Abstract
This note is about hyperkahler fourfolds $X$ admitting a non-symplectic involution $\iota $. The Bloch-Beilinson conjectures predict the way $\iota $ should act on certain pieces of the Chow groups of $X$. The main result of this note is a verification of this prediction for Fano varieties of lines on certain cubic fourfolds. This has some interesting consequences for the Chow ring of the quotient $X/\iota $.
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