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ISSN 1088-6850(e) ISSN 0002-9947(p)

Local equivalence of symmetric hypersurfaces in $\mathbb{C}^2$

Author(s): Martin Kolár
Journal: Trans. Amer. Math. Soc. 362 (2010), 2833-2843.
MSC (2010): Primary 32V35, 32V40
Posted: January 21, 2010
MathSciNet review: 2592937
Retrieve article in: PDF

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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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