Degenerate Hermitian geometry and curvatures of holomorphic fibrations

TL;DR

This paper develops a theory for calculating curvature tensors of metrics on holomorphic fibrations, especially degenerate Hermitian forms, and applies it to Grassmannian bundles to analyze their curvature properties.

Contribution

It introduces a framework for curvature calculations involving degenerate Hermitian forms and extends classical equations to this setting, with applications to Grassmannian bundles.

Findings

Curvature tensors for degenerate Hermitian forms are explicitly calculated.

A version of the Codazzi-Griffiths equations is established for this setting.

Grassmannian bundles are shown to have positive holomorphic sectional curvature if the base does.

Abstract

We calculate curvature tensors of metrics on the total spaces of holomorphic fibrations. Our main tool is a theory of Chern connections and curvature forms for possibly degenerate Hermitian forms on holomorphic vector bundles. We prove a version of the Codazzi-Griffiths equations for curvatures of sub- and quotient bundles in that setting and apply them to the study of honest Hermitian metrics on fibrations. We apply this to calculate the curvature of a metric on Grassmannian bundles and prove they have positive holomorphic sectional curvature if the base does.