Positive curvature operator, projective manifold and rational connectedness

Abstract

In his recent work [20], X. Yang proved a conjecture raised by Yau in 1982 ([25]), which states that any compact Kähler manifold with positive holomorphic sectional curvature must be projective. In this note, we prove that any compact Hermitian manifold X with positive real bisectional curvature, its hodge number h 1,0 = h 2,0 = h n−1,0 = h n,0 = 0. In particular, if in addition X is Kähler, then X is projective. Also, it is rationally connected manifold when n = 3. This partially confirms the conjecture 1.11 [20] which is proposed by X. Yang.