K\"ahler metrics of negative holomorphic (bi)sectional curvature on a compact relative K\"ahler fibration

TL;DR

This paper constructs K"ahler metrics with negative holomorphic sectional and bisectional curvature on the total space of certain compact relative K"ahler fibrations, extending curvature properties from fibers and base to the total space.

Contribution

It provides explicit constructions of negatively curved K"ahler metrics on total spaces of fibrations with negatively curved fibers and bases, addressing a specific open question for one-dimensional fibers.

Findings

Constructed K"ahler metrics with negative holomorphic sectional curvature on total spaces.

Constructed K"ahler metrics with negative holomorphic bisectional curvature on total spaces.

Resolved a question by To and Yeung for one-dimensional fibers.

Abstract

For a compact relative K\"ahler fibration over a compact K\"ahler manifold with negative holomorphic sectional curvature, if the relative K\"ahler form on each fiber also exhibits negative holomorphic sectional curvature, we can construct K\"ahler 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 K\"ahler metric with negative holomorphic bisectional curvature, we can also construct K\"ahler 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 K\"ahler metrics with negative holomorphic bisectional curvature, and the fibers are one-dimensional, we can explicitly construct K\"ahler 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.