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) ⊂ 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: SU n /T 1 . SU n - 2 , SU p × SU q /T 1 . U p - 2 . U q - 2 , SU n /T 1 . SU 2 . SU 2 . SU n - 4 , SO 1 0 /T 1 . SO 6 , E 6 /T 1 . SO 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.