On product identities and the Chow rings of holomorphic symplectic varieties

Abstract

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