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