Families of exotic affine 3-spheres

TL;DR

This paper constructs algebraic families of exotic affine 3-spheres, which are smooth affine threefolds topologically similar to a standard quadric but algebraically distinct, revealing new diversity in affine algebraic geometry.

Contribution

It introduces a method to generate families of exotic affine 3-spheres parametrized by certain contractible surfaces, expanding understanding of their algebraic and topological properties.

Findings

Existence of canonical families of exotic affine 3-spheres

Construction parametrized by smooth contractible affine surfaces

Demonstration of non-isomorphism within these families

Abstract

We construct algebraic families of exotic affine 3-spheres, that is, smooth affine threefolds diffeomorphic to a non-degenerate smooth complex affine quadric of dimension 3 but non algebraically isomorphic to it. We show in particular that for every smooth topologically contractible affine surface S with trivial automorphism group, there exists a canonical smooth family of pairwise non isomorphic exotic affine 3-spheres parametrized by the closed points of S.