Approximation and convergence of formal CR-mappings

Abstract

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