Papers by Andrea Altomani
Let V be a complex orthogonal vector space and S an irreducible Cℓ(V )-module. A supertranslation... more Let V be a complex orthogonal vector space and S an irreducible Cℓ(V )-module. A supertranslation algebra is a Z-graded Lie superalgebra m = m−2+m−1 = V +(S+· · ·+S) whose bracket [·, ·]|m −1 ⊗m −1 is so(V )-invariant and non-degenerate. We consider the maximal transitive prolongations in the sense of Tanaka of supertranslation algebras. We prove that they are finite-dimensional for dim V ≥ 3 and classify them in terms of super-Poincaré algebras and appropriate Z-gradations of simple Lie superalgebras.
Arxiv preprint arXiv:1202.4624, Jan 1, 2012
We consider isometric immersions in arbitrary codimension of three-dimensional strongly pseudocon... more We consider isometric immersions in arbitrary codimension of three-dimensional strongly pseudoconvex pseudo-hermitian CR manifolds into the Euclidean space 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.
Arxiv preprint arXiv: …, Jan 1, 2007
Abstract We investigate the $ CR $ geometry of the orbits $ M $ of a real form $ G $ of a complex... more Abstract We investigate the $ CR $ geometry of the orbits $ M $ of a real form $ G $ of a complex simple group $ G^ C $ in a complex flag manifold $ M^ C= G^ C/Q $. We are mainly concerned with finite type, Levi non-degeneracy conditions, canonical $ G $-equivariant ...
Arxiv preprint arXiv:1106.2779, Jan 1, 2011
We consider a class of compact homogeneous CR manifolds, that we call n-reductive, which includes... more We consider a class of compact homogeneous CR manifolds, that we call n-reductive, which includes the orbits of minimal dimension of a compact Lie group K 0 in an algebraic homogeneous variety of its complexification K. For these manifolds we define canonical equivariant fibrations onto complex flag manifolds. The simplest example is the Hopf fibration S 3 → CP 1 . In general these fibrations are not CR submersions, however they satisfy a weaker condition that we introduce here, namely they are CR-deployments. Date: June 15, 2011. 2000 Mathematics Subject Classification. Primary: 32V05 Secondary: 32L05, 53C30. Key words and phrases. Compact homogeneous CR manifold, CR algebra, equivariant fibration. 1 2 A. ALTOMANI, C. MEDORI, AND M. NACINOVICH
Arxiv preprint math/0601617, Jan 1, 2006
Arxiv preprint arXiv:1106.2962, Jan 1, 2011
Using a bigraded differential complex depending on the CR and pseudohermitian structure, we give ... more Using a bigraded differential complex depending on the CR and pseudohermitian structure, we give a characterization of threedimensional strongly pseudoconvex pseudo-hermitian CR manifolds isometrically immersed in Euclidean space R n in terms of an integral representation of Weierstraß type. Restricting to the case of immersions in R 4 , we study harmonicity conditions for such immersions and give a complete classification of CR-pluriharmonic immersions.
Arxiv preprint arXiv:1201.0555, Jan 1, 2012
Let (V, (·, ·)) be a pseudo-Euclidean vector space and S an irreducible Cℓ(V )-module. An extende... more Let (V, (·, ·)) be a pseudo-Euclidean vector space and S an irreducible Cℓ(V )-module. An extended translation algebra is a graded Lie algebra m = m−2 + m−1 = V + S with bracket given by ([s, t], v) = b(v · s, t) for some nondegenerate so(V )-invariant reflexive bilinear form b on S. An extended Poincaré structure on a manifold M is a regular distribution D of depth 2 whose Levi form Lx : Dx ∧ Dx → TxM/Dx at any point x ∈ M is identifiable with the bracket [·, ·] : S ∧ S → V of a fixed extended translation algebra m. The classification of the standard maximally homogeneous manifolds with an extended Poincaré structure is given, in terms of Tanaka prolongations of extended translation algebras and of appropriate gradations of real simple Lie algebras.
