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...

A remarkable result of Peter O’Sullivan asserts that the algebra epimorphism from the rational Chow ring of an abelian variety to its rational Chow ring modulo numerical equivalence admits a (canonical) section. Motivated by Beauville’s splitting principle, we formulate a conjectural Section Property which predicts that for smooth projective holomorphic symplectic varieties there exists such a section of algebra whose image contains all the Chern classes of the variety. In this paper, we investigate this property for (not necessarily symplectic) varieties with a Chow motive of abelian type. We introduce the notion of a symmetrically distinguished abelian motive and use it to provide a sufficient condition for a smooth projective variety to admit such a section. We then give a series of examples of varieties for which our theory works. For instance, we prove the existence of such a section for arbitrary products of varieties with Chow groups of finite rank, abelian varieties, hyperelliptic curves, Fermat cubic hypersurfaces, Hilbert schemes of points on an abelian surface or a Kummer surface or a K3 surface with Picard number at least 19, and generalized Kummer varieties. The latter cases provide evidence for the conjectural Section Property and exemplify the mantra that the motives of holomorphic symplectic varieties should behave as the motives of abelian varieties, as algebra objects.

Given a smooth projective variety M endowed with a faithful action of a finite group G, following Jarvis-Kaufmann-Kimura [36] and Fantechi-Göttsche [26], we define the orbifold motive (or Chen-Ruan motive) of the quotient stack [M/G] as an algebra object in the category of Chow motives. Inspired by Ruan [51], one can formulate a motivic version of his Cohomological HyperKähler Resolution Conjecture (CHRC). We prove this motivic version, as well as its K-theoretic analogue conjectured in [36], in two situations related to an abelian surface A and a positive integer n. Case (A) concerns Hilbert schemes of points of A : the Chow motive of A [n] is isomorphic as algebra objects, up to a suitable sign change, to the orbifold motive of the quotient stack [A n /S n ]. Case (B) for generalized Kummer varieties : the Chow motive of the generalized Kummer variety K n (A) is isomorphic as algebra objects, up to a suitable sign change, to the orbifold motive of the quotient stack [A n+1is the kernel abelian variety of the summation map A n+1 → A. As a byproduct, we prove the original Cohomological HyperKähler Resolution Conjecture for generalized Kummer varieties.As an application, we provide multiplicative Chow-Künneth decompositions for Hilbert schemes of abelian surfaces and for generalized Kummer varieties. In particular, we have a multiplicative direct sum decomposition of their Chow rings with rational coefficients, which is expected to be the splitting of the conjectural Bloch-Beilinson-Murre filtration. The existence of such a splitting for holomorphic symplectic varieties is conjectured by Beauville [10]. Finally, as another application, we prove that over a non-empty Zariski open subset of the base, there exists a decomposition isomorphism Rπ * Q ≃ ⊕R i π * Q[−i] in D b c (B) which is compatible with the cup-products on both sides, where π : K n (A) → B is the relative generalized Kummer variety associated to a (smooth) family of abelian surfaces A → B.Meta-conjecture 1.2 (MHRC). Let X be a smooth proper complex Deligne-Mumford stack with underlying coarse moduli space X being a (singular) symplectic variety. If there is a symplectic resolution Y → X, then we have an isomorphism h(Y) ≃ h orb (X) of commutative algebra objects in CHM C , hence in particular an isomorphism of graded C-algebras : CH * (Y) C ≃ CH * orb (X) C .See Definition 3.1 for generalities on symplectic singularities and symplectic resolutions. The reason why it is only a meta-conjecture is that the definition of orbifold Chow motive for a smooth proper Deligne-Mumford stack in general is not available in the literature and we will not develop the theory in this generality in this paper (see however Remark 2.10). From now on, let us restrict ourselves to the case where the Deligne-Mumford stack in question is of the form of a global quotient X = [M/G], where M is a smooth projective variety with a faithful action of a finite group G, in which case we will define the orbifold Chow motive h orb (X) in a very explicit way in Definition 2.5.The Motiv...