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 ${\mathbb{C}}^3$ of 2-dimensional general quadrics with common conjugate system given by the spherical coordinates on the complex sphere ${\mathbb{S}}^2\subset {\mathbb{C}}^3$ to an explicit $(n-1)$-dimensional family of deformations in ${\mathbb{C}}^{2n-1}$ of $n$-dimensional general quadrics with common conjugate system given by the spherical coordinates on the complex sphere ${\mathbb{S}}^n\subset {\mathbb{C}}^{n+1}$ and non-degenerate joined second fundamental forms. It is then proven that this family is maximal.