Local symmetries of finite type hypersurfaces in ℂ2

Abstract

The first part of this paper gives a complete description of local automorphism groups for Levi degenerate hypersurfaces of finite type in C 2 . It is also proved that, with the exception of hypersurfaces of the form v = |z| k , local automorphisms are always determined by their 1-jets. Using this result, the second part describes special normal forms which by an additional normalization eliminate the nonlinear symmetries of the model and allows to decide effectively about local equivalence of two hypersurfaces given in this normal form.

Higher order invariants of Levi degenerate hypersurfaces

“…A weight Λ will be called distinguished if there exist coordinates (z, w) in which the defining equation has form (16) v = P (z, z) + o wt (1),…”

“…Coordinates corresponding to a distinguished weight Λ, in which the local description of M has form (16), with P being Λ -homogeneous, will be called Λ -adapted. Λ M will be called the multitype weight.…”

On the CR automorphism group of a certain hypersurface of infinite type in ℂ2

“…For real hypersurfaces in C n , the study of stability groups of various hypersurfaces with further assumptions is given in [1][2][3][4][5][6][7][8][9]. Recently, Kolář and Meylan [3] and Kolář, Meylan and Zaitsev [7] obtained a precise description of the derivatives needed to characterize an automorphism of a general hypersurface.…”

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

“…those fixing the given point) of finite type hypersurfaces in C 2 are linear in some normal coordinates. The symmetries are then immediately visible from the defining equation (in the nondegenerate case it follows from a result of Kruzhilin and Loboda [16], and in the degenerate case from [15]). …”