Approximation and convergence of formal CR-mappings

Abstract

Let M ? CN be a minimal real-analytic CR-submanifold and M? ? CN? a realalgebraic subset through points p ? M and p? ? M?. We show that that any formal (holomorphic) mapping f : (CN, p) ? (CN? , p?), 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 (in an appropriate sense) M into the set E? ? M? of points of 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.