K{ä}hler Einstein manifolds and the Calabi curvature operator

Abstract

In this paper, we study the Calabi curvature operator on K{ä}hler manifolds. First, we prove that if the Calabi curvature operator on K{ä}hler manifolds satisfies $\frac{n\left(n+1\right)}{2}$-positive (nonnegative), $\frac{n+1}{2}$-positive (nonnegative), and $\left(n-1\right)$-positive (nonnegative), then the scalar curvature, Ricci curvature, and orthogonal Ricci curvature are positive (nonnegative), respectively. Second, we show that any compact K{ä}hler Einstein manifold satisfying the condition $${\lambda }_1+\cdots +{\lambda }_{\alpha }\ge -\alpha \theta (n,\alpha )\bar{\lambda },\alpha \le \frac{n}{2}$$ must have nonnegative constant holomorphic sectional curvature.