The generalized Franchetta conjecture for some hyper-K\"ahler varieties, II

We prove the generalized Franchetta conjecture for the locally complete family of hyper-K\"ahler eightfolds constructed by Lehn-Lehn-Sorger-van Straten (LLSS). 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-Macr\`i-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.


Introduction
The Franchetta property. -Let f : X → B be a smooth projective morphism between smooth schemes of finite type over the field of complex numbers. For any fiber X of f over a closed point of B, we define GDCH * B (X) := Im CH * (X ) −→ CH * (X) , the image of the Gysin restriction map. Here and in the sequel, Chow groups are always considered with rational coefficients. The elements of GDCH * B (X) are called the generically defined cycles (with respect to the deformation family B) on X. The morphism f : X → B is said to satisfy the Franchetta property for codimension-i cycles if the restriction of the cycle class map GDCH i B (X) −→ H 2i (X, Q) is injective for all (or equivalently, for very general) fibers X. It is said to satisfy the Franchetta property if it satisfies the Franchetta property for codimension-i cycles for all i. At this point, we note that if B is a smooth locally closed subscheme of B, then there is no a priori implication between the Franchetta properties for X → B and for the restricted family X B → B : informally, GDCH i B (X) is generated by more elements than GDCH i B (X); on the other hand, specializing to B creates new relations among cycles. However, if B → B is a dominant morphism, the Franchetta property for X B → B implies the Franchetta property for X → B; see [FLVS19, Rem. 2.6].
The Franchetta property is a property about the generic fiber X η . Indeed it is equivalent to the condition that the composition is injective, where the first map, which is always injective, is the pull-back to the geometric generic fiber and the second one is the cycle class map to some Weil cohomology of X η . in the case m = 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,Th. 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,Prop. 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: Indeed, any subring of CH * (Z) generated by generically defined cycles injects in cohomology, provided Z satisfies the Franchetta property. As such, one further obtains as a direct consequence of Theorem 1 that for an LLSS eightfold Z, the subring of CH * (Z) generated by the polarization h, the Chern classes c j (Z) and the classes of the (generically defined) co-isotropic subvarieties described in [FLVS19,Cor. 1.12] injects in cohomology via the cycle class map. This provides new evidence for Voisin's refinement in [Voi16] of the Beauville-Voisin conjecture. In order to deal with positive-dimensional cycles on Z, we take a completely different approach: we consider the recent description of LLSS eightfolds as certain moduli spaces of semistable objects in the Kuznetsov component of the derived category of cubic fourfolds [LLMS18,LPZ18], together with its generalization due to Bayer-Lahoz-Macrì-Nuer-Perry-Stellari [BLM + 21] to the relative setting. Our first task, which is carried out in Section 1, consists then in relating the Chow motives of the moduli space M of semistable objects in the Kuznetsov component of the derived category of a smooth cubic fourfold Y to the Chow motives of powers of Y . By adapting and refining an argument of Bülles [Bül20], we show in Theorem 1.1 that the motive of M belongs to the thick subcategory generated by Tate twists of the motive of Y m , where dim M = 2m. Since all the data involved in the above are generically defined, the Franchetta property for LLSS eightfolds is thus reduced to the Franchetta property for fourth powers of smooth cubic fourfolds (Theorem 2 below). The proof of Theorem 1 is then given in Section 3; see Theorem 3.1.
Powers of smooth cubic hypersurfaces. -The following theorem, in the case of cubic fourfolds, suggests that the Franchetta property could hold for powers of Fano varieties of cohomological K3-type; that (conjectural) motivic properties of hyper-Kähler varieties could transfer to Fano varieties of cohomological K3-type was already pinpointed in [FLV21]. Theorem 2 is established in Section 2.5. Its proof relies on the existence of a multiplicative Chow-Künneth decomposition for cubic hypersurfaces (see Theorem 2.7), and on an analogue in the case of cubic hypersurfaces of a result of Yin [Yin15a] concerning K3 surfaces (which itself is analogous to a result of Tavakol [Tav14] concerning hyperelliptic curves). The latter is embodied in Corollary 2.13. In the particular case of cubic fourfolds, it admits also the following refined form, which is the analogue of Voisin's [Voi08, Conj. 1.6]: Proposition 1 (see Proposition 2.16). -Let Y be a smooth cubic fourfold and m ∈ N.
Fano varieties of lines on smooth cubic hypersurfaces. -Combining Theorem 2 with our previous work [FLV21,Th. 4.2] where we established the Franchetta property for the square of the Fano variety of lines on a smooth cubic hypersurface, we can compute explicitly the Chow motive of the Fano variety of lines on a smooth cubic hypersurface in terms of the Chow motive of the cubic hypersurface, without resorting to Kimura-O'Sullivan finite-dimensionality arguments. The following is the main result; see Theorem 2.20 for more precise statement and stronger results.
Theorem 3. -Let Y be a smooth cubic hypersurface in P n+1 and F the associated Fano variety of lines on Y . We have an isomorphism of Chow motives In particular, we have an isomorphism of Chow motives Remark. -Let us explain the relations of Theorem 3 to earlier works and open questions in the literature.
(i) The isomorphism (1) lifts the isomorphism in cohomology due to Galkin-Shinder [GS14, Th. 6.1] to the level of Chow motives.
(iii) Theorem 3 refines the main result of [Lat17b] and, in fact, our method of proof, which goes through the Franchetta property for F × F established in [FLV21,Th. 4.2] and a cancellation property described in Proposition 2.19, provides a new, independent, and more conceptual, proof of the main result of [Lat17b].
