Remarks and questions on coisotropic subvarieties and 0-cycles of hyper-Kähler varieties

Abstract: This paper proposes a conjectural picture for the structure of the Chow ring of a (projective) hyper-K\"ahler variety, and the construction of a Beauville decomposition, with emphasis on the Chow group of $0$-cycles, which is endowed with a natural filtration of Brill-Noether type. Some of the conjectures are proved in the case of Hilbert schemes of K3 surfaces and Fano varieties of lines of cubic fourfolds.