A Lie algebra action on the Chow ring of the Hilbert scheme of points of a K3 surface

TL;DR

This paper constructs a Lie algebra action on the Chow ring of Hilbert schemes of points on K3 surfaces, simplifying proofs of injectivity of the cycle class map and providing explicit formulas for monodromy actions.

Contribution

It introduces a Lie algebra action on the Chow ring of Hilbert schemes of K3 surfaces, connecting geometric and algebraic structures in a novel way.

Findings

Lie algebra action on Chow ring constructed

Simplifies proof of cycle class map injectivity

Provides explicit formulas for monodromy actions

Abstract

We construct an action of the Neron--Severi part of the Looijenga-Lunts-Verbitsky Lie algebra on the Chow ring of the Hilbert scheme of points on a K3 surface. This yields a simplification of Maulik and Negut's proof that the cycle class map is injective on the subring generated by divisor classes as conjectured by Beauville. The key step in the construction is an explicit formula for Lefschetz duals in terms of Nakajima operators. Our results also lead to a formula for the monodromy action on Hilbert schemes in terms of Nakajima operators.