On Grauert–Riemenschneider Type Criteria

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 ?■ω~l= for all l, i.e., if T is a closed positive current on X such that ∫_XT_(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 ?■ω = 0, and T is a closed positive current with analytic singularities,such that ∫_XT_(ac)~n 0, then the class {T} is big and X is Kahler.

On Grauert—Riemenschneider Type Criteria

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 ?√?ω^l=for all l, i.e., if T is a closed positive current on X such that ?_X T_acⁿ>0, then the class {T} is big and X is Kähler. Finally, as an easy observation, we point out that Nguyen's result can be generalized as follows:if ?√?ω=0, and T is a closed positive current with analytic singularities, such that ?_X T_acⁿ>0, then the class {T} is big and X is Kähler.