The Journal of Geometric Analysis

, Volume 12, Issue 4, pp 543–580 | Cite as

Reflection Ideals and mappings between generic submanifolds in complex space

  • M. S. Baouendi
  • Nordine Mir
  • Linda Preiss Rothschild
Article

Abstract

Results on finite determination and convergence of formal mappings between smooth generic submanifolds in ℂN are established in this article. The finite determination result gibes sufficient conditions to guarantee that a formal map is uniquely determined by its jet, of a preassigned order, at a point. Convergence of formal mappings for real-analytic generic submanifolds under appropriate assumptions is proved, and natural geometric conditions are given to assure that if two germs of such submanifolds are formally equivalent, then, they are necessarily biholomorphically equivalent. It is also shown that if two real-algebraic hypersurfaces in ℂN are biholomorphically equivalent, then, they are 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.

Math Subject Classifications

32H02 

Key Words and Phrases

generie submanifold biholomorphic equivalence formal equivalence algebraic equivalence formal mapping reflection ideal finite type holomorphically nondegenerate 

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Copyright information

© Mathematica Josephina, Inc. 2002

Authors and Affiliations

  • M. S. Baouendi
    • 1
  • Nordine Mir
    • 2
  • Linda Preiss Rothschild
    • 1
  1. 1.Department of Mathematics, 0112University of California at San DiegoLa Jolla
  2. 2.Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRSUniversité de RouenMont-Saint-Aignan CedexFrange

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