On Artin approximation for formal CR mappings

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Abstract

Let M be a real-analytic CR submanifold of CN and S′ be a realanalytic subset of CN+N′ We say that the pair (M,S′) has the Artin approximation property if for every point p ∈ M and every positive integerℓif H: (CN, p) → CN′ is a formal holomorphic map such that GraphH ∩ (M × CN′ ) ⊂S′, there exists a germ at p of a holomorphic map hl (CN, p) → CN′ which agrees with H at p up to order lsatisfying Graph h ∩ (M × CN′ ) ⊂S′. In this paper, we give some sufficient conditions on a pair (M,S′) to have the Artin approximation property. We show that if the CR orbits of M are all of the same dimension and at most of codimension one in M and if S′ is any partially algebraic subset of CN × CN′ then (M,S′) has the Artin approximation property.

Original languageEnglish
Pages (from-to)221-244
Number of pages24
JournalMathematical Research Letters
Volume23
Issue number1
Publication statusPublished - 2016

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Keywords

  • Artin approximation
  • CR manifold
  • Formal map

ASJC Scopus subject areas

  • Mathematics(all)

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