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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Author: Martin Kolár
Journal: Trans. Amer. Math. Soc. 362 (2010), 2833-2843
MSC (2010): Primary 32V35, 32V40
Published electronically: January 21, 2010
MathSciNet review: 2592937
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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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Additional Information

Martin Kolár
Affiliation: Department of Mathematics and Statistics, Masaryk University, Kotlarska 2, 611 37 Brno, Czech Republic
Email: mkolar@math.muni.cz

DOI: http://dx.doi.org/10.1090/S0002-9947-10-05058-0
PII: S 0002-9947(10)05058-0
Received by editor(s): October 5, 2007
Published electronically: January 21, 2010
Additional Notes: The author was supported by a grant of the GA ČR no. 201/08/0397
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.