Adrien Dubouloz


Exotic affine spheres


If it exists, an exotic affine sphere is a smooth complex affine variety which is diffeomorphic to a smooth affine quadric when equipped with euclidean topology but not algebraically isomorphic to it. In dimension three, total spaces of nontrivial principal homogeneous bundles under the additive group scheme over the punctured affine plane are natural candidates for being exotic spheres, but it turns out to be a challenging problem to find invariants to distinguish them from the standard 3-dimensional smooth affine quadric. After a brief survey of the history of the problem, I will explain how to distinguish certain families of total spaces of principal homogeneous bundles from each others, up to isomorphism of abstract varieties, in terms of properties of their algebraic De Rham cohomology. Even though a complete classification of exotic spheres remains quite elusive with such arguments, the reward is the existence of at least one exotic affine sphere of dimension 3. If time permits, I will review some higher dimensional aspects of this question, in relation with Zariski Cancellation Problem for affine spaces.