Kähler metrics of negative holomorphic (bi)sectional curvature on a compact relative Kähler fibration

Xueyuan Wan Mathematical Science Research Center, Chongqing University of Technology, Chongqing 400054, China [email protected]

Abstract.

For a compact relative Kähler fibration over a compact Kähler manifold with negative holomorphic sectional curvature, if the relative Kähler form on each fiber also exhibits negative holomorphic sectional curvature, we can construct Kähler metrics with negative holomorphic sectional curvature on the total space. Additionally, if this form induces a Griffiths negative Hermitian metric on the relative tangent bundle, and the base admits a Kähler metric with negative holomorphic bisectional curvature, we can also construct Kähler metrics with negative holomorphic bisectional curvature on the total space. As an application, for a non-trivial fibration where both the fibers and base have Kähler metrics with negative holomorphic bisectional curvature, and the fibers are one-dimensional, we can explicitly construct Kähler metrics of negative holomorphic bisectional curvature on the total space, thus resolving a question posed by To and Yeung for the case where the fibers have dimension one.

Key words and phrases:

Kähler metrics, Negative holomorphic (bi)sectional curvature, Griffiths negative, relative tangent bundle, relative Kähler fibration

2020 Mathematics Subject Classification:

32Q05, 32G05, 53C55

1. Introduction

Negative curvature has long been a significant focus of research in complex geometry. It is well known that any compact complex manifold with negative holomorphic sectional curvature must be Kobayashi hyperbolic. Moreover, when such a manifold possesses a fibration structure, studying its negative curvature properties becomes even more intriguing. In this paper, we investigate the curvature properties of such manifolds. Let $p:\mathcal{X}\to\mathcal{B}$ be a compact holomorphic fibration, where $p$ is a proper holomorphic submersion between two compact complex manifolds $\mathcal{X}$ and $\mathcal{B}$ .

In complex geometry, the notions of holomorphic sectional curvature and holomorphic bisectional curvature are two fundamental concepts (see Definition 2.2). A natural question arises: under what conditions does the total space $\mathcal{X}$ admit Hermitian (or Kähler) metrics with negative holomorphic (bi)sectional curvature?

The simplest nontrivial compact holomorphic fibration is the Kodaira surface, defined as a smooth compact complex surface admitting a Kodaira fibration [Kod75]. The study of negative curvature properties on Kodaira surfaces has been rich. Cheung [Che89] constructed a Kähler metric with negative holomorphic sectional curvature, while Tsai [Tsa89] developed a Hermitian metric with negative holomorphic bisectional curvature. By defining a local immersion from a Kodaira surface into the Teichmüller space $\mathcal{T}_{g,1}$ , To and Yeung [TY11] demonstrated the existence of a Kähler metric with negative holomorphic bisectional curvature on the Kodaira surface. Independently, Tsui [Tsu06] constructed a Kähler metric with negative holomorphic bisectional curvature on certain special Kodaira surfaces using a completely different approach.

For general compact holomorphic fibrations, if both the base and the fiber admit Hermitian metrics with negative holomorphic sectional curvature, Cheung [Che89, Theorem 1] proved that the entire compact holomorphic fibration also admits a Hermitian metric with negative holomorphic sectional curvature. Consequently, it is natural to consider the existence of Kähler metrics with negative holomorphic sectional curvature on such fibrations. Regarding this question, we have obtained the following result for compact relative Kähler fibrations. A relative Kähler fibration is a holomorphic fibration equipped with a relative Kähler form, that is, a smooth closed $(1,1)$ -form that is positive along each fiber (see Definition 3.1).

Theorem 1.1.

Let $p:\mathcal{X}\to\mathcal{B}$ be a compact relative Kähler fibration such that the restriction of the relative Kähler form to each fiber has negative holomorphic sectional curvature. If the base manifold $\mathcal{B}$ also admits a Kähler metric of negative holomorphic sectional curvature, then there exist Kähler metrics on $\mathcal{X}$ with negative holomorphic sectional curvature.

On the other hand, finding a Kähler metric with negative holomorphic bisectional curvature on a general compact holomorphic fibration is a challenging problem. In [TY11, Theorem 1], To and Yeung successfully proved the existence of such Kähler metrics on the Kodaira surface. Subsequently, in [TY11, Remark (i)], they posed a related question in the higher-dimensional setting:

Problem 1.2.

In general given a non-trivial fibration for which fibers and base are all equipped with Kähler metrics of negative holomorphic bisectional curvature, one may ask whether the total space of the fibration admits a Kähler metric with a similar curvature property. [TY11, Theorem 1] provides an affirmative example to such a problem.

In response to this question, we have obtained the following result, addressing the case where the relative tangent bundle is Griffiths negative (see Definition 2.1 for Griffiths negative vector bundles).

Theorem 1.3.

Let $p:\mathcal{X}\to\mathcal{B}$ be a compact relative Kähler fibration over a compact Kähler manifold $\mathcal{B}$ with negative holomorphic bisectional curvature. Suppose the relative Kähler form induces a Griffiths negative Hermitian metric on the relative tangent bundle $T_{\mathcal{X}/\mathcal{B}}$ . Then there exist Kähler metrics on $\mathcal{X}$ with negative holomorphic bisectional curvature.

In particular, when the fiber is one-dimensional, we can solve Problem 1.2 by explicitly constructing a family of Kähler metrics with negative holomorphic bisectional curvature. The specific result is as follows:

Corollary 1.4.

Let $p:\mathcal{X}\to\mathcal{B}$ be a compact holomorphic fibration over a compact Kähler manifold $\mathcal{B}$ with negative holomorphic bisectional curvature. Suppose the fibration is a holomorphic family of compact Riemann surfaces of genus $\geq 2$ and is effectively parametrized (i.e., the Kodaira-Spencer map is injective at each point). Then there exist Kähler metrics on $\mathcal{X}$ with negative holomorphic bisectional curvature.

Remark 1.5.

For the Kodaira surface, defined as a smooth compact complex surface $\mathcal{X}$ that admits a Kodaira fibration, there exists a connected fibration $p:\mathcal{X}\rightarrow\mathcal{B}$ over a smooth compact Riemann surface $\mathcal{B}$ , where $p$ is a proper holomorphic submersion and the associated Kodaira-Spencer map is injective at each point (see [TY11]). By [Sch12, Application], the genus of $\mathcal{B}$ is $\geq 2$ , and so $\mathcal{B}$ is a negatively curved Riemann surface. Based on the above corollary, we can explicitly construct a smooth family of Kähler metrics $\Omega=k(p^{*}\omega_{\mathcal{B}})+\omega_{\mathcal{X}}$ on the Kodaira surface with negative holomorphic bisectional curvature for sufficiently large $k$ . Our approach is notably different from the method in [TY11].

This article is organized as follows:

In Section 2, we will review the definitions of Griffiths negative vector bundles and the negativity of holomorphic (bi)sectional curvature. Section 3 focuses on constructing a family of Kähler metrics on the total space and analyzing its curvature properties. In this section, we will prove Theorems 1.1 and 1.3. Finally, in Section 4, we will explore a special case of relative Kähler fibrations, specifically holomorphic families of canonically polarized manifolds, and we will prove Corollary 1.4.

Acknowledgments. The author would like to thank Ya Deng and Xu Wang for many helpful discussions. Xueyuan Wan is partially supported by the National Natural Science Foundation of China (Grant No. 12101093) and the Natural Science Foundation of Chongqing (Grant No. CSTB2022NSCQ-JQX0008).

2. Griffiths negativity and holomorphic (bi)sectional curvature

This section will review the definitions of the Chern connection and its curvature for a Hermitian holomorphic vector bundle. One can refer to [Kob87, Chapter 1] for more details. Throughout this paper, we will adopt the Einstein summation convention.

