On compact complex manifolds with nonnegative mean curvature

In this paper, we establish a structure theorem for projective manifolds with semipositive mean curvature. We prove that the rational dimension of a projective manifold with k k -positive mean curvature is at least k k. We give an estimate of Hodge number h p, 0 h^p, 0 for compact complex manifolds with mean curvature bounded below. As an application, we get a maximal Betti number rigidity for compact HN-nonnegative Kähler manifolds.