Local equivalence of symmetric hypersurfaces in ℂ²

Kolář, Martin

doi:10.1090/S0002-9947-10-05058-0

Local equivalence of symmetric hypersurfaces in $\mathbb C^2$
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by Martin Kolář

Trans. Amer. Math. Soc. 362 (2010), 2833-2843

DOI: https://doi.org/10.1090/S0002-9947-10-05058-0

Published electronically: January 21, 2010

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Abstract:

The Chern-Moser normal form and its analog on finite type hypersurfaces in general do not respect symmetries. Extending the work of N. K. Stanton, we consider the local equivalence problem for symmetric Levi degenerate hypersurfaces of finite type in $\mathbb C^2$. The results give complete normalizations for such hypersurfaces, which respect the symmetries. In particular, they apply to tubes and rigid hypersurfaces, providing an effective classification. The main tool is a complete normal form constructed for a general hypersurface with a tube model. As an application, we describe all biholomorphic maps between tubes, answering a question posed by N. Hanges. Similar results for hypersurfaces admitting nontransversal symmetries are obtained.

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