Degenerate complex Monge-Amp\`ere type equations on compact Hermitian manifolds and applications II

Abstract

Let $(X,\omega)$ be a compact Hermitian manifold of complex dimension $n$, equipped with a Hermitian metric $\omega$. Let $\beta$ be a possibly non-closed smooth $(1,1)$-form on $X$ such that $\int_X\beta^n>0$. Assume that there is a bounded $\beta$-plurisubharmonic function $\rho$ on $X$ and $\underline{\mathrm{Vol}}(\beta) > 0$. In this paper, we establish solutions to the degenerate complex Monge-Amp\`ere equations on $X$ within the Bott-Chern space of $\beta$ (as introduced by Boucksom-Guedj-Lu) and derive stability results for these solutions. As applications, we provide partial resolutions to the extended Tosatti-Weinkove conjecture and Demailly-P\u aun conjecture.