On product identities and the Chow rings of holomorphic symplectic varieties

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 $C{H}_{\star }(M\times X^{\ell }),\ell \ge 1,$ generalizing the classic Beauville-Voisin identity for a $K3$ surface. We emphasize consequences of the conjecture for the structure of the tautological subring $R_{\star }(M)\subset C{H}_{\star }(M).$ The conjecture places all tautological classes in the lowest piece of a natural filtration emerging on $C{H}_{\star }(M)$, which we also discuss. We prove the proposed identities when $M$ is the Hilbert scheme of points on a $K3$ surface.