Cubic fourfolds, Kuznetsov components and Chow motives

We prove that the Chow motives of two smooth cubic fourfolds whose Kuznetsov components are Fourier-Mukai derived-equivalent are isomorphic as Frobenius algebra objects. As a corollary, we obtain that there exists a Galois-equivariant isomorphism between their l-adic cohomology Frobenius algebras. We also discuss the case where the Kuznetsov component of a smooth cubic fourfold is Fourier-Mukai derived-equivalent to a K3 surface.


Introduction
In [FV21], we asked whether the bounded derived category of coherent sheaves on a hyper-Kähler variety X encodes the intersection theory on X and its powers.Precisely, given two hyper-Kähler varieties X and X ′ that are derived-equivalent, i.e.D b (X) ≃ D b (X ′ ), we asked whether the Chow motives with rational coefficients of X and X ′ are isomorphic as algebra objects.The main result of [FV21] establishes this in the simplest case where X and X ′ are K3 surfaces.The above expectation refines, in the special case of hyper-Kähler varieties, a general conjecture of Orlov [Orl03], predicting that two derived-equivalent smooth projective varieties have isomorphic Chow motives with rational coefficients.
Like hyper-Kähler varieties, the so-called K3-type varieties also behave in many ways like K3 surfaces.By definition [FLV21b], those are Fano varieties X of even dimension 2n with Hodge numbers h p,q (X) = 0 for all p = q except for h n−1,n+1 (X) = h n+1,n−1 (X) = 1.Some basic examples of such varieties are cubic fourfolds, Gushel-Mukai fourfolds and sixfolds [Muk89,KP18], and Debarre-Voisin 20-folds [DV10].As an important interplay between Fano varieties of K3 type and hyper-Kähler varieties, many hyper-Kähler varieties are constructed as moduli spaces of stable objects on some admissible subcategories of the derived categories of such Fano varieties [BLM + 17, LLMS18,LPZ23b,LPZ22]. Due to these links, in [FLV21b], we asked whether the Chow motives, considered as algebra objects, of Fano varieties of K3 type had similar properties as K3 surfaces (and what is expected for hyper-Kähler varieties).
Based on the above, we may ask whether two derived-equivalent Fano varieties of K3 type have isomorphic Chow motives as algebra objects.However, this question is uninteresting : due to the celebrated result of Bondal-Orlov [BO01], any two derived-equivalent Fano varieties are isomorphic.In the case of a cubic fourfold X, Kuznetsov [Kuz10] has identified an interesting admissible subcategory A X of D b (X), called the Kuznetsov component, consisting of objects E such that Hom(O X (i), E[m]) = 0 for i = 0, 1, 2 and any m ∈ Z.The Kuznetsov component is a K3-like triangulated category, see §4.1.Our first main result gives the correct analog of the aforementioned results on K3 surfaces for cubic fourfolds: two cubic fourfolds with Fourier-Mukai equivalent Kuznetsov components have isomorphic Chow motives as algebra objects.More precisely, we have the following.
Theorem 1.Let X and X ′ be two smooth cubic fourfolds over a field K with Fourier-Mukai equivalent Kuznetsov components A X ≃ A X ′ .Then X and X ′ have isomorphic Chow motives, as Frobenius algebra objects, in the category of rational Chow motives over K.
We refer to §5.4 for the notion of Fourier-Mukai equivalence for Kuznetsov components.By [LPZ23a], if K = C and if A X and A X ′ are equivalent as C-linear triangulated categories, then they are Fourier-Mukai equivalent.
Following our previous work [FV21,§2], a Frobenius algebra object in a rigid tensor category is an algebra object together with an extra structure, namely an isomorphism to its dual object (which we call a non-degenerate quadratic space structure, see §2.3) with a compatibility condition.The Chow motive of any smooth projective variety carries a natural structure of Frobenius algebra object in the category of Chow motives, lifting the classical Frobenius algebra structure on the cohomology ring (which essentially consists of the cup-product ⌣ together with the degree map X ).We refer to Section 2 for more details.An immediate concrete application of Theorem 1 is the following result.
