On Artin approximation for formal CR mappings

Research output: Contribution to journalArticle

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

Fingerprint

CR Mappings
Approximation Property
Holomorphic Maps
Approximation
CR-submanifold
Subset
Codimension
Orbit
Sufficient Conditions
Graph in graph theory

Keywords

  • Artin approximation
  • CR manifold
  • Formal map

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

On Artin approximation for formal CR mappings. / Mir, Nordine.

In: Mathematical Research Letters, Vol. 23, No. 1, 2016, p. 221-244.

Research output: Contribution to journalArticle

@article{a1ec90b419eb425da3dea6ea030b2495,
title = "On Artin approximation for formal CR mappings",
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.",
keywords = "Artin approximation, CR manifold, Formal map",
author = "Nordine Mir",
year = "2016",
language = "English",
volume = "23",
pages = "221--244",
journal = "Mathematical Research Letters",
issn = "1073-2780",
publisher = "International Press of Boston, Inc.",
number = "1",

}

TY - JOUR

T1 - On Artin approximation for formal CR mappings

AU - Mir, Nordine

PY - 2016

Y1 - 2016

N2 - 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.

AB - 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.

KW - Artin approximation

KW - CR manifold

KW - Formal map

UR - http://www.scopus.com/inward/record.url?scp=84973473718&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84973473718&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84973473718

VL - 23

SP - 221

EP - 244

JO - Mathematical Research Letters

JF - Mathematical Research Letters

SN - 1073-2780

IS - 1

ER -