Approximation and convergence properties of formal CR-maps

Meylan, Francine; Mir, Nordine; Zaitsev, Dmitri

Comptes Rendus. Mathématique, Tome 335 (2002) no. 8, pp. 671-676.

Soient $\mathrm{M}\subset {ℂ}^N$ une sous-variété CR analytique réelle minimale et $\mathrm{M}\text{'}\subset {ℂ}^{\mathrm{N}\text{'}}$ un sous-ensemble algébrique réel avec p∈M et p′∈M′. On montre que pour toute application (holomorphe) formelle $\mathrm{f}:({ℂ}^N,\mathrm{p})\to ({ℂ}^{\mathrm{N}\text{'}},\mathrm{p}\text{'})$, envoyant M dans M′, et pour tout entier positif k donné, il existe un germe d'application holomorphe en p, envoyant M dans M′ et dont le jet en p d'ordre k correspond à celui de f. Si M est de plus générique, on montre qu'une telle application f, non convergente, envoie nécessairement M (en un sens approprié) dans le sous-ensemble $ℰ\text{'}\subset \mathrm{M}\text{'}$ des points de type infini au sens de D'Angelo. Ceci implique en particulier la convergence de toutes les applications formelles envoyant M dans M′, si M′ ne contient pas de sous-ensemble analytique complexe irréductible de dimension positive passant par p′.

Let $\mathrm{M}\subset {ℂ}^N$ be a minimal real-analytic CR-submanifold and $\mathrm{M}\text{'}\subset {ℂ}^{\mathrm{N}\text{'}}$ a real-algebraic subset through points p∈M and p′∈M′ respectively. We show that that any formal (holomorphic) mapping $\mathrm{f}:({ℂ}^N,\mathrm{p})\to ({ℂ}^{\mathrm{N}\text{'}},\mathrm{p}\text{'})$, 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 $ℰ\text{'}\subset \mathrm{M}\text{'}$ 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.

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