Complete Intersections of Quadrics

Nicolas Addington (Imperial College London)

Monday 27th September, 2010 16:00-17:00 204

Abstract

There is a long-studied correspondence between complete intersections of two even-dimensional quadrics and hyperelliptic curves. It was first noticed in the '50s by Weil, who used it to produce evidence for the Weil conjectures, and has since been a testbed for many fashionable theories - Hodge theory and motives in the '70s, derived categories in the '90s, mirror symmetry today. The two spaces are connected by some moduli problems with a very classical flavor, involving lots of lines on quadrics, or more fashionably by matrix factorizations. The analogous correspondence between an intersection of three quadrics and a branched double cover of P^2 works almost as well, but one needs to bring in twisted sheaves. With four quadrics, the branched cover of P^3 is singular, which is a bigger problem. Guided by the geometry of the moduli problem, I produce a non-Kaehler resolution of singularities and relate its derived category to that of the complete intersection. As a special case I get a pair of derived-equivalent Calabi-Yau 3-folds, which are of interest in mirror symmetry.

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