(iv) Specializing to the case of cubic fourfolds, (1) implies an isomorphism of Chow motives is the K3-surface-like Chow motive. Recall that for a smooth projective surface S, the Hilbert square of S is isomorphic to the blow-up of the symmetric square Sym 2 S along the diagonal and that h(Hi'b 2 (S)) Sym 2 h(S) ⊕ h(S)(−1). Therefore, (3) can be interpreted as saying that the Chow motive of F is the Hilbert square of the Chow motive M of a "non-commutative" K3 surface; this is the motivic analogue of the following folklore conjecture (cf. Further outlooks. -The strategy for proving Theorem 1 has potential beyond the case of LLSS eightfolds. Indeed, once suitable stability conditions are constructed for other non-commutative K3 surfaces (cf. Section 1), one may hope our strategy can be employed to prove the generalized Franchetta conjecture for the associated hyper-Kähler varieties. In Section 4, we exemplify the above by establishing the Franchetta property for many (non locally complete) families of hyper-Kähler varieties.
Conventions. -All algebraic varieties are defined over the field of complex numbers. We work with Chow groups with rational coefficients. The categories of motives we consider are the categories of pure Chow motives with rational coefficients M rat and of pure numerical motives with rational coefficients M num , as reviewed in [And04]. We write h(X) for the Chow motive of a smooth projective variety X. The set of non-negative integers will be denoted by N.  [Yos01]. Recently, Bülles [Bül20] showed that the Chow motive of such a (smooth projective) moduli space is in the thick tensor subcategory generated by the motive of the surface. By allowing non-commutative "Calabi-Yau surfaces", we get even more examples of hyper-Kähler varieties as moduli spaces of stable objects in a 2-Calabi-Yau category equipped with stability conditions [BM14a] [FFZ21,Th. 5.3]. In this section, we provide a refinement of these results following an observation of Laterveer [Lat21].
Let Y be a smooth projective variety and let D b (Y ) be its bounded derived category of coherent sheaves. Let A be an admissible triangulated subcategory of is generated by an exceptional collection (with ⊥ A = 0 in (i)); see [MS19] or [FFZ21,Ex. 5.1] for more details.
Let us now proceed to review the notions of Mukai lattice and Mukai vector. For that purpose, recall that the topological K-theory of Y is naturally equipped with the Euler pairing: Following [AT14], the Mukai lattice of A is defined as the free abelian group Assume that A admits stability conditions in the sense of Bridgeland [Bri07]; this has by now been established for examples (i) ∼ (iii) [Bri08] [YY14] [BLM + 17] [PPZ19], and is also expected for example (iv). We denote the distinguished connected component of the stability manifold by Stab † (A ). Recall that if v is a primitive element in the Mukai lattice of A , a stability condition σ ∈ Stab † (A ) is said to be v-generic if stability coincides with semi-stability for all objects in A with Mukai vector v. General results in [Lie06] and [AHLH18] guarantee that a good moduli space M σ (A , v) of σ-semistable objects in A with Mukai vector v exists as an algebraic space of finite type. Moreover, initiated by Bayer-Macrì [BM14a,BM14b], a much deeper study shows that the moduli space M σ (A , v) is a non-empty projective hyper-Kähler manifold for examples (i), (ii) and (iii), by [BM14b], [BLM + 21], and [PPZ19] respectively (the example (iv) is also expected).
The novelty of this result with respect to [FFZ21,Th. 5.3] is the better bound on the power of Y , which will be crucial in the proof of the Franchetta property.
Proof of Theorem 1.1. -Following Bülles [Bül20], we consider the following chain of two-sided ideals of the ring of self-correspondences of M : where for any non-negative integer k, Note that I 0 = α×β | α, β ∈ CH * (M ) consists of "decomposable" cycles in M ×M . The conclusion of the theorem can be rephrased as saying that ∆ M ∈ I m .
Using Lieberman's formula, Bülles showed [Bül20, Th. 1.1] that the intersection product behaves well with respect to the grading. More precisely, for any k, k 0, The observation of Laterveer [Lat21, Lem. 2.2] is that the vanishing of the irregularity of M implies that any divisor of M × M is decomposable, that is, It was pointed out in [FFZ21,Prop. 5 with P := −Rπ 1,3, * (π * 1,2 (E ) ∨ ⊗ L π * 2,3 (E )), where E is a universal family and the π i,j 's are the natural projections from M × S × M . Therefore our goal is to show that c 2m (P) ∈ I m . We prove by induction that c i (P) ∈ I i/2 for any i ∈ N.
To this end, for any monomial c d1 Using the induction hypothesis that c j ∈ I j/2 for any j i − 1, we see that where denotes the intersection product, the second inclusion uses the multiplicativity (4), and the last inclusion follows from (6). The induction process is complete. We conclude that ∆ M = c 2m belongs to I m , as desired.
In Theorem 1.1, if the Mukai vector v is not primitive, the moduli space of semistable objects M σ (A , v) is no longer smooth (for any stability condition σ). However, in the so-called O'Grady-10 case, namely v = 2v 0 with v 2 0 = 2, for a generic stability condition σ, there exists a crepant resolution of M σ (A , v), which is a projective hyper-Kähler tenfold. In the classical case of moduli spaces of H-semistable sheaves with such a Mukai vector v on a K3 or abelian surface with the polarization H being v-generic, such a crepant resolution was constructed first by O'Grady [O'G99] for some special v, and then by Lehn-Sorger [LS06] and Perego-Rapagnetta [PR13] in general. In the broader setting where A = D b (S) is the derived category of a K3 surface and σ is a v-generic stability condition, the existence of a crepant resolution of M σ (A , v) was proved by Meachan-Zhang [MZ16, Prop. 2.2] using [BM14b]. In our general setting where A is a 2-Calabi-Yau category, Li-Pertusi-Zhao [LPZ20,§3] showed that the singularity of M σ (A , v) has the same local model as in the classical case and that the construction of the crepant resolution in [LS06] can be adapted by using [AHR20]. For details we refer to [LPZ20,§3], where the proof was written for cubic fourfolds, but works in general.
