The generalized Franchetta conjecture for some hyper-Kähler varieties

International audienceWe study the generalized Franchetta conjecture for holomorphic symplectic varieties. The conjecture predicts that the restriction of an algebraic cycle on the universal family of certain polarized hyper-Kähler varieties to a fiber is rationally equivalent to zero if and only if its cohomology class vanishes. We provide the following evidences : (1) The Beauville–Donagi family of Fano varieties of lines on cubic fourfolds ; (2) The relative square, relative cube, relative Hilbert square and relative Hilbert cube of the universal families of K3 surfaces which are complete intersections in (weighted) projective spaces ; (3) The relative product of the relative r 1 , · · · , r m-th Hilbert powers of the universal family of quartic K3 surfaces, where r 1 +.. .+r m ≤ 5 ; (4) The relative square and relative Hilbert square of the universal families of K3 surfaces of genera 6, 7, 8, 9, 10 and 12 ; (5) Relative square of the universal Fano variety of lines of the universal family of cubic fourfolds ; (6) Zero-cycles and codimension 2 cycles for the Lehn–Lehn–Sorger–van Straten family of hyper-Kähler eightfolds. We also draw many consequences in the direction of the Beauville–Voisin conjecture as well as Voisin's refinement for coisotropic subvarieties. In the appendix, we establish a new relation among tautological cycles on the square of the Fano variety of lines of a smooth cubic fourfold and provide some applications