Twisted derived equivalences and isogenies between K3 surfaces in positive characteristic

Bragg, Daniel; Yang, Ziquan

doi:10.2140/ant.2023.17.1069

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Twisted derived equivalences and isogenies between K3 surfaces in positive characteristic

Daniel Bragg and Ziquan Yang

Vol. 17 (2023), No. 5, 1069–1126

DOI: 10.2140/ant.2023.17.1069

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. Sci. Éc. Norm. Supér. $(4)$ 48:5 (2015), 1001–1033).

Keywords

derived categories, twisted sheaves, K3 surfaces, isogenies, good reduction

Mathematical Subject Classification

Primary: 11G99, 14G17, 14G35

Milestones

Received: 12 April 2021

Revised: 9 April 2022

Accepted: 25 May 2022

Published: 9 May 2023

Authors

Daniel Bragg
Department of Mathematics University of California Berkeley, CA United States

Ziquan Yang
University of Wisconsin Madison, WI United States

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Vol. 17, No. 5, 2023