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In a previous paper, we proved that a projective K\"ahler manifold ofpositive total scalar curvature is uniruled. At the other end of the spectrum,it is a well-known theorem of Campana and Koll\'ar-Miyaoka-Mori that aprojective K\"ahler manifold of positive Ricci curvature is rationallyconnected. In the present work, we investigate the intermediate notion of$k$-positive Ricci curvature and prove that for a projective $n$-dimensionalK\"ahler manifold of $k$-positive Ricci curvature the MRC fibration has genericfibers of dimension at least $n-k+1$. We also establish an analogous result forprojective K\"ahler manifolds of semi-positive holomorphic sectional curvaturebased on an invariant which records the largest codimension of maximalsubspaces in the tangent spaces on which the holomorphic sectional curvaturevanishes. In particular, the latter result confirms a conjecture of S.-T. Yauin the projective case.

On projective K\"ahler manifolds of partially positive curvature and rational connectedness