Let $\pi:(E,h^{E})\to X$ be a Hermitian holomorphic vector bundle over a complex manifold $X$ , where $\mathrm{rank}E=r$ and $\dim X=n$ . The Chern connection of $(E,h^{E})$ , denoted by $\nabla^{E}$ , preserves the metric $h^{E}$ and is of $(1,0)$ -type. Given a local holomorphic frame $\{e_{i}\}_{1\leq i\leq r}$ of $E$ , the connection satisfies

\nabla^{E}e_{i}=\theta^{j}_{i}e_{j},

where $\theta=(\theta^{j}_{i})$ represents the connection form of $\nabla^{E}$ . More precisely,

\theta^{j}_{i}=\partial h_{i\bar{k}}h^{\bar{k}j},

where $h_{i\bar{k}}:=h(e_{i},e_{k})$ . The Chern curvature of $(E,h^{E})$ is denoted by $R^{E}=(\nabla^{E})^{2}\in A^{1,1}(X,\mathrm{End}(E))$ , and can be written as

R^{E}=R^{j}_{i}e_{j}\otimes e^{i}\in A^{1,1}(X,\mathrm{End}(E)),

where $R=(R^{j}_{i})$ is the curvature matrix, whose entries are $(1,1)$ -forms. Here, $\{e^{i}\}_{1\leq i\leq r}$ represents the dual frame of $\{e_{i}\}_{1\leq i\leq r}$ . The curvature matrix $R=(R^{j}_{i})$ is given by

R^{j}_{i}=d\theta^{j}_{i}+\theta^{j}_{k}\wedge\theta^{k}_{i}=\bar{\partial}\theta^{j}_{i}.

Let $\{z^{\alpha}\}_{1\leq\alpha\leq n}$ be local holomorphic coordinates of $X$ . The components of the curvature can be expressed as

R^{j}_{i}=R^{j}_{i\alpha\bar{\beta}}dz^{\alpha}\wedge d\bar{z}^{\beta},

with

R_{i\bar{j}}:=R^{k}_{i}h_{k\bar{j}}=R_{i\bar{j}\alpha\bar{\beta}}dz^{\alpha}\wedge d\bar{z}^{\beta}.

Thus, we have

R_{i\bar{j}\alpha\bar{\beta}}=R^{k}_{i\alpha\bar{\beta}}h_{k\bar{j}}=-\partial_{\alpha}\partial_{\bar{\beta}}h_{i\bar{j}}+h^{\bar{l}k}\partial_{\alpha}h_{i\bar{l}}\partial_{\bar{\beta}}h_{k\bar{j}},

where $\partial_{\alpha}:=\partial/\partial z^{\alpha}$ and $\partial_{\bar{\beta}}:=\partial/\partial\bar{z}^{\beta}$ .

Griffiths positivity and negativity are defined as follows:

Definition 2.1.

A Hermitian holomorphic vector bundle $\pi:(E,h^{E})\to X$ is said to be Griffiths positive (resp. negative) if

R_{i\bar{j}\alpha\bar{\beta}}v^{i}\bar{v}^{j}\xi^{\alpha}\bar{\xi}^{\beta}>0\quad(\text{resp.}<0)

for any non-zero elements $v=v^{i}e_{i}\in E|_{z}$ and $\xi=\xi^{\alpha}\frac{\partial}{\partial z^{\alpha}}\in T_{X}|_{z}$ at any point $z\in X$ .

In particular, when $E=T_{X}$ , the holomorphic tangent bundle of $X$ , the holomorphic (bi)sectional curvatures are defined as follows, see also [GK67].

Definition 2.2.

For any two non-zero $(1,0)$ -type tangent vectors $\xi=\xi^{\alpha}\frac{\partial}{\partial z^{\alpha}}$ and $\eta=\eta^{\alpha}\frac{\partial}{\partial z^{\alpha}}$ , the holomorphic sectional curvature along the direction $\xi$ is given by

H(\xi):=\frac{R_{\xi\bar{\xi}\xi\bar{\xi}}}{\|\xi\|^{4}}=\frac{R_{\gamma\bar{\delta}\alpha\bar{\beta}}\xi^{\alpha}\bar{\xi}^{\beta}\xi^{\gamma}\bar{\xi}^{\delta}}{(h_{\alpha\bar{\beta}}\xi^{\alpha}\bar{\xi}^{\beta})^{2}}.

The holomorphic bisectional curvature along $\xi$ and $\eta$ is

H(\xi,\eta):=\frac{R_{\xi\bar{\xi}\eta\bar{\eta}}}{\|\xi\|^{2}\|\eta\|^{2}}=\frac{R_{\gamma\bar{\delta}\alpha\bar{\beta}}\eta^{\gamma}\bar{\eta}^{\delta}\xi^{\alpha}\bar{\xi}^{\beta}}{(h_{\alpha\bar{\beta}}\xi^{\alpha}\bar{\xi}^{\beta})(h_{\gamma\bar{\delta}}\eta^{\gamma}\bar{\eta}^{\delta})}.

The holomorphic sectional (resp. bisectional) curvature is said to be negative if $H(\xi)<0$ (resp. $H(\xi,\eta)<0$ ).

3. Curvature of Kähler metrics on a fibration

This section will prove the negativity of holomorphic sectional curvature and holomorphic bisectional curvature for a compact relative Kähler fibration.

3.1. Definition of Kähler metrics

Let $p:\mathcal{X}\to\mathcal{B}$ be a holomorphic fibration with compact fibers, that is, a proper holomorphic submersion between two compact complex manifolds $\mathcal{X}$ and $\mathcal{B}$ . Let $(z,v)=(z^{1},\dots,z^{m},v^{1},\dots,v^{n})$ denote the local holomorphic coordinates of the total space $\mathcal{X}$ , where $p(z,v)=z$ . Here, $z=(z^{\alpha})_{1\leq\alpha\leq m}$ , $m=\dim\mathcal{B}$ , represents the local holomorphic coordinates on $\mathcal{B}$ , and $v=(v^{i})_{1\leq i\leq n}$ , $n=\dim X_{z}$ ( $X_{z}:=p^{-1}(z)$ ), represents the local holomorphic coordinates on the fibers.

First, we will define a relative Kähler fibration. One can refer to [WW23, Section 1] for more details.

Definition 3.1.

We call a holomorphic fibration $p:\mathcal{X}\rightarrow\mathcal{B}$ a relative Kähler fibration if there exists a real, smooth, $d$ -closed (1,1)-form $\omega_{\mathcal{X}}$ on $\mathcal{X}$ such that $\omega_{\mathcal{X}}$ is positive on each fiber $X_{z}$ .

Now we assume that $p:(\mathcal{X},\omega_{\mathcal{X}})\to\mathcal{B}$ is a relative Kähler fibration. By $\bar{\partial}$ -Poincaré Lemma, there exists a local weight, say $\phi$ , such that

\omega_{\mathcal{X}}=\sqrt{-1}\partial\bar{\partial}\phi.

By utilizing this relative Kähler form, we can obtain a canonical horizontal-vertical decomposition of the holomorphic tangent bundle of $\mathcal{X}$ . See for example [FLW19, Section 1]. With respect to $\phi$ , the canonical lift of $\frac{\partial}{\partial z^{\alpha}}$ is given by

\tfrac{\delta}{\delta z^{\alpha}}:=\tfrac{\partial}{\partial z^{\alpha}}-\phi_{\alpha\bar{j}}\phi^{i\bar{j}}\tfrac{\partial}{\partial v^{i}}.

Here $(\phi^{i\bar{j}})$ denotes the inverse matrix of $(\phi_{i\bar{j}})$ . Similarly, we define

\delta v^{i}=dv^{i}+\phi^{i\bar{j}}\phi_{\bar{j}\alpha}dz^{\alpha}.

It can be verified that

\mathcal{H}=\mathrm{Span}_{\mathbb{C}}\left\{\tfrac{\delta}{\delta z^{\alpha}}\right\}

forms a horizontal subbundle of $T_{\mathcal{X}}$ . The holomorphic vertical subbundle is

\mathcal{V}=\mathrm{Span}_{\mathbb{C}}\left\{\tfrac{\partial}{\partial v^{i}}\right\}

which corresponds to the relative tangent bundle $T_{\mathcal{X}/\mathcal{B}}$ .

