Arithmetic Geometry and Number Theory RTG Seminar: A Crystalline Torelli Theorem for Supersingular Varieties of $K3^{[n]}$ type

Seminar | November 18 | 3:10-5 p.m. | 740 Evans Hall

 Ziquan Yang, Harvard

 Department of Mathematics

The classical global Torelli theorem for complex $K3$ surfaces states that complex $K3$ surfaces are determined up to isomorphism by their integral Hodge structures. Since its discovery, the theorem has been generalized in various directions. For example, in 1983 Ogus proved a crystalline Torelli theorem for supersingular $K3$ surfaces in positive characteristics. In 2010 Verbitsky found a generalization to higher dimensional complex irreducible symplectic manifolds.

I will explain how to simultaneously generalize Ogus' and Verbitsky's theorems to irreducible symplectic varieties of $K3^{[n]}$-type in positive characteristic. Typical examples of such varieties are moduli spaces of sheaves on $K3$ surfaces and Fano varieties of lines on cubic fourfolds. I will also give a Serre-Tate theory for such varieties, building on the work of Nygaard and Ogus.

In the pre-talk, I will give some background on Verbitsky's theorem and explain how to interpret Ogus' theorem from the perspective of integral p-adic Hodge theory, as a motivation for our methods.