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

Abstract: 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