The local frame $\left\{\tfrac{\delta}{\delta z^{\alpha}},\tfrac{\partial}{\partial v^{i}}\right\}$ of $T_{\mathcal{X}}$ is dual to $\{dz^{\alpha},\delta v^{i}\}$ . Furthermore, we have the following decomposition

\omega_{\mathcal{X}}=\sqrt{-1}\partial\bar{\partial}\phi=c(\phi)+\sqrt{-1}\phi_{i\bar{j}}\delta v^{i}\wedge\delta\bar{v}^{j},

where $c(\phi)$ is the geodesic curvature form given by

c(\phi)=\sqrt{-1}c(\phi)_{\alpha\bar{\beta}}dz^{\alpha}\wedge d\bar{z}^{\beta},\quad c(\phi)_{\alpha\bar{\beta}}=\phi_{\alpha\bar{\beta}}-\phi_{\alpha\bar{j}}\phi^{\bar{j}i}\phi_{i\bar{\beta}}.

Let $\omega_{\mathcal{B}}=\sqrt{-1}\psi_{\alpha\bar{\beta}}dz^{\alpha}\wedge d\bar{z}^{\beta}$ be a Kähler metric on the base manifold $\mathcal{B}$ . We define

(3.1)

\Omega=k(p^{*}\omega_{\mathcal{B}})+\omega_{\mathcal{X}},

which is a Kähler metric on the total space $\mathcal{X}$ for large $k$ . In terms of local coordinates, we have

\Omega=\sqrt{-1}\Omega_{\alpha\bar{\beta}}dz^{\alpha}\wedge d\bar{z}^{\beta}+\sqrt{-1}\phi_{i\bar{j}}\delta v^{i}\wedge\delta\bar{v}^{j},

where

\Omega_{\alpha\bar{\beta}}=k\psi_{\alpha\bar{\beta}}+c(\phi)_{\alpha\bar{\beta}}.

3.2. Negativity of holomorphic sectional curvature

In this subsection, we will prove the Kähler metrics in (3.1) have holomorphic sectional curvature and prove Theorem 1.1.

Let $\nabla=\nabla^{1,0}+\bar{\partial}$ denote the Chern connection associated with $\Omega$ , and define the Chern curvature as

R=\nabla^{2}=\nabla^{1,0}\circ\bar{\partial}+\bar{\partial}\circ\nabla^{1,0}\in A^{1,1}(\mathcal{X},\mathrm{End}(T_{\mathcal{X}})).

Then we have the following expressions

(3.2)

\displaystyle\begin{split}\nabla^{1,0}\tfrac{\delta}{\delta z^{\alpha}}&=\left\langle\nabla^{1,0}\tfrac{\delta}{\delta z^{\alpha}},\tfrac{\delta}{\delta z^{\beta}}\right\rangle\Omega^{\bar{\beta}\gamma}\tfrac{\delta}{\delta z^{\gamma}}+\left\langle\nabla^{1,0}\tfrac{\delta}{\delta z^{\alpha}},\tfrac{\partial}{\partial v^{j}}\right\rangle\phi^{\bar{j}i}\tfrac{\partial}{\partial v^{i}}\\ &=\partial\Omega_{\alpha\bar{\beta}}\Omega^{\bar{\beta}\gamma}\tfrac{\delta}{\delta z^{\gamma}}\end{split}

and

(3.3)

\displaystyle\begin{split}\nabla^{1,0}\tfrac{\partial}{\partial v^{i}}&=\left\langle\nabla^{1,0}\tfrac{\partial}{\partial v^{i}},\tfrac{\delta}{\delta z^{\beta}}\right\rangle\Omega^{\bar{\beta}\alpha}\tfrac{\delta}{\delta z^{\alpha}}+\left\langle\nabla^{1,0}\tfrac{\partial}{\partial v^{i}},\tfrac{\partial}{\partial v^{j}}\right\rangle\phi^{\bar{j}k}\tfrac{\partial}{\partial v^{k}}\\ &=\partial(\phi_{\bar{\beta}k}\phi^{k\bar{j}})\phi_{i\bar{j}}\Omega^{\bar{\beta}\alpha}\tfrac{\delta}{\delta z^{\alpha}}+\partial\phi_{i\bar{j}}\phi^{\bar{j}k}\tfrac{\partial}{\partial v^{k}}.\end{split}

From (3.2) and (3.3), the Chern curvature $R$ of $\Omega$ can be expressed as

\displaystyle\begin{split}R\tfrac{\delta}{\delta z^{\alpha}}&=(\nabla^{1,0}\circ\bar{\partial}+\bar{\partial}\circ\nabla^{1,0})(\tfrac{\delta}{\delta z^{\alpha}})\\ &=\nabla^{1,0}(\bar{\partial}(-\phi_{\alpha\bar{j}}\phi^{\bar{j}i})\tfrac{\partial}{\partial v^{i}})+\bar{\partial}\left(\partial\Omega_{\alpha\bar{\beta}}\Omega^{\bar{\beta}\gamma}\tfrac{\delta}{\delta z^{\gamma}}\right)\\ &=\left[\bar{\partial}\left(\partial\Omega_{\alpha\bar{\beta}}\Omega^{\bar{\beta}\gamma}\right)+\bar{\partial}(\phi_{\alpha\bar{j}}\phi^{\bar{j}i})\wedge\partial(\phi_{\bar{\beta}k}\phi^{k\bar{l}})\phi_{i\bar{l}}\Omega^{\bar{\beta}\gamma}\right]\tfrac{\delta}{\delta z^{\gamma}}\\ &\quad+\left[-\partial\bar{\partial}(\phi_{\alpha\bar{j}}\phi^{\bar{j}i})+\bar{\partial}(\phi_{\alpha\bar{j}}\phi^{\bar{j}k})\wedge\partial\phi_{k\bar{j}}\phi^{\bar{j}i}\right.\\ &\quad\left.+\partial\Omega_{\alpha\bar{\beta}}\Omega^{\bar{\beta}\gamma}\wedge\bar{\partial}(\phi_{\gamma\bar{j}}\phi^{\bar{j}i})\right]\tfrac{\partial}{\partial v^{i}}\end{split}

and

\displaystyle\begin{split}R\tfrac{\partial}{\partial v^{i}}&=\bar{\partial}\circ\nabla^{1,0}(\tfrac{\partial}{\partial v^{i}})\\ &=\bar{\partial}\left[\partial(\phi_{\bar{\beta}k}\phi^{k\bar{j}})\phi_{i\bar{j}}\Omega^{\bar{\beta}\alpha}\tfrac{\delta}{\delta z^{\alpha}}+\partial\phi_{i\bar{j}}\phi^{\bar{j}k}\tfrac{\partial}{\partial v^{k}}\right]\\ &=\bar{\partial}\left(\partial(\phi_{\bar{\beta}k}\phi^{k\bar{j}})\phi_{i\bar{j}}\Omega^{\bar{\beta}\alpha}\right)\tfrac{\delta}{\delta z^{\alpha}}\\ &\quad+\left[\bar{\partial}(\partial\phi_{i\bar{j}}\phi^{\bar{j}k})+\partial(\phi_{\bar{\beta}s}\phi^{s\bar{j}})\phi_{i\bar{j}}\Omega^{\bar{\beta}\alpha}\wedge\bar{\partial}(\phi_{\alpha\bar{l}}\phi^{\bar{l}k})\right]\tfrac{\partial}{\partial v^{k}}.\end{split}

By taking the inner product of $R\frac{\delta}{\delta z^{\alpha}}$ and $R\frac{\partial}{\partial v^{i}}$ with $\frac{\delta}{\delta z^{\beta}}$ and $\frac{\partial}{\partial v^{j}}$ , we obtain the following proposition.

Proposition 3.2.

