Derived equivalences of hyperkähler varieties

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 $\mathbf{Q}$-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.

(v5: reverted BBF form to standard normalisation; as was pointed out by Markman: the non-standard version did in general not take rational values )

References in corpus (2)

• Infinitesimal Derived Torelli Theorem for K3 surfaces
• On the Hodge structures of compact hyperkähler manifolds

Cited by in corpus (3)

• The LLV decomposition of hyper-Kaehler cohomology
• Rational Hodge isometries of hyper-Kahler varieties of K3[n]-type are algebraic
• Towards a modular construction of OG10