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

TL;DR

This paper explores the geometric implications of intermediate positive curvature conditions on projective K"ahler manifolds, establishing new bounds on the dimensions of fibers in the MRC fibration and confirming a conjecture of Yau.

Contribution

It introduces new results linking $k$-positive Ricci curvature and semi-positive holomorphic sectional curvature to the structure of rationally connected fibrations.

Findings

For $k$-positive Ricci curvature, the MRC fibration has fibers of dimension at least $n-k+1.

Semi-positive holomorphic sectional curvature implies certain bounds on the maximal subspaces with zero curvature.

Confirms Yau's conjecture in the projective case regarding holomorphic sectional curvature.

Abstract

In a previous paper, we proved that a projective K\"ahler manifold of positive 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 a projective K\"ahler 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 K\"ahler 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 K\"ahler 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.