On Grauert—Riemenschneider Type Criteria

Abstract

Let (X, ω) be a compact Hermitian manifold of complex dimension n. In this article, we first survey recent progress towards Grauert-Riemenschneider type criteria. Secondly, we give a simplified proof of Boucksom’s conjecture given by the author under the assumption that the Hermitian metric ω satisfies \(\partial \overline \partial {\omega ^l} = {\rm{for}}\;{\rm{all}}\;l\), i.e., if T is a closed positive current on X such that ∫X\(T_{ac}^n>0\), then the class {T} is big and X is Kahler. 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 ∫X\(T_{ac}^n>0\), then the class {T} is big and X is Kahler.