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Algebraic Geometry Seminar

A Crystalline Torelli Theorem for Supersingular K3^n-type Varieties
Wednesday, October 21, 2020
4:00 PM-12:00 AM
Zoom Off Campus Location
In 1983, Ogus proved that a supersingular K3 surface in characteristic at least 5 is determined up to isomorphism by the Frobenius action and the Poincare pairing on its second crystalline cohomology. This is an analogue of the classical Torelli theorem for K3's, due to Shapiro and Shafarevich, which says that a complex algebraic K3 surface is determined up to isomorphism by the Hodge structure and the Poinare pairing on its second singular cohomology. I will explain how to re-interpret Ogus' theorem from a motivic point of view and generalize the stronger form of the theorem to a class of higher dimensional analogues of K3 surfaces, called K3^[n]-type varieties. This is also an analogue of Verbitsky's global Torelli theorem for general irreducible symplectic manifolds. A new feature in Verbitsky's theorem, which did not appear in the classical Torelli theorem for K3's, is the notion of "parallel transport operators". I will explain how to work with this notion in an arithmetic setting.

As an application, I will also present a similar crystalline Torelli theorem for supersingular cubic fourfolds, the Hodge theoretic counterpart of which is a theorem of Voisin. Speaker(s): Ziquan Yang (Harvard)
Building: Off Campus Location
Location: Off Campus Location
Event Type: Workshop / Seminar
Tags: Mathematics
Source: Happening @ Michigan from Department of Mathematics