Using a codimension-1 algebraic cycle obtained from the Poincaré line bundle, Beauville defined the Fourier transform on the Chow groups of an abelian variety A and showed that the Fourier transform induces a decomposition of the Chow ring CH * (A). By using a codimension-2 algebraic cycle representing the Beauville-Bogomolov class, we give evidence for the existence of a similar decomposition for the Chow ring of hyperkähler varieties deformation equivalent to the Hilbert scheme of length-2 subschemes on a K3 surface. We indeed establish the existence of such a decomposition for the Hilbert scheme of length-2 subschemes on a K3 surface and for the variety of lines on a very general cubic fourfold.

We prove the generalized Franchetta conjecture for the locally complete family of hyper-Kähler eightfolds constructed by Lehn-Lehn-Sorger-van Straten. As a corollary, we establish the Beauville-Voisin conjecture for very general LLSS eightfolds. The strategy consists in reducing to the Franchetta property for relative fourth powers of cubic fourfolds, by using the recent description of LLSS eightfolds as moduli spaces of semistable objects in the Kuznetsov component of the derived category of cubic fourfolds, together with its generalization to the relative setting due to Bayer-Lahoz-Macrì-Nuer-Perry-Stellari. As a by-product, we compute the Chow motive of the Fano variety of lines on a smooth cubic hypersurface in terms of the Chow motive of the cubic hypersurface. 1 2 LIE FU, ROBERT LATERVEER, AND CHARLES VIAL Hyper-Kähler varieties. It was first conjectured by O'Grady [O'G13] that the universal family of K3 surfaces of given genus over the corresponding moduli space satisfies the Franchetta property. By using Mukai models, this was proved for certain families of K3 surfaces of low genus by Pavic-Shen-Yin [PSY17]. By investigating the case of the Beauville-Donagi family [BD85] of Fano varieties of lines on smooth cubic fourfolds, we were led in [FLVS19] to ask whether O'Grady's conjecture holds more generally for hyper-Kähler varieties : Conjecture 1 (Generalized Franchetta conjecture for hyper-Kähler varieties [FLVS19]). Let F be the moduli stack of a locally complete family of polarized hyper-Kähler varieties. Then the universal family X → F satisfies the Franchetta property.It might furthermore be the case that, for some positive integers n, the relative n-powers X n /F → F satisfy the Franchetta property. This was proved for instance in the case n = 2 in [FLVS19] for the universal family of K3 surfaces of genus ≤ 12 (but different from 11) and for the Beauville-Donagi family of Fano varieties of lines on smooth cubic fourfolds.The first main object of study of this paper is about the locally complete family of hyper-Kähler eightfolds constructed by Lehn-Lehn-Sorger-van Straten [LLSvS17], subsequently referred to as LLSS eightfolds. An LLSS eightfold is constructed from the space of twisted cubic curves on a smooth cubic fourfold not containing a plane. The following result, which is the first main result of this paper, completes our previous work [FLVS19, Theorem 1.11] where the Franchetta property was established for 0-cycles and codimension-2 cycles on LLSS eightfolds. Theorem 1. The universal family of LLSS hyper-Kähler eightfolds over the moduli space of smooth cubic fourfolds not containing a plane satisfies the Franchetta property.As already observed in [FLVS19, Proposition 2.5], the generalized Franchetta conjecture for a family of hyper-Kähler varieties implies the Beauville-Voisin conjecture [Voi08] for the very general member of the family : Corollary 1. Let Z be an LLSS hyper-Kähler eightfold. Then the Q-subalgebra R * (Z) := h, c j (Z) ⊂ CH * (Z) generated by the natural polarization h and the C...

The Hilbert scheme X [3] of length-3 subschemes of a smooth projective variety X is known to be smooth and projective. We investigate whether the property of having a multiplicative ChowKünneth decomposition is stable under taking the Hilbert cube. This is achieved by considering an explicit resolution of the rational map X 3 X [3] . The case of the Hilbert square was taken care of in Shen and Vial [Mem. Amer. Math. Soc. 240(1139) (2016), vii+163 pp]. The archetypical examples of varieties endowed with a multiplicative Chow-Künneth decomposition is given by abelian varieties. Recent work seems to suggest that hyperKähler varieties share the same property. Roughly, if a smooth projective variety X has a multiplicative Chow-Künneth decomposition, then the Chow rings of its powers X n have a filtration, which is the expected Bloch-Beilinson filtration, that is split.

In this paper we give two explicit relations among 1-cycles modulo rational equivalence on a smooth cubic hypersurface X. Such a relation is given in terms of a (pair of) curve(s) and its secant lines. As the first application, we reprove Paranjape's theorem that CH 1 (X) is always generated by lines and that it is isomorphic to Z if the dimension of X is at least 5. Another application is to the intermediate jacobian of a cubic threefold X. To be more precise, we show that the intermediate jacobian of X is naturally isomorphic to the Prym-Tjurin variety constructed from the curve parameterizing all lines meeting a given rational curve on X. The incidence correspondences play an important role in this study. We also give a description of the Abel-Jacobi map for 1-cycles in this setting.