We have

(3.4)	$\displaystyle\left\langle R\tfrac{\delta}{\delta z^{\alpha}},\tfrac{\delta}{\delta z^{\beta}}\right\rangle$	$\displaystyle=\left\langle R^{\mathcal{H}}\tfrac{\delta}{\delta z^{\alpha}},\tfrac{\delta}{\delta z^{\beta}}\right\rangle+\bar{\partial}(\phi_{\alpha\bar{j}}\phi^{\bar{j}i})\wedge\partial(\phi_{\bar{\beta}k}\phi^{k\bar{l}})\phi_{i\bar{l}}.$
(3.5)	$\displaystyle\left\langle R\tfrac{\partial}{\partial v^{i}},\tfrac{\partial}{\partial v^{j}}\right\rangle$	$\displaystyle=\left\langle R^{\mathcal{V}}\tfrac{\partial}{\partial v^{i}},\tfrac{\partial}{\partial v^{j}}\right\rangle+\partial(\phi_{\bar{\beta}s}\phi^{s\bar{q}})\phi_{i\bar{q}}\Omega^{\bar{\beta}\alpha}\wedge\bar{\partial}(\phi_{\alpha\bar{l}}\phi^{\bar{l}k})\phi_{k\bar{j}}.$
(3.6)	$\displaystyle\left\langle R\tfrac{\delta}{\delta z^{\alpha}},\tfrac{\partial}{\partial v^{l}}\right\rangle$	$\displaystyle=\left[-\partial\bar{\partial}(\phi_{\alpha\bar{j}}\phi^{\bar{j}i})+\bar{\partial}(\phi_{\alpha\bar{j}}\phi^{\bar{j}k})\wedge\partial\phi_{k\bar{j}}\phi^{\bar{j}i}\right.$
	$\displaystyle\left.\quad\,+\partial\Omega_{\alpha\bar{\beta}}\Omega^{\bar{\beta}\gamma}\wedge\bar{\partial}(\phi_{\gamma\bar{j}}\phi^{\bar{j}i})\right]\phi_{i\bar{l}}.$

Here, $R^{\mathcal{H}}$ denotes the Chern curvature of the Hermitian vector bundle $(\mathcal{H},(\Omega_{\alpha\bar{\beta}}))$ , and $R^{\mathcal{V}}$ is the Chern curvature of the Hermitian vector bundle $(\mathcal{V},(\phi_{i\bar{j}}))$ .

Recall that $\Omega_{\alpha\bar{\beta}}=k\psi_{\alpha\bar{\beta}}+c(\phi)_{\alpha\bar{\beta}}$ . We now derive the following estimates.

Proposition 3.3.

As $k\to\infty$ , the following estimates hold:

(3.7)	$\displaystyle R_{\gamma\bar{\sigma}\alpha\bar{\beta}}$	$\displaystyle:=\left\langle R\left(\tfrac{\delta}{\delta z^{\gamma}},\tfrac{\delta}{\delta\bar{z}^{\sigma}}\right)\tfrac{\delta}{\delta z^{\alpha}},\tfrac{\delta}{\delta z^{\beta}}\right\rangle=R^{T_{\mathcal{B}}}_{\gamma\bar{\sigma}\alpha\bar{\beta}}k+O(1),$
(3.8)	$\displaystyle R_{\gamma\bar{\sigma}i\bar{j}}$	$\displaystyle:=\left\langle R\left(\tfrac{\delta}{\delta z^{\gamma}},\tfrac{\delta}{\delta\bar{z}^{\sigma}}\right)\tfrac{\partial}{\partial v^{i}},\tfrac{\partial}{\partial v^{j}}\right\rangle=R^{\mathcal{V}}_{\gamma\bar{\sigma}i\bar{j}}+O\left(\tfrac{1}{k}\right)=O(1),$
(3.9)	$\displaystyle R_{k\bar{l}i\bar{j}}$	$\displaystyle:=\left\langle R\left(\tfrac{\partial}{\partial v^{k}},\tfrac{\partial}{\partial\bar{v}^{l}}\right)\tfrac{\partial}{\partial v^{i}},\tfrac{\partial}{\partial v^{j}}\right\rangle=R^{\mathcal{V}}_{k\bar{l}i\bar{j}}+O\left(\tfrac{1}{k}\right),$
(3.10)	$\displaystyle R_{k\bar{\beta}i\bar{j}}$	$\displaystyle:=\left\langle R\left(\tfrac{\partial}{\partial v^{k}},\tfrac{\delta}{\delta\bar{z}^{\beta}}\right)\tfrac{\partial}{\partial v^{i}},\tfrac{\partial}{\partial v^{j}}\right\rangle=O(1),$
(3.11)	$\displaystyle R_{\alpha\bar{l}i\bar{j}}$	$\displaystyle:=\left\langle R\left(\tfrac{\delta}{\delta z^{\alpha}},\tfrac{\partial}{\partial\bar{v}^{l}}\right)\tfrac{\partial}{\partial v^{i}},\tfrac{\partial}{\partial v^{j}}\right\rangle=O(1),$
(3.12)	$\displaystyle R_{\alpha\bar{l}\gamma\bar{j}}$	$\displaystyle:=\left\langle R\left(\tfrac{\delta}{\delta z^{\alpha}},\tfrac{\partial}{\partial\bar{v}^{l}}\right)\tfrac{\delta}{\delta z^{\gamma}},\tfrac{\partial}{\partial v^{j}}\right\rangle=O(1),$
(3.13)	$\displaystyle R_{i\bar{\sigma}\alpha\bar{\beta}}$	$\displaystyle:=\left\langle R\left(\tfrac{\partial}{\partial v^{i}},\tfrac{\delta}{\delta\bar{z}^{\sigma}}\right)\tfrac{\delta}{\delta z^{\alpha}},\tfrac{\delta}{\delta z^{\beta}}\right\rangle=O(1).$

Here, $f(k)=O(g(k))$ as $k\to\infty$ means that there exist constants $k_{0}>0$ and $C>0$ such that $|f(k)|<Cg(k)$ for any $k\geq k_{0}$ .

Proof.

First, note that $\frac{1}{k}\Omega_{\alpha\bar{\beta}}=\psi_{\alpha\bar{\beta}}+\frac{1}{k}c(\phi)_{\alpha\bar{\beta}}$ , so $k\Omega^{\alpha\bar{\beta}}=\psi^{\alpha\bar{\beta}}+O\left(\frac{1}{k}\right)$ . Using Proposition 3.2 (3.4), we have

	$\displaystyle\left\langle R\tfrac{\delta}{\delta z^{\alpha}},\tfrac{\delta}{\delta z^{\beta}}\right\rangle$	$\displaystyle=\bar{\partial}\partial\Omega_{\alpha\bar{\beta}}+\partial\Omega_{\alpha\bar{\sigma}}\wedge\bar{\partial}\Omega_{\gamma\bar{\beta}}\Omega^{\bar{\sigma}\gamma}+O(1)$
		$\displaystyle=k\bar{\partial}\partial\psi_{\alpha\bar{\beta}}+O(1)$
		$\displaystyle\quad+\left(k\partial\psi_{\alpha\bar{\sigma}}+O(1)\right)\wedge\left(k\bar{\partial}\psi_{\gamma\bar{\beta}}+O(1)\right)\left(\tfrac{1}{k}\psi^{\bar{\sigma}\gamma}+O\left(\tfrac{1}{k^{2}}\right)\right)$
		$\displaystyle=k\left(\bar{\partial}\partial\psi_{\alpha\bar{\beta}}+\partial\psi_{\alpha\bar{\sigma}}\wedge\bar{\partial}\psi_{\gamma\bar{\beta}}\psi^{\bar{\sigma}\gamma}\right)+O(1)$
		$\displaystyle=k\left\langle R^{T_{\mathcal{B}}}\tfrac{\delta}{\delta z^{\alpha}},\tfrac{\delta}{\delta z^{\beta}}\right\rangle+O(1).$

Thus, (3.7) and (3.13) are proved. Now, using Proposition 3.2 (3.5), we get

(3.14)

\displaystyle\left\langle R\tfrac{\partial}{\partial v^{i}},\tfrac{\partial}{\partial v^{j}}\right\rangle

\displaystyle=\left\langle R^{\mathcal{V}}\tfrac{\partial}{\partial v^{i}},\tfrac{\partial}{\partial v^{j}}\right\rangle+O\left(\tfrac{1}{k}\right)=O(1),

which proves (3.8) through (3.11). Finally, Proposition 3.2 (3.6) gives the proof of (3.12). ∎

Next, we prove Theorem 1.1.

Proof of Theorem 1.1.

For any vector $X$ of type $(1,0)$ , we decompose it as follows

X=Y+Z,\quad Y=a^{\alpha}\tfrac{\delta}{\delta z^{\alpha}},\quad Z=b^{i}\tfrac{\partial}{\partial v^{i}}.

