Special Seminar in Arithmetic Geometry: A Crystalline Torelli Theorem for Derived Categories of Twisted Supersingular K3 Surfaces

Seminar | April 13 | 11:10 a.m.-12 p.m. | 891 Evans Hall

 Daniel Bragg, University of Washington

 Department of Mathematics

The derived category of a variety is an important and subtle invariant, and it is in general a difficult question of when two varieties have equivalent derived categories. For K3 surfaces over the complex numbers, the derived Torelli theorem of Mukai and Orlov gives a satisfying answer to this question using Hodge theory. Much less is known in positive characteristic. In this talk, we will use crystalline cohomology to prove a Torelli theorem for derived categories of twisted sheaves on supersingular K3 surfaces. In the process, we will discuss the Brauer group of a supersingular K3 surface and extend the Artin invariant to twisted supersingular K3 surfaces. We will show how our theorem can be applied to answer various natural questions about derived equivalences. Finally, we will use our constructions to explain some features of the moduli space of supersingular K3 surfaces.