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 $$\mathfrak {t}(S)(1) \cong \mathfrak {t}(X)$$.