Uniform diameter estimates for Kaehler metrics in big cohomology classes

TL;DR

This paper extends diameter bounds and volume non-vanishing results for Kähler metrics to big cohomology classes, introducing a uniform diameter estimate based on integrability and stability of Monge-Ampère equations.

Contribution

It generalizes diameter estimates to big cohomology classes and develops new stability results for complex Monge-Ampère equations with prescribed singularities.

Findings

Established a uniform diameter estimate for families of Kähler metrics.

Proved local non-vanishing of volumes in big cohomology classes.

Utilized stability properties of complex Monge-Ampère equations.

Abstract

We generalize previous diameter estimates and local non-vanishing of volumes for Kaehler metrics to the case of big cohomology classes. In our proof, among other things, we will prove a uniform diameter estimate for a family of smooth Kaehler metrics only involving an integrability condition. We also have to use fine stability properties of complex Monge-Ampere equations with prescribed singularities.