The Chow ring of hyperkähler varieties of $K{3}^{[2]}$-type via Lefschetz actions

Abstract

We propose an explicit conjectural lift of the Neron-Severi Lie algebra of a hyperkähler variety $X$ of $K{3}^{[2]}$-type to the Chow ring of correspondences ${\mathrm{CH}}^{\ast }(X\times X)$ 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 $X$ of Shen and Vial agrees with the eigenspace decomposition of a canonical lift of the grading operator.

24 pages, comments are welcome!