On product identities and the Chow rings of holomorphic symplectic varieties

Abstract

Abstract For a moduli space

$${\mathsf M}$$

M

of stable sheaves over a K3 surface X, we propose a series of conjectural identities in the Chow rings

$$CH_\star ({\mathsf M}\times X^\ell ),\, \ell \ge 1,$$

C

H ⋆

( M ×

X ℓ

)

,

ℓ ≥ 1 ,

generalizing the classic Beauville–Voisin identity for a K3 surface. We emphasize consequences of the conjecture for the structure of the tautological subring

$$R_\star ({\mathsf M}) \subset CH_\star ({\mathsf M}).$$

R ⋆

( M )

⊂ C

H ⋆

( M )

.

The conjecture places all tautological classes in the lowest piece of a natural filtration emerging on

$$CH_\star ({\mathsf M})$$

C

H ⋆

( M )

, which we also discuss. We prove the proposed identities when

$${\mathsf M}$$

M

is the Hilbert scheme of points on a K3 surface.