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We prove that for any polarized symplectic automorphism of the Fano varietyof lines of a cubic fourfold (equipped with the Pl\"ucker polarization), theinduced action on the Chow group of 0-cycles is identity, as predicted byBloch-Beilinson conjecture.

On the action of symplectic automorphisms on the $CH_0$-groups of some hyper-K\"ahler fourfolds