Hermitian Metrics of Positive Holomorphic Sectional Curvature on Fibrations

TL;DR

This paper proves that the total space of a compact fibration inherits a Hermitian metric with positive holomorphic sectional curvature if both base and fibers have such metrics, using a warped product approach.

Contribution

It establishes a new result linking positive holomorphic sectional curvature of base and fibers to the total space without relying on subadditivity properties.

Findings

Total space inherits positive holomorphic sectional curvature

Uses warped product metric construction

Does not rely on Grauert-Reckziegel and Wu's subadditivity

Abstract

The main result of this note essentially is that if the base and fibers of a compact fibration carry Hermitian metrics of positive holomorphic sectional curvature, then so does the total space of the fibration. The proof is based on the use of a warped product metric as in the work by Cheung in case of negative holomorphic sectional curvature, but differs in certain key aspects, e.g., in that it does not use the subadditivity property for holomorphic sectional curvature due to Grauert-Reckziegel and Wu.