摘要
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.
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.
基金
supported by the Fundamental Research Funds for the Central Universities and by the NSFC(Grant No.11701031)
