On exotic affine 3-spheres

Adrien Dubouloz (1) , David R. Finston (2)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]
2 Department of Mathematical siences

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.

Mots clés: exotic affine spheres; homogeneous principal bundles

Domaines: Géométrie algébrique [math.AG]

Adrien Dubouloz, David R. Finston. On exotic affine 3-spheres. 2011. ⟨hal-00600542⟩