UMR CNRS 6085 Publication 0104

Auteurs : BAOUENDI M.S., MIR Nordine et ROTHSCHILD Linda Preiss

Titre : Reflection ideals and mappings between generic submanifolds in complex space.

Année : 2001

Référence : Soumis

Mots-clefs : Variétés génériques analytiques et algébriques réélles, non dégénérescence holomorphe, détermination finie, convergence d'applications formelles, équivalence formelle, holomorphe et algébrique de variétés génériques.

Key-words : real-analytic and algebraic submanifolds, holomorphic nondegeneracy, finite jet determination, convergence of formal mappings, formal, biholomorphic and algebraic equivalence of generic submanifolds.

Classification AMS : 32H02.

Résumé :
Dans cet article, nous étudions les applications (holomorphes) formelles entre sous-variétés lisses des espaces complexes et établissons certains résultats sur la détermination finie et convergence de telles applications ainsi que l'équivalence holomorphe et algébrique des sous-variétés correspondantes.

Abstract :
In this paper, we study formal mappings between smooth generic submanifolds in multidimensional complex space and establish results on finite determination, convergence and local biholomorphic and algebraic equivalence. Our finite determination result gives sufficient conditions to guarantee that a formal map as above is uniquely determined by its jet at a point of a preassigned order. For real-analytic generic submanifolds, we prove convergence of formal mappings under appropriate assumptions and also give natural geometric conditions to assure that if two germs of such submanifolds are formally equivalent, then they are necessarily biholomorphically equivalent. If the submanifolds are moreover real-algebraic, we address the question of deciding when biholomorphic equivalence implies algebraic equivalence. In particular, we prove that if two real-algebraic hypersurfaces in CN are biholomorphically equivalent, then they are in fact algebraically equivalent. All the results are first proved in the more general context of "reflection ideals" associated to formal mappings between formal as well as real-analytic and real-algebraic manifolds.

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