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An analog of the Fefferman construction

Florian Wisser (2006)

Archivum Mathematicum

The Fefferman construction associates to a manifold carrying a CR–structure a conformal structure on a sphere bundle over the manifold. There are some analogs to this construction, with one giving a Lie contact structure, a refinement of the contact bundle on the bundle of rays in the cotangent bundle of a manifold with a conformal metric. Since these structures are parabolic geometries, these constructions can be dealt with in this setting.

Analytic regularity for the Bergman kernel

Gabor Françis, Nicholas Hanges (1998)

Journées équations aux dérivées partielles

Let Ω ⊂ ℝ 2 be a bounded, convex and open set with real analytic boundary. Let T Ω ⊂ ℂ 2 be the tube with base Ω , and let ℬ be the Bergman kernel of T Ω . If Ω is strongly convex, then ℬ is analytic away from the boundary diagonal. In the weakly convex case this is no longer true. In this situation, we relate the off diagonal points where analyticity fails to the Trèves curves. These curves are symplectic invariants which are determined by the CR structure of the boundary of T Ω . Note that Trèves curves exist only...

Analytic torsions on contact manifolds

Michel Rumin, Neil Seshadri (2012)

Annales de l’institut Fourier

We propose a definition for analytic torsion of the contact complex on contact manifolds. We show it coincides with Ray–Singer torsion on any 3 -dimensional CR Seifert manifold equipped with a unitary representation. In this particular case we compute it and relate it to dynamical properties of the Reeb flow. In fact the whole spectral torsion function we consider may be interpreted on CR Seifert manifolds as a purely dynamical function through Selberg-like trace formulae, that hold also in variable...

Attracting divisors on projective algebraic varieties

Małgorzata Stawiska (2007)

Annales Polonici Mathematici

We obtain sufficient and necessary conditions (in terms of positive singular metrics on an associated line bundle) for a positive divisor D on a projective algebraic variety X to be attracting for a holomorphic map f:X → X.

Bounded symmetric domains and derived geometric structures

Wilhelm Kaup (2002)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Every homogeneous circular convex domain D ⊂ C n (a bounded symmetric domain) gives rise to two interesting Lie groups: The semi-simple group G = A ⁢ u ⁢ t ⁢ D of all biholomorphic automorphisms of D and its isotropy subgroup K ⊂ G ⁢ L ⁢ n , C at the origin (a maximal compact subgroup of G ). The group G acts in a natural way on the compact dual X of D (a certain compactification of C n that generalizes the Riemann sphere in case D is the unit disk in C ). Various authors have studied the orbit structure of the G -space X , here we are interested...

Contact geometry and CR-structures on spheres

John Bland, Tom Duchamp (1995)

Banach Center Publications

A normal form for small CR-deformations of the standard CR-structure on the (2n+1)-sphere is presented. The space of normal forms is parameterized by a single function on the sphere. For n>1, the normal form is used to obtain explicit embeddings into ℂ n + 1 . For n=1, the cohomological obstruction to embeddability is identified.

CR-structures on a real Lie algebra

Giuliana Gigante, Giuseppe Tomassini (1991)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Given the notion of C ⁢ R -structures without torsion on a real 2 ⁢ n + 1 dimensional Lie algebra L 0 we study the problem of their classification when L 0 is a reductive algebra.

Decomposition of CR-manifolds and splitting of CR-maps

Atsushi Hayashimoto, Sung-Yeon Kim, Dmitri Zaitsev (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We show the uniqueness of local and global decompositions of abstract CR-manifolds into direct products of irreducible factors, and a splitting property for their CR-diffeomorphisms into direct products with respect to these decompositions. The assumptions on the manifolds are finite non-degeneracy and finite-type on a dense subset. In the real-analytic case, these are the standard assumptions that appear in many other questions. In the smooth case, the assumptions cannot be weakened by replacing...

Deformations of CR-structures on a real Lie-algebra

Daniele Gouthier (1999)

Bollettino dell'Unione Matematica Italiana

Sia g 0 un’algebra di Lie e (p, J) una sua struttura di Cauchy-Riemann, vale a dire J è una struttura complessa integrabile del sottospazio vettoriale p. Come è stato fatto per il caso delle strutture complesse, cfr. [GT], introduciamo il concetto di deformazione di una struttura CR. Per mezzo dei gruppi di coomologia H k ⁢ g , q vengono provati risultati di rigidità. In particolare ogni struttura di Lie- CR che è semisemplice è rigida. Alcuni esempi chiariscono le soluzioni particolari esposte.

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