Arxiv preprint arXiv: …, Jan 1, 2009
Let M be a smooth locally embeddable CR manifold, having some CR dimension m and some CR codimens... more Let M be a smooth locally embeddable CR manifold, having some CR dimension m and some CR codimension d. We find an improved local geometric condition on M which guarantees, at a point p on M, that germs of CR distributions are smooth functions, and have extensions to germs of holomorphic functions on a full ambient neighborhood of p. Our condition is a form of weak pseudoconcavity, closely related to essential pseudoconcavity as introduced in [HN1]. Applications are made to CR meromorphic functions and mappings. Explicit examples are given which satisfy our new condition, but which are not pseudoconcave in the strong sense. These results demonstrate that for codimension d > 1, there are additional phenomena which are invisible when d = 1.
Journal of Geometric Analysis, Jan 1, 2010
We study CR quadrics satisfying a symmetry property (S) which is slightly weaker than the symmetr... more We study CR quadrics satisfying a symmetry property (S) which is slightly weaker than the symmetry property (S), recently introduced by W. Kaup, which requires the existence of an automorphism reversing the gradation of the Lie algebra of infinitesimal automorphisms of the quadric.
Abstract We investigate the CR geometry of the orbits M of a real form G0 of a complex simple gro... more Abstract We investigate the CR geometry of the orbits M of a real form G0 of a complex simple group G in a complex flag manifold X= G/Q. We are mainly concerned with finite type, Levi non-degeneracy conditions, canonical G0-equivariant and Mostow fibrations, and ...
Arxiv preprint math/0702845, Jan 1, 2007
Tohoku Mathematical …, Jan 1, 2008
We compute the Euler-Poincaré characteristic of the homogeneous compact manifolds that can be des... more We compute the Euler-Poincaré 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.
Arxiv preprint arXiv:0910.4531, Jan 1, 2009
We consider canonical fibrations and algebraic geometric structures on homogeneous CR manifolds, ... more We consider canonical fibrations and algebraic geometric structures on homogeneous CR manifolds, in connection with the notion of CR algebra. We give applications to the classifications of left invariant CR structures on semisimple Lie groups and of CR-symmetric structures on complete flag varieties. Proposition 1.3. Let G 0 be a Lie group, I 0 a closed subgroup of G 0 , g 0 = Lie(G 0 ) and i 0 = Lie(I 0 ) their Lie algebras. Then (1.8) establishes a one-to-one correspondence between G 0 -homogeneous CR structures on M = G 0 /I 0 and complex Lie subalgebras q of g with q ∩ g 0 = i 0 .
Arxiv preprint math/0307184, Jan 1, 2003
We study, from the point of view of CR geometry, the orbits M of a real form G of a complex semis... more We study, from the point of view of CR geometry, the orbits M of a real form G of a complex semisimple Lie groupĜ in a complex flag manifold G/Q. In particular we characterize those that are of finite type and satisfy some Levi nondegeneracy conditions. These properties are also graphically described by attaching to them some cross-marked diagrams that generalize those for minimal orbits that we introduced in a previous paper. By constructing canonical fibrations over real flag manifolds, with simply connected complex fibers, we are also able to compute their fundamental group.
Arxiv preprint math/0611755, Jan 1, 2006
Arxiv preprint math/0510635, Jan 1, 2005
Arxiv preprint arXiv: …, Jan 1, 2008
We prove a subelliptic estimate for systems of complex vector fields under some assumptions that ... more We prove a subelliptic estimate for systems of complex vector fields under some assumptions that generalize the essential pseudoconcavity for CR manifolds, that was first introduced by two of the Authors, and the Hörmander's bracket condition for real vector fields. Applications are given to prove the hypoellipticity of first order systems and second order partial differential operators. Finally we describe a class of compact homogeneous CR manifolds for which the distribution of (0, 1) vector fields satisfies a subelliptic estimate.
Arxiv preprint math/0507272, Jan 1, 2005
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Papers by Andrea Altomani