Weak convergence of complex Monge-Ampère operators on compact Hermitian manifolds

Abstract

Let $(X,\omega )$ be a compact Hermitian manifold and let $\{\beta \}\in H^{1,1}(X,\mathbb{R})$ be a real $(1,1)$-class with a smooth representative $\beta$, such that ${\int }_X{\beta }^n>0$. Assume that there is a bounded $\beta$-plurisubharmonic function $\rho$ on $X$. First, we provide a criterion for the weak convergence of non-pluripolar complex Monge-Ampère measures associated to a sequence of $\beta$-plurisubharmonic functions. Second, this criterion is utilized to solve a degenerate complex Monge-Ampère equation with an $L^1$-density. Finally, an $L^{\infty }$-estimate of the solution to the complex Monge-Ampère equation for a finite positive Radon measure is given.

36pages, Comments welcome!