Approximation and convergence properties of formal CR-maps
[Propriétés d'approximation et de convergence des applications CR formelles]
Comptes Rendus. Mathématique, Tome 335 (2002) no. 8, pp. 671-676.

Soient M N une sous-variété CR analytique réelle minimale et M' N' un sous-ensemble algébrique réel avec pM et p′∈M′. On montre que pour toute application (holomorphe) formelle f:( N ,p)( 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 '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 N be a minimal real-analytic CR-submanifold and M' N' a real-algebraic subset through points pM and p′∈M′ respectively. We show that that any formal (holomorphic) mapping f:( N ,p)( 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 '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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DOI : 10.1016/S1631-073X(02)02552-9
Meylan, Francine 1 ; Mir, Nordine 2 ; Zaitsev, Dmitri 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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Meylan, Francine; Mir, Nordine; Zaitsev, Dmitri. Approximation and convergence properties of formal CR-maps. Comptes Rendus. Mathématique, Tome 335 (2002) no. 8, pp. 671-676. doi : 10.1016/S1631-073X(02)02552-9. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02552-9/

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