Slope Stable Sheaves and Hermitian-Einstein Metrics on Normal Varieties with Big Cohomology Classes

TL;DR

This paper extends the concepts of slope stability and Hermitian-Einstein metrics to normal varieties with big cohomology classes, establishing a Kobayashi-Hitchin correspondence and proving related inequalities.

Contribution

It introduces slope stability and Hermitian Einstein metrics for big classes on normal varieties, establishing a Kobayashi-Hitchin correspondence and invariance properties.

Findings

Kobayashi-Hitchin correspondence for big classes on normal spaces

Proof of Bogomolov Gieseker inequality for slope stable sheaves

Bimeromorphic invariance of slope stability and Hermitian Einstein metrics

Abstract

In this paper, we introduce the notions of slope stability and the Hermitian Einstein metric for big cohomology classes. The main result is the Kobayashi Hitchin correspondence on compact normal spaces with big classes admitting the birational Zariski decomposition with semiample positive part. We also prove the Bogomolov Gieseker inequality for slope stable sheaves with respect to big and nef classes. Through this paper, the bimeromorphic invariance of slope stability and the existence of Hermitian Einstein metrics plays an essential role.