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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
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 -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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