Abstract
Let H:(M,p)→(M ′,p ′) be a formal mapping between two germs of real-analytic generic submanifolds in ℂN with nonvanishing Jacobian. Assuming M to be minimal at p and M ′ holomorphically nondegenerate at p ′, we prove the convergence of the mapping H. As a consequence, we obtain a new convergence result for arbitrary formal maps between real-analytic hypersurfaces when the target does not contain any holomorphic curve. In the case when both M and M ′ are hypersurfaces, we also prove the convergence of the associated reflection function when M is assumed to be only minimal. This allows us to derive a new Artin type approximation theorem for formal maps of generic full rank.
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Communicated by Steven Bell.
The author is partially supported by the Amadeus program of the ‘Partenariat Hubert Curien’.
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Sunyé, JC. On Formal Maps Between Generic Submanifolds in Complex Space. J Geom Anal 19, 944–962 (2009). https://doi.org/10.1007/s12220-009-9085-8
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DOI: https://doi.org/10.1007/s12220-009-9085-8
Keywords
- Formal mappings
- Generic submanifolds
- Holomorphic mappings
- Artin approximation theorem
- Minimality
- Holomorphic nondegeneracy