Slope stability and the Kobayashi-Hitchin correspondence for Big cohomology classes

In this paper, we introduce the notions of slope stability and the Hermitian-Einstein metric for big cohomology classes. Our 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 nef and big classes. Through this paper, the “bimeromorphic invariance” of slope stability and the existence of Hermitian-Einstein metrics play an essential role.