Degenerate complex Monge-Ampère type equations on compact Hermitian manifolds and applications II

Abstract

Let $(X,ω)$ be a compact Hermitian manifold of complex dimension $n$, equipped with a Hermitian metric $ω$. Let $β$ be a possibly non-closed smooth $(1,1)$-form on $X$ such that $\int_Xβ^n>0$. Assume that there is a bounded $β$-plurisubharmonic function $ρ$ on $X$ and $\underline{\mathrm{Vol}}(β) > 0$. In this paper, we establish solutions to the degenerate complex Monge-Ampère equations on $X$ within the Bott-Chern space of $β$ (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.