{V})

in the derived category of sheaves of $\mathbb{C}$ -vector spaces which induces an isomorphism

IH^{n+r,q}(X,\mathbb{V})\to H^{q}(X,S_{X}(\mathbb{V})),\quad\forall q\geq 0.

Proof.

The first statement is a consequence of [SC2021_CGM, Theorem 1.4]. Roughly speaking, there is a complete Kähler metric $ds^{2}$ on $X^{o}$ whose $L^{2}$ -de Rham complex $\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}$ is quasi-isomorphic to $IC_{X}(\mathbb{V})$ . As a consequence, there is a canonical isomorphism

IH^{k}(X,\mathbb{V})\simeq H^{k}_{(2)}(X^{o},\mathbb{V};ds^{2},h),\quad\forall k.

The $(p,q)$ -decomposition of forms in $\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}$ provides the Hodge structure on $IH^{k}(X,\mathbb{V})$ . See [SC2021_CGM, §8.3] for details. The second claim follows from the diagram

where $\tau$ is taking the projection to the $S(\mathbb{V})$ -valued $(n,\bullet)$ -component. ∎

Remark 5.11 (Remarks on the direct image package).

In [Kollar1986_2, §5.8], Kollár also predicts that

R^{q}f_{\ast}S(IC_{X}(\mathbb{V}))\simeq S_{Y}(R^{q}f_{\ast}IC_{X}(\mathbb{V})|_{Y^{o}})

where $Y^{o}$ is the Zariski open subset of $Y$ so that $R^{q}f_{\ast}IC_{X}(\mathbb{V})|_{Y^{o}}$ is a local system whose fiber at $y\in Y^{o}$ is canonically isomorphic to $\mathbb{H}^{q}(X_{y},IC_{X_{y}}(\mathbb{V}|_{X_{y}\cap X^{o}}))$ . This is also a consequence of the $L^{2}$ -representation of $IC_{X}(\mathbb{V})$ ([SC2021_CGM, Theorem 1.4]).

(2.8)	$\displaystyle\\|s\\|^{2}_{\pi^{\ast}ds^{2},h}$	$\displaystyle=\int\|\delta_{1}\wedge\cdots\wedge\delta_{n}\otimes\xi\|^{2}_{\pi^{\ast}ds^{2},h}\prod_{i=1}^{n}\lambda_{i}\delta_{i}\wedge\overline{\delta_{i}}$
	$\displaystyle=\int\|\xi\|^{2}_{h}\prod_{i=1}^{n}\delta_{i}\wedge\overline{\delta_{i}}$
	$\displaystyle=\\|s\\|^{2}_{ds^{2}_{\tilde{X}},h}.$

L2L^{2}-Extension of Adjoint bundles and Kollár’s Conjecture

Abstract.

1. Introduction

Theorem 1.1.

2. L2L^{2}-extension and its L2L^{2}-Dolbeault resolution

2.1. L2L^{2}-Dolbeault cohomology and L2L^{2}-Dolbeault complex

Definition 2.1.

Lemma 2.2.

Proof.

2.2. SX​(E,h)S_{X}(E,h) and its basic properties

Definition 2.3.

Proposition 2.4.

Proof.

Proposition 2.5 (Functoriality).

Proof.

Lemma 2.6.

Proof.

Lemma 2.7.

Proof.

Definition 2.8.

Proposition 2.9.

Proof.

2.3. L2L^{2}-Dolbeault resolution of SX​(E,h)S_{X}(E,h)

Theorem 2.10.

Theorem 2.11.

Proof.

Definition 2.12.

Lemma 2.13.

Proof.

3. Harmonic Representation of the derived pushforwards

3.1. Harmonic forms on weakly 1-complete spaces

Proposition 3.1.

Proof.

Proposition 3.2.

Proof.

3.2. Harmonic Representation

Proposition 3.3.

Proof.

Theorem 3.4.

Proof.

Proposition 3.5.

Proof.

Lemma 3.6.

Proof.

4. An abstract Kollár package

Definition 4.1.

Theorem 4.2 (Torsion Freeness).

Proof.

Theorem 4.3 (Injectivity Theorem).

Proof.

Theorem 4.4 (Decomposition).

Remark 4.5.

Proof.

Theorem 4.6 (Vanishing Theorem).

Proof.

5. Non-abelian Hodge theory and Kollár package

5.1. Harmonic bundle and variation of Hodge structure

Definition 5.1.

Definition 5.2.

Remark 5.3.

Definition 5.4.

Proposition 5.5.

Proof.

5.2. Harmonic bundle and Kollár package

Proposition 5.6.

Proof.

Proposition 5.7.

Proof.

Theorem 5.8.

Proof.

Remark 5.9 (Remarks on the Intersection cohomology package).

Theorem 5.10.

Proof.

Remark 5.11 (Remarks on the direct image package).

References

$L^{2}$ -Extension of Adjoint bundles and Kollár’s Conjecture

2. $L^{2}$ -extension and its $L^{2}$ -Dolbeault resolution

2.1. $L^{2}$ -Dolbeault cohomology and $L^{2}$ -Dolbeault complex

2.2. $S_{X}(E,h)$ and its basic properties

2.3. $L^{2}$ -Dolbeault resolution of $S_{X}(E,h)$