Corollary 1.Let X and X ′ be two smooth cubic fourfolds over a field K. Assume that their Kuznetsov components are Fourier-Mukai equivalent A X ≃ A X ′ .Then there exists a correspondence Γ ∈ CH 4 (X × K X ′ ) ⊗ Q such that for any Weil cohomology H * with coefficients in a field of characteristic zero, Γ * : H * (X) ) is an isomorphism of Frobenius algebras.In particular, (i) for any prime number ℓ = char K, there exists a Galois-equivariant isomorphism H * (X K , Q ℓ ) ≃ H * (X ′ K , Q ℓ ) of ℓ-adic cohomology Frobenius algebras ; (ii) there exists an isocrystal isomorphism H * cris (X) ≃ H * cris (X ′ ) of crystalline cohomology Frobenius algebras ; (iii) if K = C, there exists a Hodge isomorphism H * (X, Q) ≃ H * (X ′ , Q) of Betti cohomology Frobenius algebras.
We note that item (iii) can also be directly deduced from arguments due to Addington-Thomas [AT14] and Huybrechts [Huy17] ; see Remark 6.2.The proof of Theorem 1 is given in §6 and employs essentially two different sources of techniques.On the one hand, we proceed to a refined Chow-Künneth decomposition ( §5.2), thereby cutting the motive of a cubic fourfold into the sum of its transcendental part and its algebraic part.The transcendental part, as well as its relation to the algebraic part, is then dealt with via a weight argument ( §5.3), while the algebraic part is dealt with via considering the Chow ring modulo numerical equivalence (Proposition 6.1).On the other hand, our proof also relies on some cycle-theoretic properties of cubic fourfolds, in particular those recently established in [FLV21b,FLV21a].First, the so-called Franchetta property for cubic fourfolds and their squares (Proposition 3.2) is used to establish the following.
Theorem 2 (Theorem 5.6).Let X and X ′ be two smooth cubic fourfolds over a field K with Fourier-Mukai equivalent Kuznetsov components A X ≃ A X ′ .Then the transcendental motives h 4 tr (X)(2) and h 4 tr (X ′ )(2), as defined in §5.2, are isomorphic as quadratic space objects in the category of rational Chow motives over K.
Concretely, this involves exhibiting an isomorphism Γ tr : h 4 tr (X) → h 4 tr (X ′ ) with inverse given by its transpose.Precisely, we show in Theorem 5.6 that such an isomorphism is induced by the degree-4 part of the Mukai vector of the Fourier-Mukai kernel inducing the equivalence A X ≃ A X ′ .Such an isomorphism is then upgraded in Proposition 6.1 to an isomorphism Γ : h(X) → h(X ′ ) with inverse given by its transpose, or equivalently, to a quadratic space object isomorphism Γ : h(X)(2) → h(X ′ )(2).
The next step towards the proof of Theorem 1 consists in showing that this isomorphism Γ : h(X) → h(X ′ ) respects the algebra structure.This is achieved in Proposition 6.3, the proof of which relies on the recently established multiplicative Chow-Künneth relation (3) for cubic fourfolds (Theorem 3.1).
To make the analogy with our previous work [FV21] even more transparent, we also investigate the case of cubic fourfolds with associated (twisted) K3 surfaces, resulting in the following strengthening of [Bül20, Theorem 0.4].
Theorem 3 (Theorem 7.2).Let X be a smooth cubic fourfold over a field K and let S be a K3 surface over K equipped with a Brauer class α.Assume that A X and D b (S, α) are Fourier-Mukai equivalent.Then the transcendental motives h 4 tr (X)(2) and h 2 tr (S)(1) are isomorphic as quadratic space objects in the category of rational Chow motives over K.
Note that by Orlov's result, any equivalence between A X and D b (S, α) is a Fourier-Mukai equivalent, at least when α = 0.

Chow motives and Frobenius algebra objects
In this section, we fix a commutative ring R.