Using Proposition 3.3, the holomorphic sectional curvature along the direction $X$ is given by

(3.15)

\displaystyle\begin{split}R_{X\bar{X}X\bar{X}}&=\left\langle R(Y+Z,\bar{Y}+\bar{Z})(Y+Z),Y+Z\right\rangle\\ &=R_{Y\bar{Y}Y\bar{Y}}+2R_{Y\bar{Y}Y\bar{Z}}+2R_{Y\bar{Y}Z\bar{Y}}+4R_{Y\bar{Y}Z\bar{Z}}+R_{Y\bar{Z}Y\bar{Z}}\\ &\quad+R_{Z\bar{Y}Z\bar{Y}}+2R_{Y\bar{Z}Z\bar{Z}}+2R_{Z\bar{Z}Z\bar{Y}}+R_{Z\bar{Z}Z\bar{Z}}\\ &\leq R^{T_{\mathcal{B}}}_{p_{*}Y\overline{p_{*}Y}p_{*}Y\overline{p_{*}Y}}\cdot k+R^{\mathcal{V}}_{Z\bar{Z}Z\bar{Z}}+O\left(\tfrac{1}{k}\right)\|b\|^{4}\\ &\quad+O(1)(\|a\|^{4}+\|a\|^{3}\|b\|+\|a\|^{2}\|b\|^{2}+\|a\|\|b\|^{3}),\end{split}

where $a=(a^{1},\dots,a^{m})$ and $b=(b^{1},\dots,b^{n})$ , and the norms of $a$ and $b$ are defined as follows

(3.16)

\|a\|^{2}:={a^{\alpha}\overline{a^{\beta}}\psi_{\alpha\bar{\beta}}}\text{ and }\|b\|^{2}:={b^{i}\overline{b^{j}}\phi_{i\bar{j}}}.

By assumption, both the base and the fibers have negative holomorphic sectional curvature. Therefore,

(3.17)

R^{T_{\mathcal{B}}}_{p_{*}Y\overline{p_{*}Y}p_{*}Y\overline{p_{*}Y}}\leq-\epsilon_{0}\|p_{*}Y\|^{4}=-\epsilon_{0}|a^{\alpha}\overline{a^{\beta}}\psi_{\alpha\bar{\beta}}|^{2}=-\epsilon_{0}\|a\|^{4},

for some small positive constant $\epsilon_{0}$ , where $-\epsilon_{0}$ is taken to be the maximum holomorphic sectional curvature of $(\mathcal{B},\omega_{\mathcal{B}})$ . Similarly, we have

(3.18)

R^{\mathcal{V}}_{Z\bar{Z}Z\bar{Z}}\leq-\epsilon_{1}\|b\|^{4}

for some small positive constant $\epsilon_{1}$ . Substituting (3.17) and (3.18) into (3.15) yields

(3.19)

\displaystyle\begin{split}R_{X\bar{X}X\bar{X}}&\leq-\epsilon_{0}k\|a\|^{4}-\epsilon_{1}\|b\|^{4}+O\left(\tfrac{1}{k}\right)\|b\|^{4}\\ &\quad+O(1)(\|a\|^{4}+\|a\|^{3}\|b\|+\|a\|^{2}\|b\|^{2}+\|a\|\|b\|^{3}).\end{split}

Using Young’s inequality, we have the following estimates:

\displaystyle\begin{split}\|a\|^{3}\|b\|&\leq\tfrac{3}{4}k^{\tfrac{1}{6}}\|a\|^{4}+\tfrac{1}{4}\tfrac{1}{\sqrt{k}}\|b\|^{4}.\\ \|b\|^{3}\|a\|&\leq\tfrac{3}{4}k^{-\tfrac{1}{6}}\|b\|^{4}+\tfrac{1}{4}\sqrt{k}\|a\|^{4}.\\ \|a\|^{2}\|b\|^{2}&\leq\tfrac{\sqrt{k}}{2}\|a\|^{4}+\tfrac{1}{2\sqrt{k}}\|b\|^{4}.\end{split}

Thus, (3.19) simplifies to

\displaystyle\begin{split}R_{X\bar{X}X\bar{X}}&\leq\|a\|^{4}(-\epsilon_{0}k+O(1)(1+\tfrac{3}{4}k^{\tfrac{1}{6}}+\tfrac{3}{4}\sqrt{k}))\\ &\quad+\|b\|^{4}(-\epsilon_{1}+O(\tfrac{1}{k})+O(1)(\tfrac{1}{4\sqrt{k}}+\tfrac{3}{4}k^{-\tfrac{1}{6}}+\tfrac{1}{2\sqrt{k}}))\\ &\leq\|a\|^{4}(-\epsilon_{0}k+O(\sqrt{k}))+\|b\|^{4}(-\epsilon_{1}+O(k^{-\tfrac{1}{6}})).\end{split}

Hence, there exists some $k_{0}>0$ such that for any $k\geq k_{0}$ ,

-\epsilon_{0}k+O(\sqrt{k})<0\quad\text{and}\quad-\epsilon_{1}+O(k^{-\tfrac{1}{6}})<0.

Therefore,

R_{X\bar{X}X\bar{X}}<0,

for any $\|a\|\neq 0$ or $\|b\|\neq 0$ , i.e., for any non-zero vector $X$ . This concludes the proof of the negativity of the holomorphic sectional curvature of the Kähler metric $\Omega=k\omega_{\mathcal{B}}+\omega_{\mathcal{X}}$ for any $k\geq k_{0}$ . ∎

Remark 3.4.

By [WY16] and [TY17], if each fiber has a Kähler metric with negative holomorphic sectional curvature, then the canonical bundle of each fiber is ample. Hence, this relative Kähler fibration is a holomorphic family of compact, canonically polarized manifolds.

3.3. Negativity of holomorphic bisectional curvature

In this section, we will prove that the Kähler metrics in (3.1) have negative holomorphic bisectional curvature and prove Theorem 1.3.

Recall that the relative Kähler form $\omega_{\mathcal{X}}$ is a real $(1,1)$ -form on $\mathcal{X}$ , which is positive when restricted to each fiber. We express it as follows

\omega_{\mathcal{X}}=\sqrt{-1}\partial\bar{\partial}\phi=c(\phi)+\sqrt{-1}\phi_{i\bar{j}}\delta v^{i}\wedge\delta\bar{v}^{j}.

The relative tangent bundle $T_{\mathcal{X}/\mathcal{B}}=\mathcal{V}$ is spanned by $\{\frac{\partial}{\partial v^{i}}\}_{1\leq i\leq n}$ . Hence the Kähler form $\omega_{\mathcal{X}}$ induces a canonical Hermitian metric on $\mathcal{V}$ by

\left\langle\tfrac{\partial}{\partial v^{i}},\tfrac{\partial}{\partial v^{j}}\right\rangle:=\phi_{i\bar{j}}.

In this section, we assume that the induced Hermitian metric has Griffiths negative curvature. Denote

\Omega_{0}:=p^{*}\omega_{\mathcal{B}}+\sqrt{-1}\phi_{i\bar{j}}\delta v^{i}\wedge\delta\bar{v}^{j},

which is a Hermitian metric on $\mathcal{X}$ . Since $\mathcal{X}$ is compact, there exist two uniform positive constants $c_{0}$ and $C_{0}$ such that

(3.20)

c_{0}\|X\|_{\Omega_{0}}^{2}\|V\|^{2}\leq-R^{\mathcal{V}}_{X\bar{X}V\bar{V}}\leq C_{0}\|X\|_{\Omega_{0}}^{2}\|V\|^{2}

for any tangent vector $X$ and $V\in\mathcal{V}$ .

For any non-zero vector $X$ of type $(1,0)$ , we can decompose it as

X=Y+Z

where $Y=a^{\alpha}\tfrac{\delta}{\delta z^{\alpha}}$ and $Z=b^{i}\tfrac{\partial}{\partial v^{i}}$ . For any $t\in\mathbb{R}$ , we have

R^{\mathcal{V}}_{Y+tZ\overline{Y+tZ}V\bar{V}}=t^{2}R^{\mathcal{V}}_{Z\bar{Z}V\bar{V}}+t(R^{\mathcal{V}}_{Y\bar{Z}V\bar{V}}+R^{\mathcal{V}}_{Z\bar{Y}V\bar{V}})+R^{\mathcal{V}}_{Y\bar{Y}V\bar{V}}.

