Approximation and convergence properties of formal CR-maps

Let M ⊂ ℂ N be a minimal real-analytic CR-submanifold and M′ ⊂ ℂ N′ a real-algebraic subset through points p ∈ M and p′ ∈ M′ respectively. We show that that any formal (holomorphic) mapping f: (ℂ N, p) → (ℂ N′ 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 sending M into M′ is necessarily convergent. © 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS.

Approximation and convergence properties of formal CR-maps Academic Article