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 S^1-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 $S(N)\subset TN$ of a compact rank one symmetric space N = G/H, with the CR structure induced by the natural complex structure of $TN = G^\C/H^\C$; c) the following manifolds: $S{U}_n/T^1\cdot S{U}_{n-2}$, $S{U}_p\times S{U}_q/T^1\cdot U_{p-2}\cdot U_{q-2}$, $S{U}_n/T^1\cdot S{U}_2\cdot S{U}_2\cdot S{U}_{n-4}$, $S{O}_{10}/T^1\cdot S{O}_6$, $E_6/T^1\cdot S{O}_8$; these manifolds admit canonical holomorphic fibrations over a flag manifold (F,J_F) with typical fiber S(S^k), where k = 2, 3, 5, 7 or 9, respectively; the CR structure is determined by the invariant complex structure J_F on F and by an invariant CR structure on the typical fiber, depending on one complex parameter.

In this new version, there are no structural changes from the previous. Some mistakes in the tables of Theorem 1.4, of Theorem 1.5 and of Definition 1.7 have been corrected