On the convergence of formal mappings

Nordine Mir (1)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem

Let $f : (M,p)\rightarrow (M',p')$ 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 $\C^n$, $n\geq 2$, $p'=f(p)$. 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.

Mots clés: Algebraic field extension; Formal mapping; Segre sets; Minimality in the sense of Tumanov; Real analytic CR manifolds; Holomorphic nondegeneracy; Algebraic field extension.

Domaines: Variables complexes [math.CV]

Nordine Mir. On the convergence of formal mappings. Communications in Analysis and Geometry, 2002, 10 (1), pp.23--59. ⟨hal-00261681⟩