Degenerate real hypersurfaces in ℂ² with few automorphisms

Ebenfelt, Peter; Lamel, Bernhard; Zaitsev, Dmitri

doi:10.1090/S0002-9947-09-04626-1

Degenerate real hypersurfaces in $\mathbb {C}^2$ with few automorphisms
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by Peter Ebenfelt, Bernhard Lamel and Dmitri Zaitsev PDF

Trans. Amer. Math. Soc. 361 (2009), 3241-3267 Request permission

Abstract:

We introduce new biholomorphic invariants for real-analytic hypersurfaces in $\mathbb {C}^2$ and show how they can be used to show that a hypersurface possesses few automorphisms. We give conditions, in terms of the new invariants, guaranteeing that the stability group is finite, and give (sharp) bounds on the cardinality of the stability group in this case. We also give a sufficient condition for the stability group to be trivial. The main technical tool developed in this paper is a complete (formal) normal form for a certain class of hypersurfaces. As a byproduct, a complete classification, up to biholomorphic equivalence, of the finite type hypersurfaces in this class is obtained.

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