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 \(\mathfrak {t}(S)(1) \cong \mathfrak {t}(X)\).
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Acknowledgements
I am grateful to Daniel Huybrechts for invaluable suggestions and explanations. This work has benefited from many discussions with Thorsten Beckmann, whom I wish to thank. Finally, many thanks to Axel Kölschbach and Andrey Soldatenkov for helping to improve the exposition and to the referee for a very careful reading and several useful suggestions. This work is part of the author’s Master’s thesis.
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The author is partially supported by SFB/TR 45 ‘Periods, Moduli Spaces and Arithmetic of Algebraic Varieties’ of the DFG (German Research Foundation).
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Bülles, TH. Motives of moduli spaces on K3 surfaces and of special cubic fourfolds. manuscripta math. 161, 109–124 (2020). https://doi.org/10.1007/s00229-018-1086-0
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DOI: https://doi.org/10.1007/s00229-018-1086-0