On exotic affine 3-spheres

TL;DR

This paper explores the structure of algebraic bundles over the punctured complex affine plane, classifies certain affine 3-spheres with exotic properties, and provides examples of biholomorphic but non-isomorphic threefolds.

Contribution

It classifies total spaces of homogeneous -bundles over the punctured plane and demonstrates the existence of exotic affine 3-spheres with unique isomorphism properties.

Findings

Affine 3-sphere admits -bundle structures with homogeneity.

Classification of homogeneous -bundles over the punctured plane.

Existence of exotic spheres in dimension three.

Abstract

Every $\mathbb{A}^{1}-$bundle over the complex affine plane punctured at the origin, is trivial in the differentiable category but there are infinitely many distinct isomorphy classes of algebraic bundles. Isomorphy types of total spaces of such algebraic bundles are considered; in particular, the complex affine 3-sphere admitts such a structure with an additional homogeneity property. Total spaces of nontrivial homogeneous $\mathbb{A}^{1}$-bundles over the punctured plane are classified up to $\mathbb{G}_{m}$-equivariant algebraic isomorphism and a criterion for nonisomorphy is given. In fact the affine 3-sphere is not isomorphic as an abstract variety to the total space of any $\mathbb{A}^{1}$-bundle over the punctured plane of different homogeneous degree, which gives rise to the existence of exotic spheres, a phenomenon that first arises in dimension three. As a by product, an example is given of two biholomorphic but not algebraically isomorphic threefolds, both with a trivial Makar-Limanov invariant, and with isomorphic cylinders.