Motives of moduli spaces on K3 surfaces and of special cubic fourfolds

Abstract

For any smooth projective moduli space M of Gieseker stable sheaves on a complex projective K3 surface (or an abelian surface) S, we prove that the Chow motive $\mathfrak {h}(M)$ becomes a direct summand of a motive $\bigoplus \mathfrak {h}(S^{k_{i}})(n_i)$ with $k_i\le \dim (M)$. The result implies that finite dimensionality of $\mathfrak {h}(M)$ follows from finite dimensionality of $\mathfrak {h}(S)$. The technique also applies to moduli spaces of twisted sheaves and to moduli spaces of stable objects in $\mathrm{D}^{\mathrm{b}}(S,\alpha )$ for a Brauer class $\alpha \in \mathrm{Br}(S)$. In a similar vein, we investigate the relation between the Chow motives of a K3 surface S and a cubic fourfold X when there exists an isometry $\widetilde{H}(S,\alpha ,\mathbb {Z}) \cong \widetilde{H}(\mathcal{A}_X,\mathbb {Z})$. In this case, we prove that there is an isomorphism of transcendental Chow motives $\mathfrak {t}(S)(1) \cong \mathfrak {t}(X)$.