On first and second bm$k$-Ricci curvatures for Hermitian manifold

Abstract

This paper studies the basic properties of the non-positive or non-negative first and second $k$-Ricci curvatures.Firstly, we obtain the inequality for the Ricci curvature and the holomorphic sectional curvature. Secondly, we give vanishing theorems on compact Hermitian manifolds with positive first or second $k$-Ricci curvature.In particular, we prove that if a compact Kähler manifold has a Hermitian metric with a positive second $k$-Ricci curvature ($k\geq2$), then it must be a projective manifold.