On the Chow ring of certain algebraic hyper-K\"ahler manifolds

We study a generalization of a conjecture made by Beauville on the Chow ring of hyper-K"ahler algebraic varieties. Namely we prove in a number of cases that polynomial cohomological relations involving only CH^1(X) and the Chern classes of X are satisfied in CH(X). These cases are : punctual Hilbert schemes of a K3 surface S parameterizing subschemes of length n, for n<2b_2(S)_tr+5; Fano varieties of lines in a cubic fourfold.