Rigidity of pseudo-Hermitian homogeneous spaces of finite volume

TL;DR

This paper proves that pseudo-Hermitian homogeneous spaces with finite volume are necessarily compact, with their isometry groups also compact, and analyzes their structure via the Tits fibration, revealing new rigidity properties.

Contribution

It establishes the compactness of pseudo-Hermitian homogeneous spaces of finite volume and analyzes their automorphism groups and fibrations, extending known results on their structure.

Findings

Such spaces are compact and have compact isometry groups.

The Tits fibration of these spaces has a torus as fiber.

Examples show these spaces do not always split as products.

Abstract

Let $M$ be a pseudo-Hermitian homogeneous space of finite volume. We show that $M$ is compact and the identity component $G$ of the group of holomorphic isometries of $M$ is compact. If $M$ is simply connected, then even the full group of holomorphic isometries is compact. These results stem from a careful analysis of the Tits fibration of $M$, which is shown to have a torus as its fiber. The proof builds on foundational results on the automorphisms groups of compact almost pseudo-Hermitian homogeneous spaces. It is known that a compact homogeneous pseudo-K\"ahler manifold splits as a product of a complex torus and a rational homogeneous variety, according to the Levi decomposition of $G$. Examples show that compact homogeneous pseudo-Hermitian manifolds in general do not split in this way.