Date: Wednesday, April 08, 2015
Location: 4096 East Hall (4:00 PM to 6:00 PM)
Title: Algebraizing topological vector bundles on smooth complex affine varieties
Abstract: Suppose X is a smooth complex variety and E is a topological complex vector bundle on the underlying complex analytic space X(C). We can ask: is E algebraizable, i.e., is E the topological complex vector bundle associated with an algebraic vector bundle over X? If E is algebraizable, then the Chern classes of E are necessarily algebraic in the sense that they lie in the image of the cycle class map from Chow groups to cohomology, Thus, algebraicity of the Chern classes is a necessary condition for E to be algebraizable.
Some old remarks of Griffiths suggest that if X is a smooth complex affine variety, then algebraicity of the Chern classes might be the only obstruction to algebraizability. We will explain some recent joint work with Jean Fasel and Mike Hopkins that explores this problem. In the positive direction, we can show that if X is a smooth complex affine variety of dimension d, then algebraicity of Chern classes is sufficient to guarantee algebraizability of vector bundles if d is at most 3. If d is at least 4, then we will show that further obstructions to algebraizability arise and try to describe the first one.
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Speaker: Aravind Asok
Institution: University of Southern California
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