A note on the almost one half holomorphic pinching

TL;DR

This paper investigates pinching constants in compact Kähler manifolds with positive holomorphic sectional curvature, establishing a gap theorem analogous to results in Riemannian geometry.

Contribution

It extends the understanding of curvature pinching by proving a new gap theorem for Kähler manifolds, inspired by prior work on Riemannian manifolds.

Findings

Proves a gap theorem for Kähler manifolds with almost quarter-pinched holomorphic sectional curvature.

Establishes bounds on pinching constants for positive holomorphic sectional curvature.

Links curvature pinching phenomena in Kähler geometry to classical Riemannian results.

Abstract

Motivated by a previous work of Zheng and the second named author, we study pinching constants of compact K\"ahler manifolds with positive holomorphic sectional curvature. In particular we prove a gap theorem following the work of Petersen and Tao on Riemannian manifolds with almost quarter-pinched sectional curvature.