Peterson's Deformations of higher dimensional quadrics

Abstract

We provide the first explicit examples of deformations of higher dimensional quadrics: a straightforward generalization of Peterson's explicit 1-dimensional family of deformations in C³ of 2-dimensional general quadrics with common conjugate system given by the spherical coordinates on the complex sphere S² ⊂ C³ to an explicit (n–1)-dimensional family of deformations in C²ⁿ⁻¹ of n-dimensional general quadrics with common conjugate system given by the spherical coordinates on the complex sphere Sⁿ ⊂ Cⁿ⁺¹ and non-degenerate joined second fundamental forms. It is then proven that this family is maximal.