### Abstract

Let f : (M,p) → (M′,p′) be a formal (holomorphic) nondegenerate map, i.e., with formal holomorphic Jacobian J_{f} not identically vanishing, between two germs of real analytic generic submanifolds in ℂ^{n}, n ≥ 2, p′ = f(p). Assuming the target manifold to be real algebraic, and the source manifold to be minimal at p in the sense of Tumanov, we prove the convergence of the so-called reflection mapping associated to f. From this, we deduce the convergence of such mappings from minimal real analytic generic submanifolds into real algebraic holomorphically nondegenerate ones, as well as related results on partial convergence of such maps. For the proofs, we establish a principle of analyticity for formal CR power series. This principle can be used to reobtain the convergence of formal mappings of real analytic CR manifolds under a standard nondegeneracy condition.

Original language | English |
---|---|

Pages (from-to) | 23-59 |

Number of pages | 37 |

Journal | Communications in Analysis and Geometry |

Volume | 10 |

Issue number | 1 |

Publication status | Published - Jan 2002 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Analysis
- Statistics and Probability
- Geometry and Topology
- Statistics, Probability and Uncertainty

### Cite this

*Communications in Analysis and Geometry*,

*10*(1), 23-59.

**On the convergence of formal mappings.** / Mir, Nordine.

Research output: Contribution to journal › Article

*Communications in Analysis and Geometry*, vol. 10, no. 1, pp. 23-59.

}

TY - JOUR

T1 - On the convergence of formal mappings

AU - Mir, Nordine

PY - 2002/1

Y1 - 2002/1

N2 - Let f : (M,p) → (M′,p′) be a formal (holomorphic) nondegenerate map, i.e., with formal holomorphic Jacobian Jf not identically vanishing, between two germs of real analytic generic submanifolds in ℂn, n ≥ 2, p′ = f(p). Assuming the target manifold to be real algebraic, and the source manifold to be minimal at p in the sense of Tumanov, we prove the convergence of the so-called reflection mapping associated to f. From this, we deduce the convergence of such mappings from minimal real analytic generic submanifolds into real algebraic holomorphically nondegenerate ones, as well as related results on partial convergence of such maps. For the proofs, we establish a principle of analyticity for formal CR power series. This principle can be used to reobtain the convergence of formal mappings of real analytic CR manifolds under a standard nondegeneracy condition.

AB - Let f : (M,p) → (M′,p′) be a formal (holomorphic) nondegenerate map, i.e., with formal holomorphic Jacobian Jf not identically vanishing, between two germs of real analytic generic submanifolds in ℂn, n ≥ 2, p′ = f(p). Assuming the target manifold to be real algebraic, and the source manifold to be minimal at p in the sense of Tumanov, we prove the convergence of the so-called reflection mapping associated to f. From this, we deduce the convergence of such mappings from minimal real analytic generic submanifolds into real algebraic holomorphically nondegenerate ones, as well as related results on partial convergence of such maps. For the proofs, we establish a principle of analyticity for formal CR power series. This principle can be used to reobtain the convergence of formal mappings of real analytic CR manifolds under a standard nondegeneracy condition.

UR - http://www.scopus.com/inward/record.url?scp=0013186824&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0013186824&partnerID=8YFLogxK

M3 - Article

VL - 10

SP - 23

EP - 59

JO - Communications in Analysis and Geometry

JF - Communications in Analysis and Geometry

SN - 1019-8385

IS - 1

ER -