Reductive Compact Homogeneous Cr Manifolds

Abstract

We define and investigate a class of compact homogeneous CR manifolds, that we call -reductive. They are orbits of minimal dimension of a compact Lie group K (0) in algebraic affine homogeneous spaces of its complexification K. For these manifolds we obtain canonical equivariant fibrations onto complex flag manifolds, generalizing the Hopf fibration . These fibrations are not, in general, CR submersions, but satisfy the weaker condition of being CR-deployments; to obtain CR submersions we need to strengthen their CR structure by lifting the complex stucture of the base

Mostow’s fibration for canonical embeddings of compact homogeneous CR manifolds

“…For further reference, we state an easy consequence of Lemma 3.4. 1 Here and in the following we drop the subscript indicating where norms and scalar products are computed, when we feel that this simplified notation does not lead to ambiguity. Lemma 3.5.…”

“…The aim of this paper is to investigate relations between the cohomology groups of the tangential Cauchy Riemann complexes of n-reductive compact homogeneous CR manifolds and the corresponding Dolbeault cohomology groups of their canonical embeddings. The class of n-reductive compact homogeneous CR manifolds was introduced in [1]: its objects are the minimal orbits, in homogeneous spaces of reductive complex groups, of their compact forms.…”

“…We call n-reductive a homogeneous CR manifolds for which v = (v ∩ v) ⊕ n(v), i.e. for which T 0,1 p 0 M 0 can be identified to the nilradical of v. It was shown in [1] that the intersection of any pair of Matsuki-dual orbits in a complex flag manifold M, with the CR structure inherited from M, is an n-reductive compact homogeneous CR manifold. Moreover, when M 0 is n-reductive, v is the Lie algebra of a closed complex Lie subgroup V of K that contains the stabilizer of p 0 as its maximal compact subgroup, so that M 0 = K 0 /V 0 ֒→ M − = K/V is a generic CR-embedding.…”

Orbits of Real Forms, Matsuki Duality and CR-cohomology

“…Its CR structure is uniquely determined by the datum, for the choice of a base point p 0 of M, of the CR algebra (κ 0 , v), where κ 0 = Lie(K 0 ) and v = dπ −1 (T 0,1 p 0 M), for the complexification dπ of the differential of the canonical projection π : K 0 ∋ x → x · p 0 ∈ M. We recall that, by the formal integrability of the partial complex structure of M, the subspace v is in fact a Lie subalgebra of the complexification κ of κ 0 . These pairs where introduced in [22] to discuss homogeneous CR manifolds and the compact case was especially investigated in [3].…”

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This paper gives an overview on some topics concerning compact homogeneous CR manifolds, that have been developed in the last few years. The algebraic structure of compact Lie group was employed in [3] to show that a large class of compact CR manifolds can be viewed as the total spaces of fiber bundles over complex flag manifolds, generalizing the classical Hopf fibration for the odd dimensional sphere and the Boothby-Wang fibration for homogeneous compact contact manifolds (see [6]). If a compact group K 0 acts as a transitive group of CR diffeomorphisms of a CR manifold M 0 , which is n-reductive in the sense of [3], one can construct a homogeneous space M − = K/V of the complexification K of K 0 such that the map M 0 → M − associated to the inclusion K 0 ֒→ K is a generic CR embedding.

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