Topologically contractible smooth affine complex varieties: questions, conjectures, computations

A smooth complex variety X is called topologically contractible if its set of complex points, viewed as a complex manifold with analytic topology, is a contractible topological space. There exist a number of deep conjectures and questions regarding topologically contractible smooth affine complex varieties in algebraic geometry and in motivic homotopy theory. The generalized Serre question asks whether algebraic vector bundles over topologically contractible smooth affine complex varieties are always trivial. In this talk, we give an overview of existing results on the generalized Serre question in low dimensions, explain connections to other conjectures and questions, and finally present concrete cohomological computations for some well-known examples of topologically contractible varieties.

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