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

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.