Derived equivalences of twisted supersingular K3 surfaces

Abstract

We study the derived categories of twisted supersingular K3 surfaces. We prove a derived crystalline Torelli theorem for twisted supersingular K3 surfaces, characterizing Fourier-Mukai equivalences in terms of the twisted K3 crystals introduced in [1]. This is a positive characteristic analog of the Hodge-theoretic derived Torelli theorem of Orlov [2] and its extension to twisted K3 surfaces by Huybrechts and Stellari [3,4]. We give applications to various questions concerning Fourier-Mukai partners, extending results of Căldăraru [5] and Huybrechts and Stellari [3]. We also give an exact formula for the number of twisted Fourier-Mukai partners of a twisted supersingular K3 surface. Contents 1. Introduction 1.1. Notation 1.2. Acknowledgements 2. Supersingular K3 surfaces 2.1. The Brauer group 2.2. K3 crystals and Ogus's crystalline Torelli theorem 3. Twisted K3 crystals and derived equivalences 3.1. Crystalline B-fields 3.2. The twisted Mukai crystal 3.3. Twisted Chern characters and the twisted Néron-Severi group 3.4. Action on cohomology 3.5. Crystalline Torelli theorems for twisted supersingular K3 surfaces 4. Applications to Fourier-Mukai equivalences 4.1. Characteristic subspaces 4.2. Fourier-Mukai partners of twisted supersingular K3 surfaces 4.3. Interlude: the orthogonal group of a strictly characteristic subspace 4.4. Counting twisted Fourier-Mukai partners Appendix A. Twisted Chern characters Appendix B. Deformations of kernels of Fourier-Mukai transforms References

Lifts of twisted K3 surfaces to characteristic 0

“…Suppose that (X, α Br ) and (Y, β Br ) are twisted K3 surfaces over an algebraically closed field k. Given a Fourier-Mukai equivalence Φ P : D(X, α Br ) ∼ − → D(Y, β Br ) one can ask if the induced cohomological transform is orientation preserving (sometimes also called "signed"). 3 It is conjectured that this should always be the case. If k = C (or more generally if the characteristic of k is zero) then this was shown in the untwisted case by Huybrechts-Macrì-Stellari [7].…”

“…An alternative proof, which extends to the twisted case, was given by Reinecke [14]. If k has positive characteristic, various special cases were treated in [3,Appendix B]. Using a combination of standard techniques and Theorem 7.1, we can complete the proof of this conjecture in arbitrary characteristic.…”

“…Suppose that the characteristic of k is positive. Using now standard techniques introduced in [10], to prove the result it suffices to show that every twisted K3 surface admits a lift to characteristic 0, which reduces the problem to the case considered in [7,14] (this strategy is outlined for instance in [3,Appendix B]). Thus, the result follows from Theorem 7.1.…”

“…). However, it extends without difficulty to a field of arbitrary characteristic by instead using the extended Néron-Severi groups, as recorded in[3, Definition 3.4.6].…”

Derived invariants from topological Hochschild homology

“…of F -crystals, where H(X/W, B) is the twisted K3 crystal attached to (X, α) in [Bra18]. As in the classical setting, the construction of H(X/W, B) depends on a non-canonical choice of a B-field lift of α, although the isomorphism class of the resulting crystal is independent of this choice.…”