Derived equivalences of hyperkähler varieties

Taelman, Lenny

doi:10.2140/gt.2023.27.2649

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Derived equivalences of hyperkähler varieties

Lenny Taelman

Geometry & Topology 27 (2023) 2649–2693

DOI: 10.2140/gt.2023.27.2649

Abstract

We show that the Looijenga–Lunts–Verbitsky Lie algebra acting on the cohomology of a hyperkähler variety is a derived invariant, and obtain from this a number of consequences for the action on cohomology of derived equivalences between hyperkähler varieties.

This includes a proof that derived equivalent hyperkähler varieties have isomorphic $ℚ$ –Hodge structures, the construction of a rational “Mukai lattice” functorial for derived equivalences, and the computation (up to index $2$ ) of the image of the group of auto-equivalences on the cohomology of certain Hilbert squares of $K 3$ surfaces.

Keywords

hyperkähler varieties, derived equivalences, Fourier–Mukai transforms

Mathematical Subject Classification 2010

Primary: 14F05, 14J32

References

Publication

Received: 17 March 2020

Revised: 2 November 2021

Accepted: 15 January 2022

Published: 19 September 2023

Proposed: Richard P Thomas

Seconded: Jim Bryan, Dan Abramovich

Authors

Lenny Taelman
Korteweg-de Vries Institute for Mathematics University of Amsterdam Amsterdam Netherlands

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Volume 27, issue 7 (2023)