TY - JOUR

T1 - On the number of affine equivalence classes of spherical tube hypersurfaces

AU - Isaev, A. V.

PY - 2011/1

Y1 - 2011/1

N2 - We consider Levi non-degenerate tube hypersurfaces in Cn+1 that are (k, n - k)-spherical, i. e. locally CR-equivalent to the hyperquadric with Levi form of signature (k, n - k), with n ≤ 2k. We show that the number of affine equivalence classes of such hypersurfaces is infinite (in fact, uncountable) in the following cases: (i) k = n - 2, n ≥ 7; (ii) k = n - 3, n ≥ 7; (iii) k ≤ n - 4. For all other values of k and n, except for k = 3, n = 6, the number of affine classes was known to be finite. The exceptional case k = 3, n = 6 has been recently resolved by Fels and Kaup who gave an example of a family of (3, 3)-spherical tube hypersurfaces that contains uncountably many pairwise affinely non-equivalent elements. In this paper we deal with the Fels-Kaup example by different methods. We give a direct proof of the sphericity of the hypersurfaces in the Fels-Kaup family, and use the j-invariant to show that this family indeed contains an uncountable subfamily of pairwise affinely non-equivalent hypersurfaces.

AB - We consider Levi non-degenerate tube hypersurfaces in Cn+1 that are (k, n - k)-spherical, i. e. locally CR-equivalent to the hyperquadric with Levi form of signature (k, n - k), with n ≤ 2k. We show that the number of affine equivalence classes of such hypersurfaces is infinite (in fact, uncountable) in the following cases: (i) k = n - 2, n ≥ 7; (ii) k = n - 3, n ≥ 7; (iii) k ≤ n - 4. For all other values of k and n, except for k = 3, n = 6, the number of affine classes was known to be finite. The exceptional case k = 3, n = 6 has been recently resolved by Fels and Kaup who gave an example of a family of (3, 3)-spherical tube hypersurfaces that contains uncountably many pairwise affinely non-equivalent elements. In this paper we deal with the Fels-Kaup example by different methods. We give a direct proof of the sphericity of the hypersurfaces in the Fels-Kaup family, and use the j-invariant to show that this family indeed contains an uncountable subfamily of pairwise affinely non-equivalent hypersurfaces.

UR - http://www.scopus.com/inward/record.url?scp=78650762966&partnerID=8YFLogxK

U2 - 10.1007/s00208-010-0514-6

DO - 10.1007/s00208-010-0514-6

M3 - Article

SN - 0025-5831

VL - 349

SP - 59

EP - 74

JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 1

ER -