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.