Compact Kähler manifolds with partially semi-positive curvature

Abstract

In this paper, we study MRC fibrations of compact Kähler manifolds with partially semi-positive curvature. We first prove that a compact Kähler manifold is rationally connected if its tangent bundle is BC-$p$ positive for all $1\le p\le \dim X$. As applications, we confirm a conjecture that any compact Kähler manifold with positive orthogonal Ricci curvature must be rationally connected, and generalize a result of Heier-Wong and Yang to the conformally Kähler case. The second result concern structure theorems for two immediate curvature conditions. We prove that, a compact Kähler manifold with $k$-semi-positive Ricci curvature or semi-positive $k$-scalar curvature, either the rational dimension $\ge n-k+1$ or it admits a locally constant fibration $f:X\to Y$ such that the fibre is rationally connected and the image $Y$ is Ricci-flat.

29 pages, final version: typos corrected, to appear in the Transactions of the American Mathematical Society