On product identities and the Chow rings of holomorphic symplectic varieties

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