2.1.Chow motives.We refer to [And04, §4] for more details.Briefly, a Chow motive, or motive, over a field K with coefficients in R, is a triple (X, p, n) consisting of a smooth projective variety X over K, an idempotent correspondence p ∈ CH dim X (X × K X) ⊗ R, and an integer n ∈ Z.The motive of a smooth projective variety The composition of morphisms is given by the composition of correspondences (as in [Ful98,§16]).The category of Chow motives M(K) R over K with coefficients in R forms a R-linear rigid ⊗-category with unit 1 = h(Spec K), with tensor product given by (X, p, n) ⊗ (Y, q, m) = (X × K Y, p × q, n + m) and with duality given by (X, p, n) ∨ = (X, t p, dim X − n), where t p denotes the transpose of the correspondence p. Fix a homomorphism R → F to a field F and fix a Weil cohomology theory H * with field of coefficients F , i.e., a ⊗-functor H * : M(K) R → GrVec F to the category of Z-graded F -vector spaces such that H i (1(−1)) = 0 for i = 2 ; see [And04, Proposition 4.2.5.1].We also call such a ⊗-functor an H-realization.One thereby obtains the category of homological motives M H (K) R (or M hom (K) R , when H is clear from the context).
2.2.Algebra structure.We consider the general situation where C is an R-linear ⊗-category with unit 1 ; cf.[And04, §2.2.2].An algebra structure on an object M in C is the data consisting of a unit morphism ǫ : 1 → M and a multiplication morphism µ : M ⊗ M → M satisfying the associativity axiom µ • (id The algebra structure is said to be commutative if it satisfies the commutativity axiom µ • τ = µ where τ : M ⊗ M → M ⊗ M is the morphism permuting the two factors. In case C is the category of Chow motives over K, then the Chow motive h(X) of a smooth projective variety X over K is naturally endowed with a commutative algebra structure : the multiplication µ : h(X) ⊗ h(X) → h(X) is given by pulling back along the diagonal embedding δ X : X ֒→ X × X, while the unit morphism η : 1 → h(X) is given by pulling back along the structure morphism ǫ X : X → Spec K. Taking the H-realization, this algebra structure endows H * (X) with the usual super-commutative algebra structure given by cup-product.
2.3.Quadratic space structure.We now consider the general situation where C is an Rlinear rigid ⊗-category with unit 1 and equipped with a ⊗-invertible object denoted 1(1).Let d be an integer.A degree-d quadratic space structure, or by abuse a quadratic space structure, on an object M of C consists of a morphism, called quadratic form, which is commutative q • τ = q, where τ : M ⊗ M → M ⊗ M is the switching morphism.We say that an object M equipped with the quadratic form q above is a degree-d quadratic space object in C, or by abuse a quadratic space object.The quadratic form q : M ⊗ M → 1(−d) is said to be non-degenerate if the induced morphism M (d) → M ∨ is an isomorphism.Here the morphism M (d) → M ∨ is obtained by tensoring q with id M ∨ (d) and pre-composing with id M (d) ⊗ coev, where coev : 1 → M ⊗ M ∨ is the co-evaluation map.
In case C is the category of Chow motives over K, then the Chow motive h(X) of a smooth projective variety X of dimension d over K is naturally endowed with a non-degenerate degree-d quadratic space structure : the quadratic form q X : h(X) ⊗ h(X) → 1(−d) is simply given by the class of the diagonal ∆ X .In relation to the natural algebra structure on h(X), we have where ǫ : h(X) → 1(−d) is the dual of the unit morphism η : 1 → h(X).Taking the Hrealization, this degree-d quadratic structure endows H * (X), as a super-vector space, with the usual quadratic structure given by Note that when d is odd the form is anti-symmetric on H d (X), while when d is even, the form is symmetric on H d (X).
