Positive curvature operator, projective manifold and rational connectedness

Abstract

In his recent work \cite{Y1}, X. Yang proved a conjecture raised by Yau in 1982 (\cite{Yau82}), which states that any compact K\"{a}hler 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 K\"{a}hler, then $X$ is projective. Also, it is rationally connected manifold when $n=3$. This partially confirms the conjecture 1.11 \cite{Y1} which is proposed by X. Yang.