Remark on semi-positive holomorphic sectional curvature and quasi-negative $k$-Ricci curvature

Abstract

We record two remarks. First, for a compact Kähler manifold with semi-positive holomorphic sectional curvature, the rational dimension of the MRC fibration is exactly the number of non-truly-flat directions. Second, for compact Kähler manifolds with quasi-negative $k$-Ricci curvature, $1<k<n$, or more generally with quasi-negative mixed curvature $C_{a,b}$ for $a,b>0$, the canonical bundle is ample.