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

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 $\underset{‾}{\mathrm{V}\mathrm{o}\mathrm{l}}(\beta )>0$. In this paper, we establish solutions to the degenerate complex Monge-Ampère 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.