K\"{a}hler manifolds with an almost $1/4$-pinched metric

TL;DR

This paper constructs the first known examples of non-locally symmetric Kähler manifolds with an almost negatively 1/4-pinched metric, extending the concept of pinched metrics to complex hyperbolic geometry.

Contribution

It introduces a novel almost negatively 1/4-pinched metric on specific compact Kähler manifolds that are not locally symmetric, expanding the class of known examples.

Findings

First examples of non-locally symmetric Kähler manifolds with such metrics

Generalization of Gromov-Thurston's pinched metric to complex hyperbolic setting

These manifolds cannot admit negatively quarter-pinched Riemannian metrics

Abstract

In this paper we construct an almost negatively $1/4$-pinched Riemannian metric on a class of compact manifolds recently discovered by Stover and Toledo in [17]. It is known that these manifolds are K\"{a}hler and not locally symmetric. These are the first known examples of not locally symmetric K\"{a}hler manifolds admitting such a metric and, via the result of Hernandez [9] and Yau and Zheng [18], these manifolds cannot admit a negatively quarter-pinched Riemannian metric. This metric is also interesting because it is a generalization to the complex hyperbolic setting of the famous pinched metric constructed by Gromov and Thurston in [8].