On the stability group of CR manifolds

Bernhard Lamel; Nordine Mir

doi:10.1016/j.crma.2006.06.016

Complex Analysis

On the stability group of CR manifolds

Presented by: Jean-Pierre Demailly

Bernhard Lamel ¹; Nordine Mir ²

¹ Universität Wien, Fakultät für Mathematik, Nordbergstrasse 15, A-1090 Wien, Austria
² Université de Rouen, laboratoire de mathématiques Raphaël-Salem, UMR 6085 CNRS, avenue de l'université, B.P. 12, 76801 Saint Étienne du Rouvray, France

Comptes Rendus. Mathématique, Volume 343 (2006) no. 3, pp. 169-172.

Abstract
Résumé

For any essentially finite minimal real-analytic generic submanifold $M \subset C^{N}$ , $N ⩾ 2$ , we show that for every point $p \in M$ the local real-analytic CR automorphisms of M fixing p can be parametrized real-analytically by their $ℓ = ℓ (p)$ jets at p. As an application, we derive a Lie group structure for the stability group $Aut (M, p)$ . We also show that the order $ℓ = ℓ (p)$ of the jet space in which the group $Aut (M, p)$ embeds can be chosen to depend upper-semicontinuously on p. This yields that given any compact real-analytic minimal CR submanifold M in $C^{N}$ , there exists an integer k depending only on M such that for every point $p \in M$ local CR diffeomorphisms mapping a neighbourhood of p in M into another real-analytic CR submanifold in $C^{N}$ with the same CR dimension as that of M are uniquely determined by their k-jet at p.

Pour toute sous-variété analytique réelle générique essentiellement finie et minimale $M \subset C^{N}$ , $N ⩾ 2$ , nous montrons que pour tout point $p \in M$ , les automorphismes CR locaux analytiques réels de M fixant p sont paramétrés analytiquement par leur $ℓ = ℓ (p)$ -jets en p. Comme application, nous obtenons une structure de groupe de Lie sur le groupe d'isotropie $Aut (M, p)$ . Nous montrons aussi que l'ordre $ℓ = ℓ (p)$ de l'espace des jets dans lequel le groupe $Aut (M, p)$ se plonge peut être choisi de façon à ce que l'application $p \mapsto ℓ (p)$ soit semi-continue supérieurement. En corollaire, nous obtenons qu'étant donnée toute sous-variété CR compacte analytique réelle et minimale $M \subset C^{N}$ , il existe un entier positif k, dépendant uniquement de M, tel que pour tout point $p \in M$ les difféomorphismes CR locaux envoyant un voisinage de p dans M sur toute autre sous-variété CR de $C^{N}$ de même dimension CR que celle de M sont uniquement déterminés par leur k-jet en p.

Received: 2005-06-16
Accepted: 2006-06-13
Published online: 2006-07-07

DOI: 10.1016/j.crma.2006.06.016

Author's affiliations:

Bernhard Lamel ¹; Nordine Mir ²

¹ Universität Wien, Fakultät für Mathematik, Nordbergstrasse 15, A-1090 Wien, Austria
² Université de Rouen, laboratoire de mathématiques Raphaël-Salem, UMR 6085 CNRS, avenue de l'université, B.P. 12, 76801 Saint Étienne du Rouvray, France

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     title = {On the stability group of {CR} manifolds},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {169--172},
     publisher = {Elsevier},
     volume = {343},
     number = {3},
     year = {2006},
     doi = {10.1016/j.crma.2006.06.016},
     language = {en},
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Bernhard Lamel; Nordine Mir. On the stability group of CR manifolds. Comptes Rendus. Mathématique, Volume 343 (2006) no. 3, pp. 169-172. doi : 10.1016/j.crma.2006.06.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.06.016/

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