Invariant CR structures on compact homogeneous manifolds

Abstract

An explicit classification of simply connected compact homogeneous CR manifolds G/L of codimension one, with non-degenerate Levi form, is given. There are three classes of such manifolds: a) the standard CR homogeneous manifolds which are homogeneous S1-bundles over a flag manifold F, with CR structure induced by an invariant complex structure on F; b) the MorimotO-Nagano spaces, i.e. sphere bundles S(N) ⊂ TN of a compact rank one symmetric space N = G/H, with the CR structure induced by the natural complex structure of TN = GC/HC; c) the following manifolds: SUn/T1 · SUn−2, SUp × SUq/T1 · Up−2 · Uq−2, SUn/T1. SU2 · SU2 · SUn−4, SO10/T1 · SO6, E6/T1 · SO8; these manifolds admit canonical holomorphic fibrations over a flag manifold (F, JF) with typical fiber S(Sk), where k = 2,3,5,7 or 9, respectively; the CR structure is determined by the invariant complex structure JF on F and by an invariant CR structure on the typical fiber, depending on one complex parameter. © 2003 by the University of Notre Dame. All rights reserved.