Approximation and convergence properties of formal CR-maps

Francine Meylan; Nordine Mir; Dmitri Zaitsev

doi:10.1016/S1631-073X(02)02552-9

Approximation and convergence properties of formal CR-maps
[Propriétés d'approximation et de convergence des applications CR formelles]

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

Comptes Rendus. Mathématique, Volume 335 (2002) no. 8, pp. 671-676.

Résumé
Abstract

Soient $M \subset ℂ^{N}$ une sous-variété CR analytique réelle minimale et $M' \subset ℂ^{N'}$ un sous-ensemble algébrique réel avec p∈M et p′∈M′. On montre que pour toute application (holomorphe) formelle $f : (ℂ^{N}, p) \to (ℂ^{N'}, p')$ , 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 $ℰ' \subset M'$ 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 $M \subset ℂ^{N}$ be a minimal real-analytic CR-submanifold and $M' \subset ℂ^{N'}$ a real-algebraic subset through points p∈M and p′∈M′ respectively. We show that that any formal (holomorphic) mapping $f : (ℂ^{N}, p) \to (ℂ^{N'}, p')$ , 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 $ℰ' \subset M'$ 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.

Reçu le : 2002-05-28
Révisé le : 2002-09-12
Publié le : 2002-09-27

DOI : 10.1016/S1631-073X(02)02552-9

Affiliations des auteurs :

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

@article{CRMATH_2002__335_8_671_0,
     author = {Francine Meylan and Nordine Mir and Dmitri Zaitsev},
     title = {Approximation and convergence properties of formal {CR-maps}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {671--676},
     publisher = {Elsevier},
     volume = {335},
     number = {8},
     year = {2002},
     doi = {10.1016/S1631-073X(02)02552-9},
     language = {en},
}

TY  - JOUR
AU  - Francine Meylan
AU  - Nordine Mir
AU  - Dmitri Zaitsev
TI  - Approximation and convergence properties of formal CR-maps
JO  - Comptes Rendus. Mathématique
PY  - 2002
SP  - 671
EP  - 676
VL  - 335
IS  - 8
PB  - Elsevier
DO  - 10.1016/S1631-073X(02)02552-9
LA  - en
ID  - CRMATH_2002__335_8_671_0
ER  -

%0 Journal Article
%A Francine Meylan
%A Nordine Mir
%A Dmitri Zaitsev
%T Approximation and convergence properties of formal CR-maps
%J Comptes Rendus. Mathématique
%D 2002
%P 671-676
%V 335
%N 8
%I Elsevier
%R 10.1016/S1631-073X(02)02552-9
%G en
%F CRMATH_2002__335_8_671_0

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/

Bibliographie
Cité par

[1] M. Artin On the solutions of analytic equations, Invent. Math., Volume 5 (1968), pp. 277-291

[2] M.S. Baouendi; P. Ebenfelt; L.P. Rothschild Algebraicity of holomorphic mappings between real algebraic sets in $ℂ^{n}$ , Acta Math., Volume 177 (1996), pp. 225-273

[3] M.S. Baouendi; P. Ebenfelt; L.P. Rothschild Convergence and finite determination of formal CR mappings, J. Amer. Math. Soc., Volume 13 (2000), pp. 697-723

[4] M.S. Baouendi; N. Mir; L.P. Rothschild Reflection ideals and mappings between generic submanifolds in complex space, J. Geom. Anal., Volume 12 (2002) no. 4, pp. 543-580

[5] M.S. Baouendi; L.P. Rothschild; D. Zaitsev Equivalences of real submanifolds in complex space, J. Differential Geom., Volume 59 (2001), pp. 301-351

[6] S.S. Chern; J.K. Moser Real hypersurfaces in complex manifolds, Acta Math., Volume 133 (1974), pp. 219-271

[7] J.P. D'Angelo Finite type and the intersection of real and complex subvarieties, Proc. Sympos. Pure Math., 52, American Mathematical Society, Providence, RI, 1991, pp. 103-117 (Part 3)

[8] B. Lamel Holomorphic maps of real submanifolds in complex spaces of different dimensions, Pacific J. Math., Volume 201 (2001) no. 2, pp. 357-387

[9] L. Lempert On the boundary behavior of holomorphic mappings, Contributions to Several Complex Variables, Aspects Math., E9, Viehweg, Braunschweig, 1986, pp. 193-215

[10] F. Meylan; N. Mir; D. Zaitsev Analytic regularity of CR-mappings, Math. Res. Lett., Volume 9 (2002), pp. 73-93

[11] F. Meylan, N. Mir, D. Zaitsev, Holomorphic extension of smooth CR-mappings between real-analytic and real-algebraic CR-manifolds, Preprint, 2002, | arXiv

[12] F. Meylan, N. Mir, D. Zaitsev, Approximation and convergence of formal CR-mappings, Preprint, 2002

[13] N. Mir Formal biholomorphic maps of real analytic hypersurfaces, Math. Res. Lett., Volume 7 (2000), pp. 343-359

[14] N. Mir On the convergence of formal mappings, Comm. Anal. Geom., Volume 10 (2002) no. 1, pp. 23-59

[15] J.K. Moser; S.M. Webster Normal forms for real surfaces in $ℂ^{2}$ near complex tangents and hyperbolic surface transformations, Acta Math., Volume 150 (1983) no. 3–4, pp. 255-296

[16] A.E. Tumanov Extension of CR-functions into a wedge from a manifold of finite type, Mat. Sb. (N.S.), Volume 136(178) (1988) no. 1, pp. 128-139 (in Russian). English translation: Math. USSR-Sb., 64, 1, 1989, pp. 129-140

[17] D. Zaitsev Germs of local automorphisms of real analytic CR structures and analytic dependence on the k-jets, Math. Res. Lett., Volume 4 (1997) no. 6, pp. 823-842

[18] D. Zaitsev Algebraicity of local holomorphisms between real-algebraic submanifolds of complex spaces, Acta Math., Volume 183 (1999), pp. 273-305

Cité par Sources :

Commentaires - Politique