# On Projective Kähler Manifolds of Partially Positive Curvature and Rational Connectedness

### Gordon Heier

Department of Mathematics, University of Houston, 4800 Calhoun Road, Houston, TX 77204, USA### Bun Wong

Department of Mathematics, UC Riverside, 900 University Avenue, Riverside, CA 92521, USA

## Abstract

In a previous paper, we proved that a projective Kähler manifold of positive total scalar curvature is uniruled. At the other end of the spectrum, it is a well-known theorem of Campana and Kollár-Miyaoka-Mori that a projective Kähler 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 Kähler 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 Kähler 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.

## Cite this article

Gordon Heier, Bun Wong, On Projective Kähler Manifolds of Partially Positive Curvature and Rational Connectedness. Doc. Math. 25 (2020), pp. 219–238

DOI 10.4171/DM/745