Approximation and convergence of formal cr-mappings

D.Zaitsev
,
N.Mir
,
F.Meylan

Let $M\subset \C^N$ be a minimal real-analytic CR-submanifold and $M'\subset \C^{N'}$ a real-algebraic subset through points $p\in M$ and $p'\in M'$, $N,N'\geq 2$. We show that that any formal mapping $f\colon \to $, sending $M$ into $M'$, can be approximated up to any given order at $p$ by a convergent map sending $M$ into $M'$. If $M$ is furthermore generic, we also show that any such map $f$, that is not convergent, must send $M$ into the set ${\mathcal E}'\subset M'$ of points of {\sc D'Angelo} infinite type. Therefore, if $M'$ does not contain any nontrivial complex-analytic subvariety through $p'$, any formal map $f$ as above is necessarily convergent.