On almost nonpositive $k$-Ricci curvature

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.