On Grauert-Riemenschneider type criterions

Abstract

Let $(X,\omega)$ be a compact Hermitian manifold of complex dimension $n$. In this article, we first survey recent progress towards Grauert-Riemenschneider type criterions. Secondly, we give a simplified proof of Boucksom's conjecture given by the author under the assumption that the Hermitian metric $\omega$ satisfies $\partial\overline{\partial}\omega^l=$ for all $l$, i.e., if $T$ is a closed positive current on $X$ such that $\int_XT_{ac}^n>0$, then the class $\{T\}$ is big and $X$ is K\"{a}hler. Finally, as an easy observation, we point out that Nguyen's result can be generalized as follows: if $\partial\overline{\partial}\omega=0$, and $T$ is a closed positive current with analytic singularities, such that $\int_XT^n_{ac}>0$, then the class $\{T\}$ is big and $X$ is K\"{a}hler.