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

Xueyuan Wan

TL;DR

The paper tackles when the total space of a compact relative Kähler fibration $p:\mathcal{X}\to\mathcal{B}$ can carry Kähler metrics with negative holomorphic (bi)sectional curvature, given negative curvature on the base and fibers and Griffiths negativity of the relative tangent bundle. It constructs a family of Kähler metrics $\Omega = k(p^*\omega_{\mathcal{B}}) + \omega_{\mathcal{X}}$ and analyzes their curvature as $k\to\infty$, proving that negative holomorphic sectional curvature on base and fibers implies negative HSC on the total space for large $k$, and that Griffiths negativity of $T_{\mathcal{X}/\mathcal{B}}$ combined with negative base HB curvature yields negative holomorphic bisectional curvature on the total space (with several cases covered). The results include explicit applications to holomorphic families of canonically polarized manifolds, showing that the total space admits metrics with negative HB curvature, and in the one-dimensional fiber case, resolve To–Yeung’s question by constructing such metrics explicitly. Overall, the work extends negative curvature phenomena from fibers and base to total spaces of fibrations, providing concrete geometric criteria and large-scale asymptotics that guarantee desired curvature properties.

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.

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

TL;DR

The paper tackles when the total space of a compact relative Kähler fibration can carry Kähler metrics with negative holomorphic (bi)sectional curvature, given negative curvature on the base and fibers and Griffiths negativity of the relative tangent bundle. It constructs a family of Kähler metrics and analyzes their curvature as , proving that negative holomorphic sectional curvature on base and fibers implies negative HSC on the total space for large , and that Griffiths negativity of combined with negative base HB curvature yields negative holomorphic bisectional curvature on the total space (with several cases covered). The results include explicit applications to holomorphic families of canonically polarized manifolds, showing that the total space admits metrics with negative HB curvature, and in the one-dimensional fiber case, resolve To–Yeung’s question by constructing such metrics explicitly. Overall, the work extends negative curvature phenomena from fibers and base to total spaces of fibrations, providing concrete geometric criteria and large-scale asymptotics that guarantee desired curvature properties.

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.
Paper Structure (7 sections, 7 theorems, 89 equations)

This paper contains 7 sections, 7 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Griffiths negativity and holomorphic (bi)sectional curvature
  3. Curvature of Kähler metrics on a fibration
  4. Definition of Kähler metrics
  5. Negativity/Positivity of holomorphic sectional curvature
  6. Negativity of holomorphic bisectional curvature
  7. Applications

Key Result

Theorem 1.2

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}/\mat

Theorems & Definitions (21)

  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • ...and 11 more