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

Abstract

The main result of this note is that, for each $n\in \{1,2,3,\dots \}$, 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.

to appear in Volume 654 of Contemporary Mathematics

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• Hermitian manifolds with semi-positive holomorphic sectional curvature