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Higher order symmetries of real hypersurfaces in $ \mathbb{C}^3$


Authors: Martin Kolar and Francine Meylan
Journal: Proc. Amer. Math. Soc. 144 (2016), 4807-4818
MSC (2010): Primary 32V35, 32V40
DOI: https://doi.org/10.1090/proc/13090
Published electronically: April 25, 2016
MathSciNet review: 3544531
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Abstract: We study nonlinear automorphisms of Levi degenerate hypersurfaces of finite multitype. By results of Kolar, Meylan, and Zaitsev in 2014, the Lie algebra of infinitesimal CR automorphisms may contain a graded component consisting of nonlinear vector fields of arbitrarily high degree, which has no analog in the classical Levi nondegenerate case, or in the case of finite type hypersurfaces in $ \mathbb{C}^2$. We analyze this phenomenon for hypersurfaces of finite Catlin multitype with holomorphically nondegenerate models in complex dimension three. The results provide a complete classification of such manifolds. As a consequence, we show on which hypersurfaces 2-jets are not sufficient to determine an automorphism. The results also confirm a conjecture about the origin of nonlinear automorphisms of Levi degenerate hypersurfaces, formulated by the first author for an AIM workshop in 2010.


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Additional Information

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

Francine Meylan
Affiliation: Department of Mathematics, University of Fribourg, CH 1700 Perolles, Fribourg, Switzerland
Email: francine.meylan@unifr.ch

DOI: https://doi.org/10.1090/proc/13090
Received by editor(s): August 10, 2015
Received by editor(s) in revised form: January 11, 2016
Published electronically: April 25, 2016
Additional Notes: The first author was supported by the project CZ.1.07/2.3.00/20.0003 of the Operational Programme Education for Competitiveness of the Ministry of Education, Youth and Sports of the Czech Republic.
The second author was supported by Swiss NSF Grant 2100-063464.00/1
Communicated by: Franc Forstneric
Article copyright: © Copyright 2016 American Mathematical Society