Metric rigidity of Kahler manifolds with lower Ricci bounds and almost maximal volume

TL;DR

This paper proves that Kahler manifolds with lower Ricci bounds and nearly maximal volume are geometrically close to complex projective space, extending rigidity results to the Kahler setting using recent holomorphic and Einstein structure theorems.

Contribution

It establishes a new metric rigidity result for Kahler manifolds with Ricci bounds and almost maximal volume, combining recent holomorphic and Einstein structure theorems.

Findings

Kahler manifolds with lower Ricci bounds and near-maximal volume are Gromov-Hausdorff close to projective space.

The proof combines recent holomorphic rigidity results with structure theorems for almost Einstein manifolds.

The result is a complex analog of classical Riemannian shape theorems for manifolds with almost maximal volume.

Abstract

In this short note we prove that a Kahler manifold with lower Ricci curvature bound and almost maximal volume is Gromov-Hausdorff close to the projective space with the Fubini-Study metric. This is done by combining the recent results of Kewei Zhang and Yuchen Liu on holomorphic rigidity of such Kahler manifolds with the structure theorem of Tian-Wang for almost Einstein manifolds. This can be regarded as the complex analog of the result on Colding on the shape of Riemannian manifolds with almost maximal volume