A classification of locally Chern homogeneous Hermitian manifolds

Abstract

We apply the algebraic consideration of holonomy systems to study Hermitian manifolds whose Chern connection is Ambrose-Singer and prove structure theorems for such manifolds. The main result (Theorem 1.2) asserts that the universal cover of such a Hermitian manifold must be the product of a complex Lie group and Hermitian symmetric spaces, which was previously proved up to complex dimension four by the authors. This in some sense is the Hermitian version of Cartan's classification of Hermitian symmetric spaces. We also obtain results on Hermitian manifolds whose Bismut connection is Ambrose-Singer when the complex dimension $n\le 4$. Furthermore we discuss a project of classifying such manifolds via Alekseevski\u{i} and Kimel\!\'{}\!fel\!\'{}\!d type theorems, which we establish for the family of the Gauduchon connections on a compact Hermitian manifold except for the Bismut connection.