ON PROJECTIVE KAHLER MANIFOLDS OF PARTIALLY POSITIVE CURVATURE AND RATIONAL CONNECTEDNESS

In a previous paper, we proved that a projective Kahler manifold of positive total scalar curvature is uniruled. At the other end of the spectrum, it is a well-known theorem of Campana and Kollar-Miyaoka-Mori that a projective Kahler manifold of positive Ricci curvature is rationally connected. In the present work, we investigate the intermediate notion of k-positive Ricci curvature and prove that for a projective n-dimensional Kahler manifold of k-positive Ricci curvature the MRC fibration has generic fibers of dimension at least n - k + 1. We also establish an analogous result for projective Kahler manifolds of semi-positive holomorphic sectional curvature based on an invariant which records the largest codimension of maximal subspaces in the tangent spaces on which the holomorphic sectional curvature vanishes. In particular, the latter result confirms a conjecture of S.-T. Yau in the projective case.