Dmitry V. Alekseevsky, Andrea F. Spiro
Invariant CR Structures on Compact Homogeneous Manifolds
Preprint series: ESI preprints

MSC:

32C16 CR-manifolds

53C30 Homogeneous manifolds, See also {14M15, 14M17, 32M10,

57S25 Groups acting on specific manifolds

Abstract: An explicit classification of the simply connected
homogeneous spaces $G/L$ of a compact Lie group
$G$, admitting
a $G$-invariant CR structure
of codimension one and Levi non degenerate, is given.
For each such a homogeneous space, all
admissible $G$-invariant CR structures are listed and classified
up to CR equivalences. \par
It is also proved that if a compact homogeneous CR manifold
$G/L$ is not the covering space of a
$G$-orbit in $TS^n$, $T{\Bbb H} P^n$ or
$T\Bbb OP^2$, then
there exists a holomorphic
fibration $\pi\: G/L \to G/K$, where $G/K$ is a flag manifold
endowed with an invariant complex structure and the
typical fiber $K/L$ is $S^1$ or
it is equivalent to (the universal covering of) a $K$-orbit
in $TS^{2}$ or in $TS^{2n-1}$ with $2\leq n\leq 7$.
Keywords: homogeneous CR manifolds, real hypersurfaces, actions of compact Lie groups