The $\partial\overline{\partial}$-Bochner formulas for holomorphic mappings between Hermitian manifolds and their applications

TL;DR

This paper develops $ ext{ extbackslash}partial ext{ extbackslash}overline{ ext{ extbackslash}partial}$-Bochner formulas for holomorphic maps between Hermitian manifolds, leading to new Schwarz lemma estimates, rigidity, and degeneracy theorems that extend previous Kähler results.

Contribution

It introduces new $ ext{ extbackslash}partial ext{ extbackslash}overline{ ext{ extbackslash}partial}$-Bochner formulas for Hermitian manifolds and applies them to derive generalized Schwarz lemmas and rigidity theorems.

Findings

No non-constant holomorphic maps from certain Hermitian manifolds with positive Ricci curvature to those with non-positive curvature.

Generalization of Ni's results from Kähler to Hermitian manifolds.

An integral inequality for holomorphic maps between Hermitian manifolds.

Abstract

In this paper, we derive some $\partial\overline{\partial}$-Bochner formulas for holomorphic maps between Hermitian manifolds. As applications, we prove some Schwarz lemma type estimates, rigidity and degeneracy theorems. For instance, we show that there is no non-constant holomorphic map from a comapct Hermitian manifold with positive (resp. non-negative) $\ell$-second Ricci curvature to a Hermitian manifold with non-positive (resp. negative) real bisectional curvature. These theorems generalize the results \cite{Ni1,Ni2} proved recently by L. Ni on K\"{a}hler manifolds to Hermitian manifolds. We also derive an integral inequality for holomorphic map between Hermitian manifolds.