Almost quarter-pinched K\"ahler metrics and Chern numbers

TL;DR

This paper shows that compact Kähler manifolds with sectional curvatures nearly 1/4-pinched have Chern number ratios close to those of complex hyperbolic space, implying certain geometric obstructions.

Contribution

It establishes a quantitative link between curvature pinching and Chern number ratios, extending understanding of Kähler geometry and hyperbolic manifolds.

Findings

Chern number ratios approximate those of complex hyperbolic space

Mostow-Siu surfaces cannot admit nearly 1/4-pinched Kähler metrics

Provides new obstructions to certain Kähler metrics with specific curvature pinching

Abstract

We prove that compact K\"ahler manifolds whose sectional curvatures are close to 1/4-pinched have ratios of Chern numbers close to the corresponding ratios of a complex hyperbolic space form. We deduce that the Mostow-Siu surfaces (and their three-dimensional analogues constructed by the first author) do not admit K\"ahler metrics with pinching close to 1/4.