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...

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Given a smooth projective variety, a Chow–Künneth decomposition is called multiplicative if it is compatible with the intersection product. Following works of Beauville and Voisin, Shen and Vial conjectured that hyper-Kähler varieties admit a multiplicative Chow–Künneth decomposition. In this paper, based on the mysterious link between Fano varieties with cohomology of K3 type and hyper-Kähler varieties, we ask whether Fano varieties with cohomology of K3 type also admit a multiplicative Chow–Künneth decomposition, and provide evidence by establishing their existence for cubic fourfolds and Küchle fourfolds of type c7. The main input in the cubic hypersurface case is the Franchetta property for the square of the Fano variety of lines; this was established in our earlier work in the fourfold case and is generalized here to arbitrary dimension. On the other end of the spectrum, we also give evidence that varieties with ample canonical class and with cohomology of K3 type might admit a multiplicative Chow–Künneth decomposition, by establishing this for two families of Todorov surfaces.