The following two results exemplify the belief that the Chow motive of the crepant resolution can be controlled in the same way as in Theorem 1.1. Theorem 1.2 is for K3 and abelian surfaces, while Theorem 1.3 is in the non-commutative setting where A is the Kuznetsov component of a cubic fourfold.  Proof. -Replacing M everywhere by the stable locus M st in the proof of Theorem 1.1 yields that Indeed, let us give some details: define the chain of subgroups is no longer a ring for the composition of self-correspondences. It is however easy to see that the multiplicativity (4) and the inclusion (5) still hold. Again by [FFZ21,Prop. 5.2], ∆ M st = c 10 (P) with P defined similarly as in the proof of Theorem 1.1, by using the universal sheaf over M st . The Grothendieck-Riemann-Roch theorem implies that the Chern characters of P belong to I 1 . The same induction process as in the proof of Theorem 1.1 shows that the i-th Chern class of P lies in I i/2 for all i. In particular, ∆ M st = c 10 (P) ∈ I 5 , which is nothing but (7). The rest of the proof is as in [FFZ21,§4]. Let us give a sketch. First, as two birationally isomorphic hyper-Kähler varieties have isomorphic Chow motives [Rie14], it is enough to consider the crepant resolution constructed by O'Grady and Lehn-Sorger. By construction, there is a further blow-up M → M whose boundary ∂ M := M M st is the union of two divisors denoted by Ω and Σ. By taking closures, (7) implies that there exist It remains to show that h( Ω) and h( Σ) are both direct summands of Chow motives of the form  Proof. -Using the arguments of [FFZ21,§5], one sees that the proof of the theorem is the same as that of Theorem 1.2 by replacing S by Y .
Remark 1.4. -By way of example, let V = V (Y ) be the compactification of the "twisted intermediate jacobian filtration" constructed in [Voi18]; this V is a hyper-Kähler tenfold of O'Grady-10 type. There exists a tenfold M as in Theorem 1.3 that is birational to V [LPZ20, Th. 1.3], and so one obtains a split injection of Chow motives

The Franchetta property for fourth powers of cubic hypersurfaces
For a morphism Y → B to a smooth scheme B of finite type over a field and for Y a fiber over a closed point of B, we define, for all positive integers m, 2.1. Generically defined cycles and tautological cycles. -We adapt the stratification argument [FLVS19, Prop. 5.7] (which was for Mukai models of K3 surfaces) to its natural generality. We first record the following standard fact.
Lemma 2.1. -Let P be a smooth projective variety. The following conditions are equivalent: The Chow groups of powers of P satisfy the Künneth formula: for any m ∈ N, [Kim09] and [Via10]. The implication (i) + (ii) ⇒ (v) is also clear by the Künneth formula for cohomology. It remains to show (v) ⇒ (iv). Suppose CH * (P 2 ) ∼ = CH * (P ) ⊗ CH * (P ). Then there exist α i , β i ∈ CH * (P ) such that In particular, the identity morphism of CH * (P ) factors through an r-dimensional Q-vector space. Therefore CH * (P ) is finite dimensional.
Definition 2.2. -We say a smooth projective variety P has trivial Chow groups if P satisfies one of the equivalent conditions in Lemma 2.1.
Examples of varieties with trivial Chow groups include homogeneous varieties, toric varieties, and varieties whose bounded derived category of coherent sheaves admits a full exceptional collection [MT15]. Conjecturally, a smooth projective complex variety has trivial Chow groups if and only if its Hodge numbers h i,j vanish for all i = j.
Definition 2.3 (Tautological rings). -Let P be a smooth projective variety with trivial Chow groups and Y a smooth subvariety. The tautological ring of Y is by definition the Q-subalgebra generated by the restrictions of cycles of P and the Chern classes of the tangent bundle T Y . Note that if Y is the zero locus of a dimensionally transverse section of a vector bundle on P , the Chern classes of T Y automatically come from P . More generally, for any m ∈ N, we define the tautological ring of Y m as the Q-subalgebra generated by pull-backs of tautological classes on factors and pull-backs of the diagonal ∆ Y ⊂ Y × Y . Here, p i and p j,k denote the various projections from Y m to Y and to Y 2 . Note that by Lemma 2.1 (v), the cycles coming from the ambient space are all tautological: Similar subrings are studied for hyperelliptic curves by Tavakol [Tav14], for K3 surfaces by Voisin [Voi08] and Yin [Yin15a], and for cubic hypersurfaces by Diaz [Dia19].
Given an equivalence relation ∼ on {1, . . . , m}, we define the corresponding partial diagonal of Y m by {(y 1 , . . . , y m ) ∈ Y m | y i = y j if i ∼ j}. Natural projections and inclusions along partial diagonals between powers of Y preserve the tautological rings. More generally, we have the following fact, which implies that the system of tautological cycles in Definition 2.3 is the smallest one that is preserved by natural functorialities and contains Chern classes and cycles restricted from the ambient spaces.
Lemma 2.4 (Functoriality). -Notation is as before. Let φ : I → J be a map between two finite sets and let f : Y J → Y I be the corresponding morphism. Then Proof. -The fact that tautological rings are preserved by f * is clear from the definition. Let us show that they are preserved by f * . By writing φ as a composition of a surjective map and an injective map, it is enough to show the lemma in these two cases separately.
