Derived equivalences of hyperkähler varieties

Taelman, Lenny¹

¹ Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, Netherlands

Geometry & topology, Tome 27 (2023) no. 7, pp. 2649-2693.

Résumé

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 $\mathbb{Q}$–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 $K3$ surfaces.

Détail

DOI : 10.2140/gt.2023.27.2649

Classification : 14F05, 14J32
Keywords: hyperkähler varieties, derived equivalences, Fourier–Mukai transforms

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