The complex Monge–Ampère type equation on compact Hermitian manifolds and applications

Abstract

We prove the existence and uniqueness of continuous solutions to the complex Monge–Ampère type equation with the right hand side in Lp, p>1, on compact Hermitian manifolds. Next, we generalise results of Eyssidieux, Guedj and Zeriahi [17,18] to compact Hermitian manifolds which a priori are not in the Fujiki class. These generalisations lead to a number of applications: we obtain partial results on a conjecture of Tosatti and Weinkove [40] and on a weak form of a conjecture of Demailly and Paun [11].