When φ is surjective, f : Y J → Y I is a partial diagonal embedding. Choosing a section of φ gives rise to a projection p : We leave to the reader the proof in the case where φ is injective (f : Y J → → Y I is then a projection), which is not needed later.
-Let E be a vector bundle on a variety P . Given an integer r ∈ N, we say that the pair (P, E) satisfies condition ( r ) if for any r distinct points x 1 , . . . , x r ∈ P , the evaluation map is surjective, where E(x) denotes the fiber of E at x; or equivalently, is of codimension r · rank(E) in H 0 (P, E). Clearly, ( r ) implies ( k ) for all k < r. Note that condition ( 1 ) is exactly the global generation of E.
Proposition 2.6 (Generic vs. Tautological). -Let P be a smooth projective variety with trivial Chow groups and E a globally generated vector bundle on P . Denote Proof. -This is an adaptation of [FLVS19, Prop. 5.7], which relies on the notion of stratified projective bundle [FLVS19, Def. 5.1]. Let q : Y r /B → P r be the natural projection. The morphism q is a stratified projective bundle, where the strata of P r are defined by the different types of incidence relations of r points in P : gives that the generically defined cycles GDCH * B (Y r ) := Im CH * (Y r /B ) → CH * (Y r ) can be expressed as follows: We proceed to show inductively (just as in [FLVS19, Proof of Prop. 5.7]) that each term on the right-hand side of (9) is in the tautological ring R * (Y r ): -For i = 0, this follows simply from the fact that CH * (T 0 ) = CH * (P r ) ∼ = CH * (P ) ⊗r .
-Assume a general point of T i parameterizes r points of P with at least two of them coinciding. Then the contribution of the i-th summand of (9) factors through GDCH * B (Y r−1 ) via the diagonal push-forward. By the induction hypothesis, this is contained in the diagonal push-forward of R * (Y r−1 ), hence by Lemma 2.4 is contained in R * (Y r ).
-Assume a general point of T i parameterizes r distinct points of P . In that case, the condition ( r ) guarantees that the codimension of Y i in Y i−1 is equal to codim Ti−1 (T i ). The excess intersection formula ([Ful98, §6.3]), applied to the Cartesian square tells us that modulo the (i + 1)-th term of (9), the contribution of the i-th term is contained in the (i − 1)-th term.
-Finally, the contribution of the i = m term of (9) is the push-forward of R * (Y ), via the small diagonal embedding Y → Y m . This is contained in R * (Y m ) by Lemma 2.4.

Multiplicative Chow-Künneth decomposition for smooth cubic hypersurfaces
Let Y ⊂ P n+1 be a smooth hypersurface of degree d. Recall that if denotes the hyperplane section, then the correspondences The notion of multiplicative Chow-Künneth (MCK) decomposition was introduced by Shen-Vial [SV16a]. While the existence of a Chow-Künneth decomposition is expected for all smooth projective varieties-in which case the cycles π k • δ Y • (π i ⊗ π j ) vanish in H 4n (Y × Y × Y ) for all k = i + j due to the fact that the cohomology ring H * (Y, Q) is graded, there are examples of varieties that do not admit such a multiplicative decomposition; it was however conjectured [SV16a, Conj. 4], following the seminal work of Beauville and Voisin [BV04], that all hyper-Kähler varieties admit such a decomposition. We refer to [FLV21] for a list of hyper-Kähler varieties for which the conjecture has been established, as well as for some evidence that Fano varieties of cohomological K3 type (e.g. smooth cubic fourfolds) should also admit such a decomposition, but also for examples of varieties not admitting such a decomposition.
A new proof of Theorem 2.7 is given in [FLV21]. The strategy consists in reducing to the Franchetta property for the universal family F → B of Fano varieties of lines in smooth cubic hypersurfaces and its relative square. Note that, in that reduction step, we in fact established the Franchetta property for Y → B and for Y × B Y → B.
That the canonical Chow-Künneth decomposition (10) of a hypersurface be multiplicative can be spelled out explicitly as follows. (Note that the expression (11) with d = 3 corrects the expression for the cycle γ 3 in [Dia19, §2].) will be subsequently referred to as the MCK relation: where p i and p j,k denote the various projections from Y 3 to Y and to Y 2 .
Proof. -Let us define π 2i alg := 1 d h n−i × h i for 0 i n; these coincide with π 2i if 2i = n. A direct calculation shows that . Assume temporarily that Y has no primitive cohomology, i.e., Y has degree 1 or Y is an odd-dimensional quadric. In that case, we have ∆ Y = i π 2i alg ; in particular by (12) the Chow-Künneth decomposition (10) is multiplicative. Furthermore, from Combining the above two expressions for ∆ Y and δ Y yields the relation (11). From now on, we therefore assume that Y has non-trivial primitive cohomology. In that case, the above relations imply that the Chow-Künneth decomposition (10) is multiplicative if and only if π 2n • δ Y • (π n ⊗ π n ) = δ Y • (π n ⊗ π n ).
Substituting π n = ∆ Y − i =n π i into the identity π 2n •δ Y •(π n ⊗π n ) = δ Y •(π n ⊗π n ), and developing, yields an identity of the form where λ is a rational number to be determined and Q is a symmetric rational polynomial in 3 variables to be determined. Projecting on the first two factors yields an identity in CH n (Y × Y ) ∆ Y = dλ ∆ Y + R(p * 1 h, p * 2 h) for some symmetric rational polynomial R in 2 variables. Since ∆ Y is not of the form S(p * 1 h, p * 2 h) for some symmetric rational polynomial S in 2 variables (otherwise, the cohomology ring of Y would be generated by h), we find that λ = 1/d. The coefficients of the polynomial Q are then obtained by successively applying (p 1,2 ) * ((−) · p * 3 h n−k ) for k 0 and by symmetrizing. Note that in the above we use the identity which is obtained by applying the excess intersection formula [Ful98, Th. 6.3] to the following Cartesian diagram, with excess normal bundle O Y (d), , where by abuse we have denoted h a hyperplane section of both P n+1 and Y .
