The Chow ring of hyperkähler varieties of $K3^{[2]}$-type via Lefschetz actions

Abstract

We propose an explicit conjectural lift of the Neron-Severi Lie algebra of a hyperkähler variety X of K3 [2] -type to the Chow ring of correspondences CH * (X × X) in terms of a canonical lift of the Beauville-Bogomolov class obtained by Markman. We give evidence for this conjecture in the case of the Hilbert scheme of two points of a K3 surface and in the case of the Fano variety of lines of a very general cubic fourfold. Moreover, we show that the Fourier decomposition of the Chow ring of X from [SV16, Theorem 2] agrees with the eigenspace decomposition of a canonical lift of the grading operator.