Optimal pinching for the holomorphic sectional curvature of Hitchin's metrics on Hirzebruch surfaces

TL;DR

This paper constructs specific Hodge metrics on Hirzebruch surfaces with optimally pinched positive holomorphic sectional curvature, extending Hitchin's work and analyzing product metrics for curvature pinching.

Contribution

It demonstrates the existence of Hodge metrics with optimal curvature pinching on all Hirzebruch surfaces and generalizes curvature pinching results for product manifolds.

Findings

Existence of Hodge metrics with 1/(1+2n)^2 pinched curvature on Hirzebruch surfaces.

General curvature pinching result for product of Hermitian manifolds.

Extension of Hitchin's metrics analysis to all Hirzebruch surfaces.

Abstract

The main result of this note is that, for each $n\in \{1,2,3,\ldots\}$, there exists a Hodge metric on the $n$-th Hirzebruch surface whose positive holomorphic sectional curvature is $\frac{1}{(1+2n)^2}$-pinched. The type of metric under consideration was first studied by Hitchin in this context. In order to address the case $n=0$, we prove a general result on the pinching of the holomorphic sectional curvature of the product metric on the product of two Hermitian manifolds $M$ and $N$ of positive holomorphic sectional curvature.