The Chow ring of hyperk\"ahler varieties of $K3^{[2]}$-type via Lefschetz actions

TL;DR

This paper conjectures a lift of the Neron-Severi Lie algebra to the Chow ring for hyperk"ahler varieties of $K3^{[2]}$-type, supported by evidence from specific geometric cases and relating to Fourier and eigenspace decompositions.

Contribution

It introduces a conjectural lift of the Neron-Severi Lie algebra to the Chow ring for hyperk"ahler varieties of $K3^{[2]}$-type, linking geometric and algebraic structures.

Findings

Evidence from Hilbert schemes of points on K3 surfaces.

Evidence from Fano varieties of lines in cubic fourfolds.

Alignment of Fourier decomposition with eigenspace decomposition.

Abstract

We propose an explicit conjectural lift of the Neron-Severi Lie algebra of a hyperk\"ahler variety $X$ of $K3^{[2]}$-type to the Chow ring of correspondences ${\rm CH}^\ast(X \times X)$ 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.