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 [15], and extends results of [52]. 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 [29].

Derived isogenies and isogenies for abelian surfaces

“…Proof. This is similar to the case of K3 surfaces proved in [7]. We only need to prove the assertion for a single derived equivalence Φ : D b (X )…”

“…Similar as K3 surfaces, the action of twisted derived equivalence on abelian surfaces are isometries between the twisted Mukai lattices (cf. [7,Theorem 3.6]). For our purpose, we concentrate on the action of prime-to p twisted derived equivalence on the 2 nd crystalline cohomology of abelian surfaces.…”

“…We will investigate this problem in a sequel to this paper. We shall mention that recently Bragg and Yang have studied the twisted derived equivalence between K3 surfaces in [7] and they provided a weaker version of the statement in Conjecture 1.0.1 (cf. [7,Theorem 1.2]).…”