Remark 2.9. -Using the above-mentioned fact that the cycles π k • δ Y • (π i ⊗ π j ) vanish in H 4n (Y ×Y ×Y ) for all k = i+j, the proof of Proposition 2.8 also establishes that the MCK relation (11) holds in H 4n (Y ×Y ×Y, Q) for all smooth hypersurfaces Y in P n+1 . We note that if (Y, h) satisfies the conclusion of Question 2.10 for m = 2, then . This fails for a very general curve of genus 4 where h is taken to be the canonical divisor; see [GG03] and [Yin15b]. Furthermore, the MCK relation (11) does not hold for a very general curve of genus 3 (see [FV20,Prop. 7.2]) so that the conclusion of Question 2.10 for m = 3 fails for a very general curve of genus 3. Note however that Tavakol [Tav18] has showed that any hyperelliptic curve (in particular any curve of genus 1 or 2) satisfies the conclusion of Question 2.10 for all m 2.
Proposition 2.11 below, which parallels [Tav18] in the case of hyperelliptic curves and [Yin15a] in the case of K3 surfaces, shows that for a Fano or Calabi-Yau hypersurface Y the only non-trivial relations among tautological cycles in powers of Y are given by the MCK relation (11) and the finite-dimensionality relation. As a consequence, we obtain in Corollary 2.13 that any Kimura-O'Sullivan finite-dimensional cubic hypersurface (e.g. cubic threefolds and cubic fivefolds-conjecturally all cubic hypersurfaces in any positive dimension since smooth projective varieties are conjecturally Kimura-O'Sullivan finite-dimensional, cf. [Kim05]) satisfies the conclusion of Question 2.10 for all m 2.
Proposition 2.11. -Let Y be a Fano or Calabi-Yau smooth hypersurface in P n+1 ; i.e., a hypersurface in P n+1 of degree n + 2. Then

has a positive answer for all m if and only if the Chow-Künneth decomposition (10) is multiplicative and the Chow motive of Y is Kimura-O'Sullivan finite-dimensional.
Before giving the proof of the proposition, let us introduce some notations. Let h be the hyperplane section class. We denote o the class of (1/deg(Y ))h n ∈ CH 0 (Y ). If Y is Fano, o is represented by any point; if Y is Calabi-Yau, then o is the canonical 0-cycle studied in [Voi12]. For ease of notation, we write o i and h i for p * i o and p * i h respectively, where p i : Y m → Y is the projection on the i-th factor. Finally we define the following correspondence in CH n (Y × Y ): where π n is the Chow-Künneth projector defined in (10), and we set τ i,j := p * i,j τ , where p i,j : Y m → Y × Y is the projection on the product of the i-th and j-th factors. Note that τ is an idempotent correspondence that commutes with π n , and that cohomologically it is nothing but the orthogonal projector on the primitive cohomology for the direct summand of h(Y ) cut out by τ and call it the primitive summand of h(Y ).
We note that Proposition 2.11 is trivial in the case b pr (Y ) = 0. Indeed in that case Y is either a hyperplane (hence isomorphic to projective space) or an odd-dimensional quadric. In both cases, the Chow motive of Y is known to be of Lefschetz type, so that it is finite-dimensional and any Chow-Künneth decomposition is multiplicative (see [SV16b,Th. 2]). From now on, we will therefore assume that b pr (Y ) = 0. In order to prove Proposition 2.11, we first determine as in [Yin15a,Lem. 2.3] the cohomological relations among the cycles introduced above.
is isomorphic to the free graded Q-algebra generated by o i , h i , τ j,k , modulo the following relations: and its permutations under S 2bpr+2 if n is even, Proof. -The proof is directly inspired by, and closely follows, [Tav18, §3.1] and [Yin15a,§3].
First, we check that the above relations hold in H * (Y m , Q). The relations (15) take place in Y and are clear. The relations (16) take place in Y 2 : the relation τ i,j = τ j,i is clear, the relation τ i,j · h i = 0 follows directly from (13), while the relation τ i,j · τ i,j = (−1) n b pr o i ·o j follows from the additional general fact that deg(∆ Y ·∆ Y ) = χ(Y ), the topological Euler characteristic of Y which in our case is χ(Y ) = n+1+(−1) n b pr . The relation (17) takes place in Y 3 and follows from the relations (15) and (16) together with the fact that the cohomology algebra H * (Y m , Q) is graded (see also Remark 2.9). Finally, if n is even, the relation (18) takes place in Y 2(bpr+1) and expresses the fact that the (b pr + 1)-th exterior power of H n (Y, Q) prim vanishes; while if n is odd, the relation (18) takes place in Y bpr+2 and follows from the vanishing of the (b pr + 2)-th symmetric power of H n (Y, Q) prim viewed as a super-vector space.
Second, we check that these relations generate all relations among o i , h i , τ j,k , 1 i m, 1 j < k m. To that end, let R * (Y m ) denote the formally defined graded Q-algebra generated by symbols τ i,j , o i , h i (placed in appropriate degree) modulo relations (15), (16), (17), (18). We claim that the pairing  (14). The relations (15) hold true in R * (Y ) for any smooth hypersurface Y . The relations (16) hold true in R * (Y 2 ) for any smooth hypersurface Y ; this is a combination of (13) and of the identity The relation (17) takes place in R * (Y 3 ) and, given the relations (15) and (16)  clearly defines a cycle on Sym bpr+2 h n (Y ) prim , it vanishes. Conversely, if R n(bpr+1) (Y 2(bpr+1) ) injects in cohomology via the cycle class map, then the tautological cycle which defines an idempotent correspondence on (h n (Y ) prim ) ⊗(bpr+1) with image Sym bpr+1 h n (Y ) prim , is rationally trivial since Sym bpr+1 H n (Y, Q) prim = 0.
