Orbits of Real Forms, Matsuki Duality and CR-cohomology

Abstract

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. The manifold M − is algebraic over C and a tubular neighborhood of M 0 . For instance, if M 0 is the sphere S 2n−1 in C n and K 0 = SU(n), the embedding M 0 ֒→ M = CP n is useful to compute the maximal group of CR automorphisms of M 0 (see [10,11]), while the embedding M 0 ֒→ M − = C n \{0} better reflects the topology and the CR cohomology of M 0 . Thus, for some aspects of CR geometry, we can consider M − to be the best complex realization of M 0 . This is the essential contents of the PhD thesis of the first Author ([18]): his aim was to show that, in a range which depends on the pseudoconcavity of M 0 , the groups of tangential Cauchy-Riemann cohomology of M 0 are isomorphic to the corresponding Dolbeault cohomology groups of M − . The class of compact homogeneous CR manifolds to which this theory applies includes the intersections of Matsuki dual orbits in complex flag manifolds (cf. [21]).The paper is organized as follows. In the first section we discuss some basic facts on compact homogeneous CR manifolds, including Matsuki duality and the notion of n-reductivity. The second section describes K 0 -covariant fibrations M − → M 0 and, under a special assumption on the partial complex structure of M 0 (that we call HNR), results on the cohomology, which in part are contained in [18] and havethe corresponding positive Weyl chamber. Lemma 1.2. Every adjoint orbit M in g u intersects iC(H 0 ) in exactly one point.Proof. Let f (X) = b(X, H 0 ). Since M is compact, f has stationary points on M. A stationary point X 0 is characterized by d f (X 0 ) = 0 ⇐⇒ 0 = b([X, X 0 ], H 0 ) = b(X, [X 0 , H 0 ]), ∀X ∈ g u ⇐⇒ X 0 ∈ t u .

Higher order Levi forms on homogeneous CR manifolds

“…With the partial complex structures induced by F, these orbits make a class of homogeneous CR manifolds that were studied by many authors (see e.g. [1,2,3,8,10,11,14,18,20]). Cross-marked Dynkin diagrams.…”

On some classes of Z-graded Lie algebras

“…Our interest in this topic was fostered by our previous work on homogeneous CR manifolds (se e.g. [3,5,4,6,20,21,24,25]). The more recent [23] showed us that some of the Z-graded Lie algebra naturally arising in this context do not satisfy all standard requirements of Tanaka's theory under which e.g.…”