On product identities and the Chow rings of holomorphic symplectic varieties

TL;DR

This paper proposes conjectural identities in the Chow rings of moduli spaces of stable sheaves on K3 surfaces, generalizing known identities and exploring their implications for tautological classes, with proof for Hilbert schemes of points.

Contribution

It introduces new conjectural identities in Chow rings for moduli spaces of sheaves on K3 surfaces, extending classical results and analyzing their structural consequences.

Findings

Proposes conjectural identities generalizing Beauville-Voisin identity.

Establishes the identities for Hilbert schemes of points on K3 surfaces.

Discusses the placement of tautological classes within a natural filtration.

Abstract

For a moduli space $M$ of stable sheaves over a $K3$ surface $X$, we propose a series of conjectural identities in the Chow rings $CH_\star (M \times X^\ell),\, \ell \geq 1,$ generalizing the classic Beauville-Voisin identity for a $K3$ surface. We emphasize consequences of the conjecture for the structure of the tautological subring $R_\star (M) \subset CH_\star (M).$ The conjecture places all tautological classes in the lowest piece of a natural filtration emerging on $CH_\star (M)$, which we also discuss. We prove the proposed identities when $M$ is the Hilbert scheme of points on a $K3$ surface.