Non-collapsing volume estimate for local K\"ahler metrics in big cohomology classes

TL;DR

This paper establishes a uniform local volume estimate for singular K"ahler metrics in big cohomology classes, extending mixed energy estimates to this setting, which aids in understanding metric non-collapsing behavior.

Contribution

It introduces a novel non-collapsing volume estimate for singular metrics in big cohomology classes, generalizing energy estimates to this complex geometric context.

Findings

Proves a uniform local volume non-collapsing estimate for singular metrics.

Extends mixed energy estimates to big cohomology classes.

Provides tools for analyzing metric behavior in complex geometry.

Abstract

We prove a uniform local non-collapsing volume estimate for a large family of singular metrics in the big cohomology classes, which are K\"ahler on an open Euclidean subset of the manifold. The key ingredient is a generalization of a mixed energy estimate for functions in the complex Sobolev space to the setting of big cohomology classes.