A note on the CR cohomology of Levi-Flat minimal orbits in complex flag manifolds

We prove a relation between the $\bar\partial_M$ cohomology of a minimal orbit $M$ of a real form $G_0$ of a complex semisimple Lie group $G$ in a flag manifold $G/Q$ and the Dolbeault cohomology of the Matsuki dual open orbit $X$ of the complexification $K$ of a maximal compact subgroup $K_0$ of $G_0$, under the assumption that $M$ is Levi-flat.Comment: 7 pages, AMS-LaTeX to appear in Rend. Ist. Mat. Univ. Trieste v2: added reference in introduction + minor revisio

Classification of maximal transitive prolongations of super-Poincar\'e algebras

Let $V$ be a complex vector space with a non-degenerate symmetric bilinear form and $\mathbb S$ an irreducible module over the Clifford algebra $Cl(V)$ determined by this form. A supertranslation algebra is a $\mathbb Z$-graded Lie superalgebra $\mathfrak m=\mathfrak{m}_{-2}\oplus\mathfrak{m}_{-1}$, where $\mathfrak{m}_{-2}=V$ and $\mathfrak{m}_{-1}=\mathbb S\oplus\cdots\oplus\mathbb{S}$ is the direct sum of an arbitrary number $N\geq 1$ of copies of $\mathbb S$, whose bracket $[\cdot,\cdot]|_{\mathfrak{m}_{-1}\otimes \mathfrak{m}_{-1}}:\mathfrak{m}_{-1}\otimes\mathfrak{m}_{-1}\rightarrow\mathfrak{m}_{-2}$ is symmetric, $\mathfrak{so}(V)$-equivariant and non-degenerate (that is the condition "$s\in\mathfrak{m}_{-1}, [s,\mathfrak{m}_{-1}]=0$" implies $s=0$). We consider the maximal transitive prolongations in the sense of Tanaka of supertranslation algebras. We prove that they are finite-dimensional for $\dim V\geq3$ and classify them in terms of super-Poincar\'e algebras and appropriate $\mathbb Z$-gradings of simple Lie superalgebras.Comment: 32 pages, v2: general presentation improved, corrected several typos. Proofs and results unchanged. Final version to appear in Adv. Mat

Abelian extensions of semisimple graded CR algebras

In this paper we take up the problem of describing the CR vector bundles M over compact standard CR manifolds S, which are themselves standard CR manifolds. They are associated to special graded Abelian extensions of semisimple graded CR algebras.Comment: 25 pages, 5 figure

Associated Families of Immersions of Three Dimensional CR Manifolds in Euclidean Spaces

We consider isometric immersions in arbitrary codimension of three-dimensional strongly pseudoconvex pseudo-hermitian CR manifolds into the Euclidean space $\mathbb{R}^n$ and generalize in a natural way the notion of associated family. We show that the existence of such deformations turns out to be very restrictive and we give a complete classification.Comment: 15 page

On the topology of minimal orbits in complex flag manifolds

We compute the Euler-Poincar\'e characteristic of the homogeneous compact manifolds that can be described as minimal orbits for the action of a real form in a complex flag manifold.Comment: 21 pages v2: Major revisio

The CR structure of minimal orbits in complex flag manifolds

Let \^G be a complex semisimple Lie group, Q a parabolic subgroup and G a real form of \^G. The flag manifold \^G/Q decomposes into finitely many G-orbits; among them there is exactly one orbit of minimal dimension, which is compact. We study these minimal orbits from the point of view of CR geometry. In particular we characterize those minimal orbits that are of finite type and satisfy various nondegeneracy conditions, compute their fundamental group and describe the space of their global CR functions. Our main tool are parabolic CR algebras, which give an infinitesimal description of the CR structure of minimal orbits.Comment: AMS-TeX, 44 pages v2: minor revisio