Approximation and convergence properties of formal CR-maps

Abstract

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. To cite this article: F. Meylan et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 671–676.