In what follows, if M = (X, p, d) is a Chow motive with dim X = 2d, we view M as a quadratic space object via Proposition 2.1.Let M = (X, p, d) and M ′ = (X ′ , p ′ , d ′ ) be Chow motives in M(K) R .Assume that p = t p, p ′ = t p ′ , dim X = 2d and dim X ′ = 2d ′ , so that M = M ∨ and M ′ = M ′∨ .The following are equivalent : (i) M and M ′ are isomorphic as quadratic space objects ; (ii) There exists an isomorphism Γ : Proof.The quadratic forms q M and q M ′ are the (non-degenerate) quadratic forms associated to the identifications M = M ∨ and M ′ = M ′∨ , respectively.By definition, a morphism Γ : M → M ′ is a morphism of quadratic space objects if and only if q M ′ • (Γ ⊗ Γ) = q M .The latter is then equivalent to t Γ • Γ = id M , where we have identified Γ ∨ with t Γ via the identifications M = M ∨ and M ′ = M ′∨ .This shows that a morphism Γ : M → M ′ is a morphism of quadratic space objects if and only if Γ is split injective with left-inverse t Γ.This proves the proposition.
2.4.Frobenius algebra structure.This notion was introduced in [FV21, §2], as a generalization of the classical Frobenius algebras (cf.[Koc04]).Consider again the general situation where C is an R-linear rigid ⊗-category with unit 1 and equipped with a ⊗-invertible object denoted 1(1).Let d be an integer.A degree-d (commutative) Frobenius algebra structure on an object M of C consists of a unit morphism ǫ : 1 → M , a multiplication morphism µ : M ⊗ M → M and a non-degenerate degree-d quadratic form q : M ⊗ M → 1(−d) such that (M, µ, ǫ) is an algebra object, and the following compatibility relation, called the Frobenius condition, holds: where δ : M → M ⊗ M (d) is the dual of the multiplication µ, via the identification M (d) ≃ M ∨ provided by the non-degenerate quadratic form q.
In case C is the category of Chow motives over K, then the Chow motive h(X) of a smooth projective variety X of dimension d over K is naturally endowed with a degree-d Frobenius algebra structure.That the unit, multiplication and quadratic form given in § §2.2-2.3 above do define such a structure on h(X) is explained in [FV21, Lemma 2.7].Taking the H-realization and forgetting Tate twists, this degree-d Frobenius algebra structure endows H * (X) with the usual Frobenius algebra structure (consisting of the cup-product together with the quadratic form q X of (1)) ; see [FV21, Example 2.5].

The Chow ring of powers of cubic fourfolds
In this section, we gather the cycle-theoretic results needed about cubic fourfolds ; Proposition 3.2 is used to obtain isomorphisms as quadratic space objects as in Theorem 2, and Theorem 3.1 is used in addition to upgrade those isomorphisms to isomorphisms of algebra objects as in Theorem 1.
From now on, we fix a field K with algebraic closure K, Chow groups and motives are with rational coefficients (R = Q), and we fix a Weil cohomology theory H * with coefficients in a field of characteristic zero.
Recall that a Chow-Künneth decomposition, or weight decomposition, for a motive M is a finite grading M = i∈Z M i such that H * (M i ) = H i (M ).This notion was introduced by Murre [Mur93], who conjectured that every motive admits such a decomposition.Now, if M is a Chow motive equipped with an algebra structure (e.g., M = h(X) equipped with the intersection pairing), then we say that a Chow-Künneth decomposition M = i∈Z M i is multiplicative if it defines an algebra grading, i.e., if the composition M i ⊗ M j ֒→ M ⊗ M → M factors through M i+j for all i, j.This notion was introduced in [SV16, §8], where it was conjectured that the motive of any hyper-Kähler variety admits a multiplicative Chow-Künneth decomposition.
Let B be the open subset of PH 0 (P 5 , O(3)) parameterizing smooth cubic fourfolds, let X → B be the universal family of smooth cubic fourfolds and ev : X → P 5 be the evaluation map.If H := ev * (c 1 (O P 5 (1))) ∈ CH 1 (X ) denotes the relative hyperplane section, then X defines a relative Chow-Künneth decomposition, in the sense that its specialization to any fiber X b over b ∈ B gives a Chow-Künneth decomposition of X b .Given a smooth cubic fourfold X, we denote h X the restriction of H to X and we denote {π 0 X , π 2 X , π 4 X , π 6 X , π 8 X } the restriction of the above projectors to the fiber X.
In our previous work [FLV21b], we established the following two results : where P is an explicit symmetric rational polynomial in 3 variables.
