Homogeneous Hermitian manifolds and special metrics

TL;DR

This paper studies non-Kaehler homogeneous complex manifolds, especially Calabi-Eckmann manifolds, proving the existence and discussing the uniqueness of invariant Hermitian metrics that are Chern-Einstein.

Contribution

It establishes the existence of invariant Chern-Einstein Hermitian metrics on certain non-Kaehler homogeneous manifolds, including Calabi-Eckmann manifolds.

Findings

Existence of invariant Chern-Einstein metrics on these manifolds

Discussion of the uniqueness of such metrics

Extension of Hermitian geometry to non-Kaehler homogeneous spaces

Abstract

We consider non-Kaehler compact complex manifolds which are homogeneous under the action of a compact Lie group of biholomorphisms and we investigate the existence of special (invariant) Hermitian metrics on these spaces. We focus on a particular class of such manifolds comprising the case of Calabi-Eckmann manifolds and we prove the existence of an invariant Hermitian metric which is Chern-Einstein, namely whose second Ricci tensor of the associated Chern connection is a positive multiple of the metric itself. The uniqueness is also discussed.