Geometry of Hermitian manifolds (Q2888826)

On Hermitian manifolds, the second Ricci curvature tensors of various metric connections are closely related to the geometry of Hermitian manifolds. Using the generalized Bochner-Kodaira formulas established in this paper, for any Hermitian complex vector bundle (Riemannian real vector bundle) with an arbitrary metric connection over a compact Hermitian manifold, they derive various vanishing theorems for Hermitian manifolds and complex vector bundles by the second Ricci curvature tensors. For example, they prove that if the second Ricci-Chern curvature of a compact Hermitian manifold or the Hermitian-Ricci curvature of a compact balanced Hermitian manifold is nonnegative and positive at least at one point, then \(H^{p,0}_{\bar{\partial}}(M)=0\) for \(1\leq p\leq \text{dim}_{\mathbb C}M\). Finally the paper introduces a natural geometric flow on Hermitian manifolds by using the second Ricci curvature tensor