Comptes Rendus
no. 8
Approximation and convergence properties of formal CR-maps
[Propriétés d'approximation et de convergence des applications CR formelles]
Francine Meylan1 ; Nordine Mir2 ; Dmitri Zaitsev3
1 Institut de mathématiques, Université de Fribourg, 1700 Perolles, Fribourg, Switzerland
2 Université de Rouen, laboratoire de mathématiques Raphaël Salem, UMR 6085 CNRS, 76821 Mont-Saint-Aignan cedex, France
3 Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
Comptes Rendus. Mathématique, Volume 335 (2002) no. 8, pp. 671-676.

Soient M⊂ℂ N une sous-variété CR analytique réelle minimale et M'⊂ℂ N' un sous-ensemble algébrique réel avec p∈M et p′∈M′. On montre que pour toute application (holomorphe) formelle f:(ℂ N ,p)→(ℂ N' ,p'), envoyant M dans M′, et pour tout entier positif k donné, il existe un germe d'application holomorphe en p, envoyant M dans M′ et dont le jet en p d'ordre k correspond à celui de f. Si M est de plus générique, on montre qu'une telle application f, non convergente, envoie nécessairement M (en un sens approprié) dans le sous-ensemble ℰ'⊂M' des points de type infini au sens de D'Angelo. Ceci implique en particulier la convergence de toutes les applications formelles envoyant M dans M′, si M′ ne contient pas de sous-ensemble analytique complexe irréductible de dimension positive passant par p′.

Let M⊂ℂ N be a minimal real-analytic CR-submanifold and M'⊂ℂ N' a real-algebraic subset through points p∈M and p′∈M′ respectively. We show that that any formal (holomorphic) mapping f:(ℂ N ,p)→(ℂ N' ,p'), sending M into M′, can be approximated up to any given order at p by a convergent map sending M into M′. If M is furthermore generic, we also show that any such map f, that is not convergent, must send (in an appropriate sense) M into the set ℰ'⊂M' of points of D'Angelo infinite type. Therefore, if M′ does not contain any nontrivial complex-analytic subvariety through p′, any formal map f sending M into M′ is necessarily convergent.

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Révisé le :
Publié le :
DOI : 10.1016/S1631-073X(02)02552-9
Francine Meylan 1 ; Nordine Mir 2 ; Dmitri Zaitsev 3

1 Institut de mathématiques, Université de Fribourg, 1700 Perolles, Fribourg, Switzerland
2 Université de Rouen, laboratoire de mathématiques Raphaël Salem, UMR 6085 CNRS, 76821 Mont-Saint-Aignan cedex, France
3 Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany 
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     title = {Approximation and convergence properties of formal {CR-maps}},
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     pages = {671--676},
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Francine Meylan; Nordine Mir; Dmitri Zaitsev. Approximation and convergence properties of formal CR-maps. Comptes Rendus. Mathématique, Volume 335 (2002) no. 8, pp. 671-676. doi : 10.1016/S1631-073X(02)02552-9. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02552-9/

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