Symplectic automorphisms of Fano varieties of cubic fourfolds and its action on algebraic cycles

This talk is mainly on the Chow-theoretical aspects of projective hyper-Kähler varieties. A smooth projective complex variety is called hyper-Kähler if it is simply-connected and has a unique, up to scalar, holomorphic symplectic 2-form. Given a finite-order symplectic automorphism of such a variety, some generalization of Bloch's conjecture predicts that the induced action on its Chow group of zero-dimensional cycles is trivial. We prove this conjecture for the Fano variety of lines of a smooth cubic fourfold (which is a hyper-Kähler variety by Beauville-Donagi's result) under the extra condition that the automorphism preserves the Plücker polarization. This result partially generalizes a recent theorem of Huybrechts and Voisin in the case of projective K3 surfaces. If time permits, some related classification results will also be touched upon.
References: arXiv:1302.6531, arXiv:1303.2241