Fibrations and globalizations of compact homogeneous CR-manifolds

Abstract

Fibrations methods which were previously used for complex homogeneous spaces and CR-homogenous spaces of special types ([HO1], [AHR], [HR],[R]}) are developed in a general framework. These include the $\lie g$-anticanonical fibration in the CR-setting which reduces certain considerations to the compact projective algebraic case where a Borel-Remmert type splitting theorem is proved. This allows a reduction to spaces homogeneous under actions of compact Lie groups. General globalization theorems are proved which allow one to regard the homogeneous CR-manifold as the orbit of a real Lie group in a complex homogeneous space of a complex Lie group. In the special case of CR-codimension at most two precise classification results are proved and are applied to show that in most cases there exists such a globalization.