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

2018/06/21 by Tim-Henrik Bülles, Bülles, Tim-Henrik · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1806.08284

openalex publication_date 2018/06/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \mathfrakh(M) becomes a direct summand of a motive \bigoplus \mathfrakh(S^k_i)(n_i) with k_i≤ dim(M). The result implies that finite dimensionality of \mathfrakh(M) follows from finite dimensionality of \mathfrakh(S). The technique also applies to moduli spaces of twisted sheaves and to moduli spaces of stable objects in \rm D^{\rm b}(S,α) for a Brauer class α∈\rm 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,α,ℤ) ≃ \widetilde H(\cal A_X,ℤ). In this case, we prove that there is an isomorphism of transcendental Chow motives \mathfrakt(S)(1) ≃ \mathfrakt(X).

Cited by

• Arithmetic Period Map and Complex Multiplication for Cubic Fourfolds

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