Chow groups of surfaces of lines in cubic fourfolds

The surface of lines in a cubic fourfold intersecting a fixed line splits motivically into two parts, one of which resembles a K3 surface. We define the analogue of the Beauville-Voisin class and study the push-forward map to the Fano variety of all lines with respect to the natural splitting of the Bloch-Beilinson filtration introduced by Mingmin Shen and Charles Vial.

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