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, and some rigidity and degeneracy theorems. For instance, we show that there is no non-constant holomorphic map from a compact Hermitian manifold with positive (resp. non-negative) ℓ-second Ricci curvature to a Hermitian manifold with non-positive (resp. negative) real bisectional curvature. These theorems generalize the results [5, 6] proved recently by L. Ni on Kähler manifolds to Hermitian manifolds. We also derive an integral inequality for a holomorphic map between Hermitian manifolds.

在本文中，我们推导出了一些\(\partial \overline \partial \) -Bochner 公式，用于 Hermitian 流形之间的全纯映射。作为应用，我们证明了一些 Schwarz 引理类型估计，以及一些刚性和简并定理。例如，我们证明了从具有正（非负）ℓ -second Ricci 曲率的紧凑 Hermitian 流形到具有非正（分别为负）实二分曲率的 Hermitian 流形的非恒定全纯映射。这些定理将 L. Ni 最近在 Kähler 流形上证明的结果 [5, 6] 推广到 Hermitian 流形。我们还推导出了 Hermitian 流形之间的全纯映射的积分不等式。