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