Fibrations and globalizations of compact homogeneous CR-manifolds

Fibration methods which were previously used for complex homogeneous spaces and CR-homogeneous spaces of special types [1]–[4] are developed in a general framework. These include the $\mathfrak g$-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 leads to a reduction to spaces homogeneous under actions of compact Lie groups. General globalization theorems are proved which enable one to regard a homogeneous CR-manifold as an 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.