If $Y$ , $Z$ , and $V$ are non-zero, by taking $t=-\tfrac{R^{\mathcal{V}}_{Y\bar{Z}V\bar{V}}+R^{\mathcal{V}}_{Z\bar{Y}V\bar{V}}}{2R^{\mathcal{V}}_{Z\bar{Z}V\bar{V}}}$ , we obtain

R^{\mathcal{V}}_{Y+tZ\overline{Y+tZ}V\bar{V}}=\frac{4R^{\mathcal{V}}_{Z\bar{Z}V\bar{V}}R^{\mathcal{V}}_{Y\bar{Y}V\bar{V}}-\left(R^{\mathcal{V}}_{Y\bar{Z}V\bar{V}}+R^{\mathcal{V}}_{Z\bar{Y}V\bar{V}}\right)^{2}}{4R^{\mathcal{V}}_{Z\bar{Z}V\bar{V}}}<0.

This implies that

(3.21)

2\sqrt{R^{\mathcal{V}}_{Z\bar{Z}V\bar{V}}R^{\mathcal{V}}_{Y\bar{Y}V\bar{V}}}-\left|R^{\mathcal{V}}_{Y\bar{Z}V\bar{V}}+R^{\mathcal{V}}_{Z\bar{Y}V\bar{V}}\right|\geq c_{1}\|V\|^{2}\|Z\|_{\Omega_{0}}\|Y\|_{\Omega_{0}},

for some constant $c_{1}>0$ .

On the other hand, we also have

\sqrt{R^{\mathcal{V}}_{Z\bar{Z}V\bar{V}}R^{\mathcal{V}}_{Y\bar{Y}V\bar{V}}}\leq C_{0}\|V\|^{2}\|Z\|_{\Omega_{0}}\|Y\|_{\Omega_{0}}.

Combining this with (3.21), we obtain

	$\displaystyle\quad\left(-R^{\mathcal{V}}_{Z\bar{Z}V\bar{V}}-R^{\mathcal{V}}_{Y\bar{Y}V\bar{V}}\right)\left(1-\tfrac{c_{1}}{2C_{0}}\right)-\left\|R^{\mathcal{V}}_{Y\bar{Z}V\bar{V}}+R^{\mathcal{V}}_{Z\bar{Y}V\bar{V}}\right\|$
	$\displaystyle\geq 2\sqrt{R^{\mathcal{V}}_{Z\bar{Z}V\bar{V}}R^{\mathcal{V}}_{Y\bar{Y}V\bar{V}}}\left(1-\tfrac{c_{1}}{2C_{0}}\right)-\left\|R^{\mathcal{V}}_{Y\bar{Z}V\bar{V}}+R^{\mathcal{V}}_{Z\bar{Y}V\bar{V}}\right\|$
	$\displaystyle\geq 0.$

Thus, we conclude

(3.22)

\displaystyle\begin{split}&\quad-R^{\mathcal{V}}_{Z\bar{Z}V\bar{V}}-R^{\mathcal{V}}_{Y\bar{Y}V\bar{V}}-\left|R^{\mathcal{V}}_{Y\bar{Z}V\bar{V}}+R^{\mathcal{V}}_{Z\bar{Y}V\bar{V}}\right|\\ &\geq\epsilon_{0}\left(-R^{\mathcal{V}}_{Z\bar{Z}V\bar{V}}-R^{\mathcal{V}}_{Y\bar{Y}V\bar{V}}\right),\end{split}

where $\epsilon_{0}=\frac{c_{1}}{2C_{0}}>0$ .

For any two non-zero vectors $X$ and $W$ of type $(1,0)$ , we decompose them as follows

X=Y+Z\text{ and }W=U+V,

where $Y=a^{\alpha}\frac{\delta}{\delta z^{\alpha}}$ , $Z=b^{i}\frac{\partial}{\partial v^{i}}$ , $U=c^{\alpha}\frac{\delta}{\delta z^{\alpha}}$ , and $V=d^{i}\frac{\partial}{\partial v^{i}}$ . Then, the holomorphic bisectional curvature satisfies

(3.23)

\displaystyle\begin{split}R_{X\bar{X}W\bar{W}}&=\left\langle R(Y+Z,\overline{Y+Z})(U+V),U+V\right\rangle\\ &=R_{Y\bar{Y}U\bar{U}}+R_{Y\bar{Y}U\bar{V}}+R_{Y\bar{Y}V\bar{U}}+R_{Y\bar{Y}V\bar{V}}\\ &\quad+R_{Y\bar{Z}U\bar{U}}+R_{Y\bar{Z}U\bar{V}}+R_{Y\bar{Z}V\bar{U}}+R_{Y\bar{Z}V\bar{V}}\\ &\quad+R_{Z\bar{Y}U\bar{U}}+R_{Z\bar{Y}U\bar{V}}+R_{Z\bar{Y}V\bar{U}}+R_{Z\bar{Y}V\bar{V}}\\ &\quad+R_{Z\bar{Z}U\bar{U}}+R_{Z\bar{Z}U\bar{V}}+R_{Z\bar{Z}V\bar{U}}+R_{Z\bar{Z}V\bar{V}}.\end{split}

By Proposition 3.3 and equation (3.20), and following the same reasoning as in the proof of (3.17) and by assumption $(\mathcal{B},\omega_{\mathcal{B}})$ has negative holomorphic bisectional curvature, there exists a small positive constant $\epsilon_{1}$ such that

	$\displaystyle R_{Y\bar{Y}U\bar{U}}$	$\displaystyle\leq\\|a\\|^{2}\\|c\\|^{2}(-\epsilon_{1}k+O(1)),$
	$\displaystyle R_{Z\bar{Z}V\bar{V}}$	$\displaystyle\leq\\|b\\|^{2}\\|d\\|^{2}(-\epsilon_{1}+O(\tfrac{1}{k})),$
	$\displaystyle R_{Y\bar{Y}V\bar{V}}$	$\displaystyle\leq\\|a\\|^{2}\\|d\\|^{2}(-\epsilon_{1}+O(\tfrac{1}{k})),$
	$\displaystyle R_{Z\bar{Z}U\bar{U}}$	$\displaystyle\leq\\|b\\|^{2}\\|c\\|^{2}(-\epsilon_{1}+O(\tfrac{1}{k})),$

where the norms of $\|a\|,\|b\|,\|c\|,\|d\|$ are defined by (3.16).

The cross terms involving different combinations of $Y$ , $Z$ , $U$ , and $V$ have the following bounds

	$\displaystyle\|R_{Y\bar{Y}U\bar{V}}\|=\|R_{Y\bar{Y}V\bar{U}}\|$	$\displaystyle\leq\\|a\\|^{2}\\|c\\|\\|d\\|O(1),$
	$\displaystyle\|R_{Y\bar{Z}U\bar{U}}\|=\|R_{Z\bar{Y}U\bar{U}}\|$	$\displaystyle\leq\\|a\\|\\|b\\|\\|c\\|^{2}O(1),$
	$\displaystyle\|R_{Y\bar{Z}V\bar{V}}\|=\|R_{Z\bar{Y}V\bar{V}}\|$	$\displaystyle\leq\\|a\\|\\|b\\|\\|d\\|^{2}O(1),$
	$\displaystyle\|R_{Z\bar{Z}U\bar{V}}\|=\|R_{Z\bar{Z}V\bar{U}}\|$	$\displaystyle\leq\\|b\\|^{2}\\|c\\|\\|d\\|O(1),$
	$\displaystyle\|R_{Y\bar{Z}U\bar{V}}\|=\|R_{Z\bar{Y}V\bar{U}}\|$	$\displaystyle\leq\\|a\\|\\|b\\|\\|c\\|\\|d\\|O(1).$

Using equations (3.14) and (3.22) and the symmetry of the curvature tensor, we obtain

(3.24)

