# Positive curvature operator, projective manifold and rational connectedness

## TL;DR

This paper proves that compact Hermitian manifolds with positive real bisectional curvature have vanishing certain Hodge numbers and are projective if Kähler, additionally showing rational connectedness in dimension three, thus supporting a conjecture by X. Yang.

## Contribution

It establishes new vanishing theorems for Hodge numbers and links positive curvature conditions to projectivity and rational connectedness, advancing understanding of curvature and complex geometry.

## Key findings

- Vanishing of specific Hodge numbers for manifolds with positive real bisectional curvature
- Compact Kähler manifolds with this curvature are projective
- 3-dimensional cases are rationally connected

## 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.