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

RESUMO

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 $$\lambda_1+\dots+\lambda_{\alpha}\ge -{\alpha}\theta(n,{\alpha})\bar\lambda,\; {\alpha}\le \frac{n}{2}$$ must have nonnegative constant holomorphic sectional curvature.