On Artin approximation for formal CR mappings
Let \(M\) be a real-analytic CR submanifold of \(\mathbb{C}^N\) and \(S'\) a real-analytic subset of \(\mathbb{C}^{N+N'}\). A pair \((M,S')\) is said to have the \textit{Artin approximation property} if for every \(p\in M\) and every positive integer \(\ell\) it follows that for each formal holomorphic map \[ H:(\mathbb{C}^N,p)\to\mathbb{C}^N \] satisfying \[ \mathrm{Graph}(H) \cap (M\times\mathbb{C}^{N'})\subset S' \] there exists the germ of a holomorphic map at \(p\) \[ h^{\ell}:(\mathbb{C}^N,p)\to\mathbb{C}^N \] satisfying \[ \mathrm{Graph}(h^{\ell}) \cap (M\times\mathbb{C}^{N'})\subset S' \] and agreeing with \(H\) up to order \(\ell\). A real-analytic subset \[ S'\subset \mathbb{C}_{Z}^N\times \mathbb{C}_{Z'}^{N'} \] is called partially algebraic if it is given locally by the vanishing of finitely many functions that are real polynomials in \(Z'\) with real-analytic coefficients in \(Z\). Given a real-analytic CR submanifold \(M\subset \mathbb{C}^N\) with complex tangent bundle \(T^cM\subset TM\), it is known that at every \(p\in M\), there exists a unique germ of a real-analytic CR submanifold \(\mathcal{O}_p\) through \(p\) called the CR orbit of \(M\) at \(p\), such that every point \(q\in\mathcal{O}_p\) can be reached from \(p\) by a piecewise differentiable curve in \(M\) whose tangent vectors are in \(T^cM\). Sufficient conditions that \((M,S')\) have the Artin approximation property are given. The main result states that, given a connected real-analytic submanifold \(M\subset \mathbb{C}^N\) having the property, then all CR orbits of \(M\) are all of the same dimension and at most codimension one, then \((M,S')\) will have the Artin approximation property whenever \(S'\subset\mathbb{C}^N\times\mathbb{C}^{N'}\) is real-analytic and partially algebraic. The proof involves reduction of the original system of analytic and CR equations to a system of real analytic equations that has a nested approximation property. That is, the reduced system of equations admits a formal solution in which some of its components depend on fewer variables. This is based on a result of \textit{J. Denef} and \textit{L. Lipschitz} [Math. Ann. 253, 1--28 (1980; Zbl 0426.13010)].
scientific article; zbMATH DE number 6609433