Slope stability and the Kobayashi–Hitchin correspondence for big cohomology classes

Source abstract

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.