# Lehn's formula in Chow and Conjectures of Beauville and Voisin

## TL;DR

This paper extends Lehn's formula and the W_{1+infinity} algebra action to Chow groups, proving a weak version of the Beauville-Voisin conjecture for Hilbert schemes of points on K3 surfaces.

## Contribution

It introduces a Chow group version of Lehn's formula and extends algebraic structures, providing new evidence for the Beauville-Voisin conjecture in specific hyperk"ahler cases.

## Key findings

- Proves a weak version of the Beauville-Voisin conjecture for Hilbert schemes of points on K3 surfaces.
- Extends Lehn's formula and algebra actions from cohomology to Chow groups.
- Establishes the weak splitting conjecture for these geometries.

## Abstract

The Beauville-Voisin conjecture for a hyperk\"ahler manifold X states that the subring of the Chow ring A^*(X) generated by divisor classes and Chern characters of the tangent bundle injects into the cohomology ring of X. We prove a weak version of this conjecture when X is the Hilbert scheme of points on a K3 surface, for the subring generated by divisor classes and tautological classes. This in particular implies the weak splitting conjecture of Beauville for these geometries. In the process, we extend Lehn's formula and the Li-Qin-Wang W_{1+infinity} algebra action from cohomology to Chow groups, for the Hilbert scheme of an arbitrary smooth projective surface S