Finally, by combining Proposition 2.11(ii) with Theorem 2.7, we have the following result in the case of cubic hypersurfaces: Corollary 2.13. -Let Y be a smooth cubic hypersurface in P n+1 . Then the tauto- Remark 2.14. -We note that all smooth projective curves, and in particular cubic curves, are Kimura-O'Sullivan finite-dimensional, and that cubic surfaces have trivial Chow groups (Definition 2.2) and hence are Kimura-O'Sullivan finite-dimensional by Lemma 2.1 and by the fact that Lefschetz motives are finite-dimensional. In addition, cubic threefolds and fivefolds are known to be Kimura-O'Sullivan finite dimensional; see [GG12] and [Via13,Ex. 4.12], respectively. Hence, for n = 1, 2, 3, 5, Corollary 2.13 gives the injectivity R * (Y m ) → H 2 * (Y m , Q) for all m. We note further that in case n = 1, after fixing a closed point O in Y , the pair (Y, O) defines an elliptic curve; in this case the injectivity R * (Y m ) → H 2 * (Y m , Q) for all m is due to Tavakol [Tav11], and can also be deduced directly from O'Sullivan's theory of symmetrically distinguished cycles [O'S11].

2.4.
On the extended tautological ring of smooth cubic fourfolds. -This paragraph is not needed in the rest of the paper; its aim is to show how the arguments of Section 2.3 can be refined to establish analogues of [Yin15a, Th.] concerning K3 surfaces in the case of cubic fourfolds.
For a K3 surface S, let R * (S m ) ⊂ CH * (S m ) be the Q-subalgebra generated by CH 1 (S) and the diagonal ∆ S ∈ CH 2 (S × S). Voisin conjectures [Voi08, Conj. 1.6] that R * (S m ) injects into cohomology (by [Yin15a] this is equivalent to Kimura-O'Sullivan finite-dimensionality of S). Here is a version of Voisin's conjecture for cubic fourfolds that refines Question 2.10: Conjecture 2.15. -Let Y be a smooth cubic fourfold, and m ∈ N. Let R * (Y m ) be the Q-subalgebra where p i , p j and p k, denote the various projections from Y m to Y and to Y 2 . Then In what follows, b tr (Y ) denotes the dimension of the transcendental cohomology of the smooth cubic fourfold Y , i.e., the dimension of the orthogonal complement in H * (Y, Q) of the subspace consisting of Hodge classes. Using the multiplicative Chow-Künneth relation (11) for cubic hypersurfaces, we can adapt the method of Yin concerning K3 surfaces [Yin15a] and prove the following result. Before giving the proof of Proposition 2.16, let us introduce some notations. We fix a smooth cubic fourfold Y . First we note that the cycle class map CH 2 (Y ) → H 4 (Y, Q) is injective and an isomorphism on the Hodge classes in H 4 (Y, Q). Let { s } s be an orthogonal basis of the Hodge classes in H 4 (Y, Q) prim , i.e., {h 2 } ∪ { s } s forms a basis of CH 2 (Y ) with the property that s · s = 0 whenever s = s and s · h = 0 for all s. Note that the latter property s · h = 0 holds cohomologically (by definition of primitive cohomology) and lifts to rational equivalence since s · h must be a rational multiple of h 3 in CH 3 (Y ). Recall that all points on Y are rationally equivalent and that o denotes the class of any point on Y . For ease of notation, we write o i , s i and h i for p * i o, p * i s and p * i h respectively, where p i : Y m → Y is the projection on the i-th factor. Finally we define where π 4 is the Chow-Künneth projector π 4 Y defined in (10) (in our case, n = 4), and we set τ i,j = p * i,j τ , where p i,j : Y m → Y × Y is the projection on the product of the i-th and j-th factors. Note that τ is an idempotent correspondence, and that cohomologically it is nothing but the orthogonal projector on the transcendental cohomology of Y , i.e., on the orthogonal complement of the space of Hodge classes in H * (Y, Q).
In order to prove Proposition 2.16, we establish as in [Yin15a, Lem. 2.3] sufficiently many relations among the cycles introduced above. Central is the MCK relation (11) of Proposition 2.8.
where b tr := b tr (Y ) is the rank of the transcendental part of H 4 (Y, Q).
Proof. -The relations (19) take place in Y and were established in the discussion above the lemma. The relations (20) take place in Y 2 . The relation τ i,j = τ j,i is trivial and the relation τ i,j · h i = 0 follows directly from (13). The relation τ i,j · τ i,j = b tr o i · o j follows from the general fact that deg(∆ Y · ∆ Y ) = χ(Y ), the topological Euler characteristic of Y , and the fact that where δ Y is the small diagonal of Y 3 , seen as a correspondence from Y to Y × Y and where we used the relation h · s = 0 together with the MCK relation (11) of Proposition 2.8. Finally, the relation (21) takes place in Y 3 and follows from the relations (19) and (20) Proof. -In view of Proposition 2.6, we simply check that (P n+1 , O(3)) satisfies the condition ( 4 ). Let ∆ i,j := p −1 i,j (∆ P n+1 ) be a big diagonal in (P n+1 ) 4 . Since all the closed orbits of the natural action of PGL n+2 on (P n+1 ) 4 ( i,j ∆ i,j ) parameterize four collinear points, we only need to check ( 4 ) for four collinear points x 1 , . . . , x 4 . In this case, the needed surjectivity follows from surjectivity of the restriction and the evaluation where P 1 is the line containing these points.
