Reductive compact homogeneous CR manifolds

TL;DR

This paper introduces a class of compact homogeneous CR manifolds called $ $-reductive, explores their structure via canonical fibrations onto complex flag manifolds, and defines the new concept of CR-deployments, generalizing CR submersions.

Contribution

The paper defines $ $-reductive CR manifolds, constructs canonical fibrations onto flag manifolds, and introduces the concept of CR-deployments as a generalization of CR submersions.

Findings

$ $-reductive CR manifolds include minimal orbits of compact Lie groups.

Canonical fibrations onto complex flag manifolds are constructed.

Introduction of CR-deployments as a weaker condition than CR submersions.

Abstract

We consider a class of compact homogeneous CR manifolds, that we call $\mathfrak n$-reductive, which includes the orbits of minimal dimension of a compact Lie group $K_0$ in an algebraic homogeneous variety of its complexification $K$. For these manifolds we define canonical equivariant fibrations onto complex flag manifolds. The simplest example is the Hopf fibration $S^3\to\mathbb{CP}^1$. In general these fibrations are not $CR$ submersions, however they satisfy a weaker condition that we introduce here, namely they are CR-deployments.