Seminar Algebra and Geometry: Adrien Dubouloz (Dijon)

Family of exotic affine spheres

An exotic affine sphere is a smooth complex affine variety which is diffeomorphic to a smooth non degenerate affine quadric of the same dimension but not algebraically isomorphic to it. The smooth threefold with equation x²v + y²u = 1 is such an exotic sphere, and in fact is essentially the unique example known so far. In this talk, after a short review of the existing results and open questions around these spheres, I will discuss some strategies to construct larger families of exotic 3-spheres, as a well as higher dimensional exotic spheres. It time permits, I will indicate some connexions between the exoticity of a certain 5-dimensional sphere in the sens of A1-homotopy theory and the stable exoticity of the Russell cubic threefold.

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