The generalized Franchetta conjecture for some hyper-Kähler varieties

We 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-Kahler 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-Kahler 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.