Twisted derived equivalences and isogenies between K3 surfaces in positive characteristic

Abstract

We study isogenies between K3 surfaces in positive characteristic. Our main result is a characterization of K3 surfaces isogenous to a given K3 surface X in terms of certain integral sublattices of the second rational ℓ-adic and crystalline cohomology groups of X . This is a positive characteristic analog of a result of Huybrechts (Comment. Math. Helv. 94:3 (2019), 445-458), and extends results of Yang (Int. Math. Res. Not. 2022:6 (2022), 4407-4450). We give applications to the reduction types of K3 surfaces and to the surjectivity of the period morphism. To prove these results we describe a theory of B-fields and Mukai lattices in positive characteristic, which may be of independent interest. We also prove some results on lifting twisted Fourier-Mukai equivalences to characteristic 0, generalizing results of Lieblich and Olsson (Ann.