Vanishing theorems on compact Chern-K\"{a}hler-like Hermitian manifolds

TL;DR

This paper proves vanishing theorems for holomorphic tensor fields on compact Chern-Kähler-like Hermitian manifolds under curvature conditions, extending classical results from Kähler geometry.

Contribution

It establishes new vanishing theorems for holomorphic tensors on a broader class of Hermitian manifolds with curvature conditions, inspired by classical Kähler results.

Findings

Holomorphic tensor fields are trivial under certain curvature conditions.

Vanishing theorems extend classical Kähler results to Hermitian manifolds.

Proofs are inspired by ideas from Yang and Zheng.

Abstract

We show that, under the definiteness of holomorphic sectional curvature, the spaces of some holomorphic tensor fields on compact Chern-K\"{a}hler-like Hermitian manifolds are trivial. These can be viewed as counterparts to Bochner's classical vanishing theorems on compact K\"{a}hler manifolds under the definiteness of Ricci curvature or the existence of K\"{a}hler-Einstein metrics. Our proof is inspired by and based on some ideas due to X. Yang and L. Ni-F. Zheng.