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Alex Perry : Derived categories of cubic fourfolds and their geometric applications

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A fundamental problem in algebraic geometry is to determine whether a given algebraic variety is birational to projective space. This is most prominently open for cubic fourfolds, i.e. hypersurfaces defined by a cubic polynomial in a five-dimensional projective space. A decade ago, Kuznetsov suggested an approach to this problem using the derived category of coherent sheaves. I will explain recent applications of this perspective to fundamental questions in hyperkahler geometry and Hodge theory, which in turn shed light on the original question about cubic fourfolds.

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