On irreducible symplectic varieties of K3[n]-type in positive characteristic

Abstract

We show that there is a good notion of irreducible symplectic varieties of K3[n]-type over an arbitrary field of characteristic zero or p>n+1. Then we construct mixed characteristic moduli spaces for these varieties. Our main result is a generalization of Ogus' crystalline Torelli theorem for supersingular K3 surfaces. For applications, we answer a slight variant of a question asked by F. Charles on moduli spaces of sheaves on K3 surfaces and give a crystalline Torelli theorem for supersingular cubic fourfolds.