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 $⨁\mathfrak{h}(S^{k_i})(n_i)$ 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 $\tilde{H}(S,\alpha ,\mathbb{Z})\simeq \tilde{H}({\mathcal{A}}_X,\mathbb{Z})$. In this case, we prove that there is an isomorphism of transcendental Chow motives $\mathfrak{t}(S)(1)\simeq \mathfrak{t}(X)$.

15 pages

References in corpus (3)

• The K3 category of a cubic fourfold
• The Fano variety of lines and rationality problem for a cubic hypersurface
• On the rationality and the finite dimensionality of a cubic fourfold