Properties of compact complex manifolds carrying closed positive currents

Abstract

We show that a compact complex manifold is Moishezon if and only if it carries a strictly positive, integral (1, 1)-current. We then study holomorphic line bundles carrying singular hermitian metrics with semi-positive curvature currents, and we give some cases in which these line bundles are big. We use these cases to provide sufficient conditions for a compact complex manifold to be Moishezon in terms of the existence of certain semi-positive, integral (1,1)-currents. We also show that the intersection number of two closed semi-positive currents of complementary degrees on a compact complex manifold is positive when the intersection of their singular supports is contained in a Stein domain.