On almost nonpositive $k$-Ricci curvature

TL;DR

This paper introduces the concept of almost nonpositive $k$-Ricci curvature in K"ahler manifolds, establishes bounds for the twisted K"ahler-Ricci flow, and proves that such manifolds have nef canonical line bundles.

Contribution

It defines almost nonpositive $k$-Ricci curvature, provides flow existence bounds, and shows manifolds with this curvature condition have nef canonical bundles.

Findings

Lower bound for twisted K"ahler-Ricci flow existence time.

Manifolds with almost nonpositive $k$-Ricci curvature have nef canonical line bundle.

Abstract

Motivated by the recent work of Chu-Lee-Tam on the nefness of canonical line bundle for compact K\"{a}hler manifolds with nonpositive $k$-Ricci curvature, we consider a natural notion of {\em almost nonpositive $k$-Ricci curvature}, which is weaker than the existence of a K\"{a}hler metric with nonpositive $k$-Ricci curvature. When $k=1$, this is just the {\em almost nonpositive holomorphic sectional curvature} introduced by Zhang. We firstly give a lower bound for the existence time of the twisted K\"{a}hler-Ricci flow when there exists a K\"{a}hler metric with $k$-Ricci curvature bounded from above by a positive constant. As an application, we prove that a compact K\"{a}hler manifold of almost nonpositive $k$-Ricci curvature must have nef canonical line bundle.