\displaystyle\begin{split}&\quad R_{Z\bar{Z}U\bar{U}}+R_{Z\bar{Z}V\bar{V}}+|R_{Z\bar{Z}U\bar{V}}+R_{Z\bar{Z}V\bar{U}}|\\ &=R_{U\bar{U}Z\bar{Z}}+R_{V\bar{V}Z\bar{Z}}+|R_{U\bar{V}Z\bar{Z}}+R_{V\bar{U}Z\bar{Z}}|\\ &\leq R_{U\bar{U}Z\bar{Z}}^{\mathcal{V}}+R_{V\bar{V}Z\bar{Z}}^{\mathcal{V}}+|R^{\mathcal{V}}_{U\bar{V}Z\bar{Z}}+R^{\mathcal{V}}_{V\bar{U}Z\bar{Z}}|\\ &\quad+O(\tfrac{1}{k})\|b\|^{2}(\|c\|+\|d\|)^{2}\\ &\leq\epsilon_{0}\left(R_{U\bar{U}Z\bar{Z}}^{\mathcal{V}}+R_{V\bar{V}Z\bar{Z}}^{\mathcal{V}}\right)+O(\tfrac{1}{k})\|b\|^{2}(\|c\|+\|d\|)^{2}\\ &\leq(-\epsilon_{0}\epsilon_{1}+O(\tfrac{1}{k}))\|b\|^{2}(\|c\|^{2}+\|d\|^{2}).\end{split}

Similarly, we have

\displaystyle\begin{split}&\quad R_{Y\bar{Y}V\bar{V}}+R_{Z\bar{Z}V\bar{V}}+|R_{Z\bar{Y}V\bar{V}}+R_{Z\bar{Y}V\bar{V}}|\\ &\leq(-\epsilon_{0}\epsilon_{1}+O(\tfrac{1}{k}))\|d\|^{2}(\|a\|^{2}+\|b\|^{2}).\end{split}

Now we begin to prove Theorem 1.3.

Proof of Theorem 1.3.

We will discuss the negativity of holomorphic bisectional curvature in the following five cases.

Case I: If $\|b\|=0$ , i.e. $Z=0$ , then

\displaystyle\begin{split}R_{X\bar{X}W\bar{W}}&=R_{Y\bar{Y}U\bar{U}}+R_{Y\bar{Y}U\bar{V}}+R_{Y\bar{Y}V\bar{U}}+R_{Y\bar{Y}V\bar{V}}\\ &\leq\|a\|^{2}\|c\|^{2}(-\epsilon_{1}k+O(1))+2\|a\|^{2}\|c\|\|d\|O(1)\\ &\quad+\|a\|^{2}\|d\|^{2}(-\epsilon_{1}+O(\tfrac{1}{k}))\\ &\leq\|a\|^{2}\|c\|^{2}(-\epsilon_{1}k+O(1)+\sqrt{k}O(1))\\ &\quad+\|a\|^{2}\|d\|^{2}(-\epsilon_{1}+O(\tfrac{1}{k})+\tfrac{1}{\sqrt{k}}O(1)),\end{split}

where the last inequality by the inequality $2\|c\|\|d\|\leq\|c\|^{2}\sqrt{k}+\|d\|^{2}/\sqrt{k}$ . Hence there exists a large $k_{0}\geq 0$ such that

-\epsilon_{1}k+O(1)+\sqrt{k}O(1)<0\text{ and }-\epsilon_{1}+O(\tfrac{1}{k})+\tfrac{1}{\sqrt{k}}<0

for any $k\geq k_{0}$ . Hence $R_{X\bar{X}W\bar{W}}<0$ .

Case II: If $\|d\|=0$ , i.e. $V=0$ , this reduces to the previous case since $R_{X\bar{X}W\bar{W}}=R_{W\bar{W}X\bar{X}}$ . Hence $R_{X\bar{X}W\bar{W}}<0$ for any $k\geq k_{0}$ .

Case III: If $\|b\|=\|d\|=1$ and $\|a\|\leq\|c\|$ , using (3.24), (3.23) and the estimates on curvature tensors, then

\displaystyle\begin{split}R_{X\bar{X}W\bar{W}}&\leq\|a\|^{2}\|c\|^{2}(-\epsilon_{1}k+O(1))+2\|a\|^{2}\|c\|O(1)+2\|a\|\|c\|^{2}O(1)\\ &\quad+4\|a\|\|c\|O(1)+\|a\|^{2}(-\epsilon_{1}+O(\tfrac{1}{k}))+2\|a\|O(1)\\ &\quad+(-\epsilon_{0}\epsilon_{1}+O(\tfrac{1}{k}))(\|c\|^{2}+1)\\ &\leq\|a\|^{2}\|c\|^{2}(-\epsilon_{1}k+O(1)+4\sqrt{k}O(1))\\ &\quad+\|a\|^{2}(-\epsilon_{1}+O(\tfrac{1}{k})+O(\tfrac{1}{\sqrt{k}}))\\ &\quad+\|c\|^{2}(-\epsilon_{0}\epsilon_{1}+O(\tfrac{1}{k})+O(\tfrac{1}{\sqrt{k}}))+(-\epsilon_{0}\epsilon_{1}+O(\tfrac{1}{k})\\ &\quad+O(\tfrac{1}{\sqrt{k}}))+2\|a\|O(1).\end{split}

By choosing $k$ large enough such that

\begin{cases}-\epsilon_{1}k+O(1)+4\sqrt{k}O(1)&<-\tfrac{\epsilon_{1}}{2}k,\\ -\epsilon_{1}+O(\tfrac{1}{k})+O(\tfrac{1}{\sqrt{k}})&<0,\\ -\epsilon_{0}\epsilon_{1}+O(\tfrac{1}{k})+O(\tfrac{1}{\sqrt{k}})&<0,\\ -\epsilon_{0}\epsilon_{1}+O(\tfrac{1}{k})+O(\tfrac{1}{\sqrt{k}})&<-\tfrac{1}{2}\epsilon_{0}\epsilon_{1}.\end{cases}

We obtain

\displaystyle\begin{split}R_{X\bar{X}W\bar{W}}&\leq-\tfrac{\epsilon_{1}}{2}k\|a\|^{4}-\tfrac{1}{2}\epsilon_{0}\epsilon_{1}+2\|a\|O(1)\\ &\leq(-\tfrac{\epsilon}{2}k+\tfrac{\sqrt{k}}{2}O(1))\|a\|^{4}+(-\tfrac{1}{2}\epsilon_{0}\epsilon_{1}+\tfrac{3}{2}k^{-\tfrac{1}{6}}O(1)),\end{split}

where the last inequality by Young inequality $\|a\|\leq\frac{\|a\|^{4}\sqrt{k}}{4}+\frac{3}{4}k^{-\tfrac{1}{6}}$ . By taking $k$ sufficiently large, one has

\begin{cases}-\tfrac{\epsilon_{1}}{2}k+\tfrac{\sqrt{k}}{2}O(1)&<0,\\ -\tfrac{1}{2}\epsilon_{0}\epsilon_{1}+\tfrac{3}{2}k^{-\tfrac{1}{6}}O(1)&<0.\end{cases}

Thus, we can find $k_{1}>0$ , such that $R_{X\bar{X}W\bar{W}}<0$ for any $k\geq k_{1}$ .

Case IV: For the case where $\|b\|=\|d\|=1$ and $\|a\|\geq\|c\|$ , this follows from Case III, as

R_{X\bar{X}W\bar{W}}=R_{W\bar{W}X\bar{X}}<0

for any $k\geq k_{1}$ .

Case V: If $\|b\|\neq 0$ and $\|d\|\neq 0$ , then

R_{X\bar{X}W\bar{W}}=\|b\|^{2}\|d\|^{2}R_{X^{\prime}\overline{X^{\prime}}W^{\prime}\overline{W^{\prime}}},

where $X^{\prime}=\tfrac{1}{\|b\|}X$ and $W^{\prime}=\tfrac{1}{\|d\|}W$ . If we assume that $X^{\prime}=a^{\prime\alpha}\tfrac{\delta}{\delta z^{\alpha}}+b^{\prime i}\tfrac{\partial}{\partial v^{i}}$ , $W^{\prime}=c^{\prime\alpha}\tfrac{\delta}{\delta z^{\alpha}}+d^{\prime i}\tfrac{\partial}{\partial v^{i}}$ , then $b^{\prime}=\tfrac{b}{\|b\|}$ , and so $\|b^{\prime}\|=1$ . Similarly, $\|d^{\prime}\|=1$ . By the above two cases, we obtain $R_{X\bar{X}W\bar{W}}<0$ for any $k\geq k_{1}$ .

