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Approximation and convergence of formal CR-mappings

Francine Meylan·Nordine Mir·Dmitri Zaitsev
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Abstract

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

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Francine Meylan, Nordine Mir, Dmitri Zaitsev. 2002-05-14. Approximation and convergence of formal CR-mappings. https://arxiv.org/abs/math/0205151

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