On Artin approximation for formal CR mappings

Let $M$ be a real-analytic CR submanifold of $\mathbb{C}^N$ and $S^{\prime}$ be a real analytic subset of $\mathbb{C}^{N+N^{\prime}}$. We say that the pair $(M, S^{\prime})$ has the Artin approximation property if for every point $p \in M$ and every positive integer $\ell$, if $H : (\mathbb{C}^N , p) \to \mathbb{C}^{N^{\prime}}$ is a formal holomorphic map such that Graph $H \cap (M \times \mathbb{C}^{N^{\prime}}) \subset S^{\prime}$, there exists a germ at $p$ of a holomorphic map $h^{\ell} : (\mathbb{C}^N, p) \to \mathbb{C}^{N^{\prime}}$ which agrees with $H$ at $p$ up to order $\ell$ satisfying Graph $h^{\ell} \cap (M \times \mathbb{C}^{N^{\prime}}) \subset S^{\prime}$. In this paper, we give some sufficient conditions on a pair $(M,S^{\prime})$ 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 $S^{\prime}$ is any partially algebraic subset of $\mathbb{C}^N \times \mathbb{C}^{N^{\prime}}$, then $(M, S^{\prime})$ has the Artin approximation property.

Keywords

formal map, CR manifold, Artin approximation

2010 Mathematics Subject Classification

32C05, 32C07, 32H02, 32V05, 32V20, 32V40

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Accepted 19 December 2014

Published 25 May 2016