Monge-Amp\`ere type equation on compact Hermitian manifolds

Abstract

Given a cohomology $(1,1)$-class $\{\beta\}$ of compact Hermitian manifold $(X,\omega)$ possessing a bounded potential and fixed a model potential $\phi$, motivated by Darvas-Di Nezza-Lu and Li-Wang-Zhou's work, we show that degenerate complex Monge-Amp\`ere equation $(\beta+dd^c \varphi)^n=e^{\lambda \varphi}\mu$ has a unique solution in the relative full mass class $\mathcal{E}(X,\beta,\phi)$, where $\mu$ is a non-pluripolar measure on $X$ and $\lambda\geq0$ is a fixed constant. As an application, we give an explicit description of Lelong numbers of elements in $\mathcal{E}(X,\beta,\phi)$ which generalized a theorem of Darvas-Di Nezza-Lu in the Hermitian context.