Invariant CR Structures on Compact Homogeneous Manifolds

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: $SU_n/T^1\cdot SU_{n-2}$, $SU_p\times SU_q/T^1 \cdot U_{p-2}\cdot U_{q-2}$, $SU_n/T^1\cdot SU_2\cdot SU_2\cdot SU_{n-4}$, $SO_{10}/T^1\cdot SO_6$, $E_6/T^1\cdot SO_8$; 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.