Torelli Theorems and Isogeny Theory for Irreducible Symplectic Varieties in Positive Characteristics

Abstract

The first part treats the a class of higher dimensional analogues of K3 surfaces, called K3^n-type varieties, in positive characteristics. The notion of K3^n-type varieties is well understood in complex hyperkähler geometry. We show that this notion extends very well to characteristic p base fields when 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. The second part treats the isogeny theory for K3 surfaces. Over the field of complex numbers, this theory has become well understood thanks to the recent works of Buskin and Huybrechts. In particular, all isogenies in this case come from twisted derived equivalences. We generalize this theory again to characteristic p base fields and provide existence and uniqueness theorems modeled on the theory over the complex numbers. As an application, we show that squares of K3 surfaces over finite fields of characteristic p at least 5 satisfy the Tate conjecture and the Tate classes are spanned by the obvious ones and the cycles coming from twisted derived equivalences.