Proof of Theorem 2. -By Proposition 2.18, when m 4, GDCH * B (Y m ) is generated by tautological cycles, i.e., GDCH * B (Y m ) = R * (Y m ). We then note that Corollary 2.13 and Remark 2.14 give the injectivity of R * (Y m ) → H 2 * (Y m , Q) for all m if n = 1, 2, 3, 5, for m 45 if n is even and for m 171 if n is odd (indeed, for n > 2 even we have b pr (Y ) 22 and for n > 5 odd we have b pr (Y ) 170, see e.g. [Huy19, Cor. 1.11]).

The Franchetta property and the cancellation property for Chow motives
Due to Theorem 2, the following motivic proposition applies to cubic hypersurfaces.
Proposition 2.19. -Let X → B be a smooth projective family parameterized by a smooth quasi-projective variety B. Let X := X b be a closed fiber and consider the additive thick subcategory M X of M rat generated by the Tate twists h(X)(n), n ∈ Z, with morphisms given by generically defined correspondences. Assume that X satisfies the standard conjectures and that X 2 /B → B has the Franchetta property. Then M X is semi-simple. In particular, cancellation holds, i.e., if we have Proof. -By definition, the objects of M X are of the form (Z, p, n) with Z = i X ×P ni for finitely many n i ∈ Z 0 , p ∈ End Mrat (h(Z)) an idempotent and generically defined correspondence, and n an integer; and the morphisms Hom M X ((Z 1 , p 1 , n 1 ), (Z 2 , p 2 , n 2 )) ⊆ Hom Mrat ((Z 1 , p 1 , n 1 ), (Z 2 , p 2 , n 2 )) are given by the subspace consisting of generically defined correspondences.
By the Franchetta property for X 2 /B → B and the coincidence of homological and numerical equivalence on X × X, the restriction of the functor M rat → M num to M X is fully faithful. Let us denote M X its essential image. We have to show that M X is semi-simple. This follows simply from the fact that, for M ∈ M X , End M X (M ) is a sub-algebra of the algebra End Mnum (M ) which is semi-simple by the main theorem of [Jan92].

2.7.
Application to the motive of the Fano variety of lines on a smooth cubic hypersurface. -We start with the observation that the Chow-Künneth decomposition (10) for smooth hypersurfaces is generically defined in the following sense. Let be the open subset parameterizing smooth hypersurfaces of degree d in P n+1 , and let Y → B be the universal family. If H ∈ CH 1 (Y ) denotes the relative hyperplane section, then the relative correspondences define a relative Chow-Künneth decomposition, in the sense that they are relative idempotent correspondences and their specializations to any fiber Y b over b ∈ B gives a Chow-Künneth decomposition of Y b , and in fact restrict to the Chow-Künneth decomposition (10) for all b ∈ B. Moreover, the idempotent correspondence τ of (14) defining the direct summand h n (Y ) prim is also generically defined with respect to the family Y → B; indeed, the relative correspondence defines a relative idempotent correspondence whose specialization to any fiber Y b over b ∈ B is the idempotent correspondence τ of (14).
Let us now focus on the case where Y is a smooth cubic hypersurface in P n+1 . Let F be its Fano variety of lines, which is known to be smooth projective of dimension 2n−4. As before, we denote B ⊂ PH 0 (P n+1 , O(3)) the Zariski open subset parameterizing smooth cubic hypersurfaces of dimension n; and we denote Y → B and F → B the corresponding universal families. The first isomorphism in the following theorem is a motivic lifting of [GS14]. It refines, and gives a new proof of, the main result of [Lat17b].
Theorem 2.20. -Notation is as above.
(i) We have an isomorphism of Chow motives

(iii) We have an isomorphism of Chow motives
(iv) F and Y have canonical Chow-Künneth decompositions, and where the isomorphism is given by P * : h n (Y ) prim (1) → h n−2 (F ) and in the even case, for the i-th copy, where 1 i (n + 2)/4 , P = {(y, ) ∈ Y × F | y ∈ } is the incidence correspondence and g := −c 1 (E | F ), c := c 2 (E | F ) with E being the rank-2 tautological bundle on the Grassmannian Gr(P 1 , P n+1 ).
