Abstract
This note is about the Chow groups of a certain family of smooth cubic fourfolds. This family is characterized by the property that each cubic fourfold X in the family has an involution such that the induced involution on the Fano variety F of lines in X is symplectic and has a K3 surface S in the fixed locus. The main result establishes a relation between X and S on the level of Chow motives. As a consequence, we can prove finite-dimensionality of the motive of certain members of the family.
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