In summary, we can find $k_{0},k_{1}>0$ , for any $k\geq\max\{k_{0},k_{1}\}$ , and for any two non-zero $(1,0)$ -type vectors $X,W$ , we have $R_{X\bar{X}W\bar{W}}<0$ . This completes the proof of Theorem 1.3. ∎

4. Applications

This section will consider the relative Kähler fibration comes from a holomorphic family of compact, canonically polarized manifolds. We assume that $p:\mathcal{X}\to\mathcal{B}$ is a holomorphic family of compact, canonically polarized manifolds, and each fiber is equipped with a Kähler-Einstein metric of negative scalar curvature. One can refer to [Sch12] for more details.

Let

\omega_{\mathcal{X}/\mathcal{B}}=\sqrt{-1}g_{i\bar{j}}(z,v)dv^{i}\wedge d\bar{v}^{j}

denote the smooth family of Kähler-Einstein metrics satisfying the equation

(4.1)

\tfrac{\partial^{2}}{\partial v^{i}\partial\bar{v}^{j}}\log\det(g_{i\bar{j}})=g_{i\bar{j}}.

Next, we define

\omega_{\mathcal{X}}=\sqrt{-1}\partial\bar{\partial}\log\det(g_{i\bar{j}})=\sqrt{-1}\partial\bar{\partial}\phi,

where $\phi:=\log\det(g_{i\bar{j}})$ . Hence $\omega_{\mathcal{X}}$ is a relative Kähler form on $\mathcal{X}$ . By equation (4.1), we have

e^{\phi}=\det(g_{i\bar{j}})=\det(\phi_{i\bar{j}}).

The geodesic curvature form satisfies the equation:

(1+\Box)c(\phi)_{\alpha\bar{\beta}}=\langle\mu_{\alpha},\mu_{\beta}\rangle,

where

\mu_{\alpha}=\bar{\partial}^{V}(\tfrac{\delta}{\delta z^{\alpha}})=-\partial_{\bar{l}}(\phi_{\alpha\bar{j}}\phi^{\bar{j}k})\tfrac{\partial}{\partial v^{k}}\otimes d\bar{v}^{l}\in\mathbb{H}^{0,1}(X_{z},T_{X_{z}})

is harmonic, and $\Box:=-\phi^{i\bar{j}}\partial_{i}\partial_{\bar{j}}$ ; see [Sch12, Proposition 3]. From [Sch12, Theorem 1], $c(\phi)$ is semi-positive and strictly positive in the horizontal directions for families that are not infinitesimally trivial. If the family is effectively parametrized, then $\Omega$ defined by

\Omega=k(p^{*}\omega_{\mathcal{B}})+\omega_{\mathcal{X}},

remains a Kähler metric on $\mathcal{X}$ even for $k=0$ .

Note that the Kähler-Einstein metric on each fiber is given by $\omega_{\mathcal{X}/\mathcal{B}}=\sqrt{-1}\phi_{i\bar{j}}dv^{i}\wedge d\bar{v}^{j}$ , which induces a Hermitian metric on the relative tangent bundle $\mathcal{V}=T_{\mathcal{X}/\mathcal{B}}$ as

\left\langle\tfrac{\partial}{\partial v^{i}},\tfrac{\partial}{\partial v^{j}}\right\rangle:=\phi_{i\bar{j}}.

The Hermitian vector bundle $(T_{\mathcal{X}/\mathcal{B}},\omega_{\mathcal{X}/\mathcal{B}})$ is Griffiths negative if

(4.2)

R^{\mathcal{V}}_{X\bar{X}V\bar{V}}:=\left\langle R^{\mathcal{V}}(X,\bar{X})V,V\right\rangle<0

for any non-zero vector $X\in T^{1,0}_{(z,v)}\mathcal{X}$ and non-zero vector $V\in\mathcal{V}_{(z,v)}$ .

In particular, the geodesic curvature form satisfies

	$\displaystyle c(\phi)$	$\displaystyle=\partial\bar{\partial}\phi\left(\tfrac{\delta}{\delta z^{\alpha}},\tfrac{\delta}{\delta z^{\beta}}\right)\sqrt{-1}dz^{\alpha}\wedge d\bar{z}^{\beta}$
		$\displaystyle=\partial\bar{\partial}\log\det(\phi_{i\bar{j}})\left(\tfrac{\delta}{\delta z^{\alpha}},\tfrac{\delta}{\delta z^{\beta}}\right)\sqrt{-1}dz^{\alpha}\wedge d\bar{z}^{\beta}$
		$\displaystyle=-R^{\mathcal{V}}_{\alpha\bar{\beta}i\bar{j}}\phi^{i\bar{j}}\sqrt{-1}dz^{\alpha}\wedge d\bar{z}^{\beta},$

which is strictly positive in the horizontal directions by (4.2). From Theorem 1.3, we obtain the following theorem.

Theorem 4.1.

Let $p:\mathcal{X}\to\mathcal{B}$ be a compact holomorphic fibration over a compact Kähler manifold $\mathcal{B}$ with negative holomorphic bisectional curvature. Suppose the fiber is equipped with a smooth family of Kähler-Einstein metrics such that the induced Hermitian metric on the relative tangent bundle $T_{\mathcal{X}/\mathcal{B}}$ is Griffiths negative. Then, there exist Kähler metrics on $\mathcal{X}$ with negative holomorphic bisectional curvature.

As an application, we can solve Problem 1.2 for the case where the fibers have dimension one.

Corollary 4.2.

Let $p:\mathcal{X}\to\mathcal{B}$ be a compact holomorphic fibration over a Kähler manifold $\mathcal{B}$ with negative holomorphic bisectional curvature. Suppose the fibration is a holomorphic family of compact Riemann surfaces of genus $\geq 2$ and is effectively parametrized. Then there exists a Kähler metric on $\mathcal{X}$ with negative holomorphic bisectional curvature.

Proof.

If each fiber has dimension one, then $K_{\mathcal{X}/\mathcal{B}}^{-1}=T_{\mathcal{X}/\mathcal{B}}$ , where $K_{\mathcal{X}/\mathcal{B}}^{-1}$ denotes the anti-canonical line bundle, which is the dual of the relative canonical bundle $K_{\mathcal{X}/\mathcal{B}}$ . By [Sch12, Theorem 1], the Ricci curvature of $(T_{\mathcal{X}/\mathcal{B}},\omega_{\mathcal{X}/\mathcal{B}})$ is negative. Therefore, $T_{\mathcal{X}/\mathcal{B}}$ is Griffiths negative since each fiber has dimension one. The proof is complete. ∎

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	$\displaystyle\|R_{Y\bar{Y}U\bar{V}}\|=\|R_{Y\bar{Y}V\bar{U}}\|$	$\displaystyle\leq\\|a\\|^{2}\\|c\\|\\|d\\|O(1),$
	$\displaystyle\|R_{Y\bar{Z}U\bar{U}}\|=\|R_{Z\bar{Y}U\bar{U}}\|$	$\displaystyle\leq\\|a\\|\\|b\\|\\|c\\|^{2}O(1),$
	$\displaystyle\|R_{Y\bar{Z}V\bar{V}}\|=\|R_{Z\bar{Y}V\bar{V}}\|$	$\displaystyle\leq\\|a\\|\\|b\\|\\|d\\|^{2}O(1),$
	$\displaystyle\|R_{Z\bar{Z}U\bar{V}}\|=\|R_{Z\bar{Z}V\bar{U}}\|$	$\displaystyle\leq\\|b\\|^{2}\\|c\\|\\|d\\|O(1),$
	$\displaystyle\|R_{Y\bar{Z}U\bar{V}}\|=\|R_{Z\bar{Y}V\bar{U}}\|$	$\displaystyle\leq\\|a\\|\\|b\\|\\|c\\|\\|d\\|O(1).$