(v) If n = 4, cup-product induces an isomorphism of Chow motives Proof. -Our starting point is the isomorphism of Chow motives As in the proof of Proposition 2.19, denote M X the semisimple category that is the essential image of M X in M num . Looking at the reduction modulo numerical equivalence of the isomorphism (26) (which takes place in M X ), and using the semisimplicity of M num [Jan92], we obtain a split injective morphism Using the standard conjectures for F and Y , combined with the Franchetta property for F × F [FLV21, Th. 4.2], this lifts to a split injection of Chow motives (2) via a generically defined correspondence. It follows that the Franchetta property for F × B Y × B Y is implied by that for Y m /B with m 4, which is Theorem 2. With all conditions of Proposition 2.19 verified for X, we deduce that the category M X is semi-simple and in particular, the cancellation property holds. We obtain (24) by canceling an isomorphic direct summand from both sides of (26). Statement (ii) follows from (i), by writing Likewise, statement (iii) follows from (i) by writing For statement (iv), we observe that F has a generically defined Künneth decomposition; using the Franchetta property for F × F , this is a generically defined Chow-Künneth decomposition. The isomorphism (25) is generically defined and holds true in cohomology [FLV21,Prop. 4.8]. As such, the isomorphism (25) holds in M X ⊂ M num . But M X → M X is fully faithful, proving (iv). Statement (v) is proved similarly, using as input the well-known fact that cupproduct induces an isomorphism H 4 (F, Q) ∼ = Sym 2 H 2 (F, Q), and the Franchetta property for F × F .  For any cubic fourfold Y not containing a plane, the result of Li-Pertusi-Zhao [LPZ18, Th. 1.2] says that the LLSS hyper-Kähler eightfold Z(Y ) associated to Y is isomorphic to M σ (A Y , 2λ 1 + λ 2 ), the moduli space of σ-stable objects in A Y with Mukai vector 2λ 1 + λ 2 (alternatively, one can reduce the proof of Theorem 3.1 to the very general cubic fourfold Y , for which the modular construction of Z(Y ) was already done in [LLMS18,Main Th.].) Theorem 1.1 applied in this special case implies that there exists a split injection of Chow motives:

The generalized Franchetta conjecture for LLSS eightfolds
The theory of stability conditions in family has recently been worked out in [BLM + 21], and as such the isomorphism of Li-Pertusi-Zhao can be formulated in a relative setting. More precisely, the family Z → B • is isomorphic, as B • -scheme, to the relative (smooth and projective) moduli space M → B • , whose fiber over b is . It is also clear from the proof of Theorem 1.1 that the injection (27), as well as its left inverse, is generically defined over B • and gives rise to the following morphism of relative Chow motives over B • which is fiberwise a split injection: As a consequence, for any b ∈ B • , we have the following commutative diagram (29) , where GDCH * (Z b ) := Im(CH * (Z ) → CH * (Z b )) is the group of generically defined cycles. In the above diagram (29), the two vertical arrows are injective by (28), the bottom arrow is injective by Theorem 2. Therefore the top arrow is also injective, which is the content of the Franchetta property for the family Z → B • .

Further results
The aim of this section is to show how the results of Section 1 also make it possible to establish the Franchetta property and to deduce Beauville-Voisin type results for certain non locally complete families of hyper-Kähler varieties. These include certain moduli spaces of sheaves on K3 surfaces (Corollary 4.2), and certain O'Grady tenfolds (Theorem 4.3). We also include a Beauville-Voisin type result for Ouchi eightfolds (Corollary 4.4).

The Franchetta property for some moduli spaces of sheaves on K3 surfaces
We show how Theorem 1.1 makes it possible to extend our previous results [FLVS19, Ths 1.4 & 1.5] on the Franchetta property for certain Hilbert schemes of points on K3 surfaces of small genus to the case of certain moduli spaces of sheaves on K3 surfaces of small genus.
Let F g be the moduli stack of polarized K3 surfaces of genus g and let S → F g be the universal family. Denote by H the universal ample line bundle of fiberwise self-intersection number 2g − 2.
Theorem 4.1. -Let m be a positive integer. If S m /Fg → F g satisfies the Franchetta property, then for any r, d, s ∈ N such that 0 d 2 (g−1)−rs+1 m and gcd(r, d, s) = 1, the relative moduli space M → F g of H-stable sheaves with primitive Mukai vector v = (r, dH, s) also satisfies the Franchetta property.
Proof. -For a given K3 surface S of genus g, the moduli space M H (S, v) is a projective hyper-Kähler variety of dimension v 2 + 2 = d 2 (2g − 2) − 2rs + 2 2m. By Theorem 1.1, we have the following split injective morphism of Chow motives It is clear from the proof of Theorem 1.1 that the above split injective morphism, as well as its left inverse, is generically defined over F g . We have therefore a morphism of relative Chow motives (over F g ) that is fiberwise a split injection: As a consequence, for any b ∈ F g , we have the following commutative diagram where GDCH * (M H (S b , v)) := Im CH * (M ) → CH * (M H (S b , v)) is the group of generically defined cycles relative to F g . In the above diagram (31), the two vertical arrows are injective by (30), the bottom arrow is injective by hypothesis. Therefore the top arrow is also injective, which is the content of the Franchetta property for the family M → F g .
Combined with [FLVS19, Ths 1.4 & 1.5], we get their generalization as follows. Then for any r, d, s ∈ N such that 0 d 2 (g − 1) − rs + 1 m(g) and gcd(r, d, s) = 1, the relative moduli space M → F g of H-stable sheaves with primitive Mukai vector v = (r, dH, s) satisfies the Franchetta property. (i) Assume that S 5 /Fg → F g satisfies the Franchetta property. Then M → F g satisfies the Franchetta property.
(ii) In case g = 3 (quartic surface case), the family M → F 3 has the Franchetta property. In particular, the Beauville-Voisin conjecture is true for the very general element of M → F 3 .
Proof. -Statement (i) follows from Theorem 1.2, plus the observation that the construction of the split injection of that theorem can be performed relatively over  injects into cohomology, via the cycle class map. Here, h is the natural polarization and Y ⊂ Z is the Lagrangian embedding constructed in [Ouc17].
Proof. -Referring to the notation of the proof of Theorem 3.1, one considers the relative moduli space M → B, whose fiber over b ∈ B, denoted M b , is M σ (A Y b , 2λ 1 +λ 2 ). For b ∈ B • , M b is isomorphic to the LLSS eightfold associated to Y b by [LPZ18]; while for a very general point b ∈ B B • , M b is isomorphic to the Ouchi eightfold associated to Y b by [BLM + 21, Ex. 32.6]. Moreover, the classes h, c j (M b ) and [Y b ] on Ouchi eightfolds are specializations of the corresponding classes on LLSS eightfolds. Thanks to Theorem 3.1, the Q-subalgebra generated by h, c j and [Y ] (which are generically defined with respect to B) injects into cohomology. By specialization, the same holds true for Ouchi eightfolds.