Sur la convergence d'applications formelles entre sous-variétés analytiques réelles. (On the convergence of formal mappings between real-analytic submanifolds)
Let \(M\subset\mathbb{C}^n \simeq\mathbb{R}^{2n}\) be a real analytic submanifold of codimension \(d\geq 1\) defined in a neighborhood of \(o\) and let \(M' \subset \mathbb{C}^{n'}\) be a real analytic set in \(\mathbb{C}^{n'}\) with \(o\in M\), \(o\in M'\) and let \(f\) be a formal map between \(M\) and \(M'\) with \(f(o)=o\). The authors give new sufficient conditions for \(f\) to be convergent. In fact, inspired by a number of previous results by H. Lewy and others, the authors introduce the notion of first characteristic variety associated to \((M, M',f)\), and prove that if \(M\) is minimal at \(o\) and the first characteristic variety \({\mathcal V}_1\) is of dimension \(0\) at \(o\) then \(f\) is convergent. In the case when \({\mathcal V}_1\) is of positive dimension the authors introduce the notion of the second characteristic variety \({\mathcal V}_2\) and prove a similar result as above.
scientific article; zbMATH DE number 1886880
scientific article