Proof.That the Chow-Künneth decomposition {π 0 3) is due to Diaz [Dia21].That the two formulations are equivalent is [FLV21a, Proposition 2.8].The proof in loc.cit. is over C, but one can extend the result to arbitrary base fields as follows.By the Lefschetz principle, (3) holds for any algebraically closed field of characteristic zero.Since the pull-back morphism CH(X 3 ) → CH(X 3 Ω ) associated with the field extension from K to a universal domain Ω is injective, and all the terms in (3) are defined over K, we have the result in characteristic zero.If char(K) > 0, take a lifting X /W over some discrete valuation ring W with residue field K and fraction field of characteristic zero.Then by specialization, the validity of (3) on the generic fiber implies the same result on the special fiber.Proposition 3.2.Let X → B be the above-defined family of smooth cubic fourfolds and let X = X b be a fiber.For a positive integer n, define GDCH * B (X n ), which stands for generically defined cycles, to be the image of the Gysin restriction ring homomorphism has the Franchetta property for n ≤ 2.
Proof.This was established in [FLV21b, Proposition 5.6].The proof in loc.cit. is given for K = C but holds for any field K.
Remark 3.3.Proposition 3.2 was extended to n ≤ 4 in [FLV21a, Theorem 2].However, the cases n = 3 and n = 4 are not needed for the proof of Theorem 1 and, besides, their proofs are significantly more involved.

Kuznetsov components and primitive motives
4.1.Kuznetsov component and projectors.For the basic theory of Fourier-Mukai transforms, we refer to the book [Huy06].Let X ⊂ P 5 be a smooth cubic fourfold defined over a base field K. Following [Kuz10], the Kuznetsov component A X of X is defined to be the rightorthogonal complement of the triangulated subcategory generated by the exceptional collection O X , O X (1), O X (2) in the bounded derived category of coherent sheaves D b (X): By Serre duality, A X is also the left-orthogonal complement of the triangulated subcategory generated by the exceptional collection O X (−3), O X (−2), O X (−1) in D b (X): In other words, we have semi-orthogonal decompositions As is pointed out by Kuznetsov [Kuz10] (see also [Kuz16, Proposition 1.4]), A X is a K3like category (or sometimes called a non-commutative K3 surface), in the sense that its Serre functor S A X = [2] (see for example [Kuz19]) and its Hochschild homology, which is HH agrees with the Hochschild homology of a K3 surface, at least when char(K) = 2 or 3.The latter, which will not be used in this work, can be established by using the additivity of Hochschild homology, the HKR isomorphism [Yek02, AV20] applied to the cubic fourfold, and the computation of Hodge numbers of cubic fourfolds.
As A X is an admissible subcategory ([Bon89, BK89]), the inclusion functor i X : A X ֒→ D b (X) has both left and right adjoint functors ; these are denoted by i * X and i !X : D b (X) → A X , respectively.In addition, since i X is fully faithful, the adjunction morphisms We denote P L X and P R X the respective Fourier-Mukai kernels in D b (X × K X) of the functors p L X and p L X .Recall that, given E ∈ D b (X) an exceptional object, the Fourier-Mukai kernel of the left mutation functor L E is given by cone 2 with p, q : X × K X → X the two natural projections.Therefore the Fourier-Mukai kernel of p L X is given by the convolution of the kernels of the mutation functors : (4) The Fourier-Mukai kernel P R X of p R X admits a similar description.Remark 4.2.Consider the universal family of smooth cubic fourfolds X → B as in Section 3. Since objects of the form O X (i) are defined family-wise for X → B, by the formula (4), the Fourier-Mukai kernels P L X (and similarly P R X ) are defined family-wise.
Now we turn to the study of cohomological or Chow-theoretic Fourier-Mukai transforms.Recall that for E ∈ D b (X), its Mukai vector is defined as v(E) := ch(E) td(T X ) ∈ CH * (X), and we denote its cohomology class by [v(E)] ∈ H * (X) and its numerical class by v(E) ∈ CH * (X), where CH * (X) := CH * (X)/≡ is the Q-algebra of cycles on X modulo numerical equivalence.
The Mukai pairing on CH * (X) is given as follows: where The same formula defines the Mukai pairing on H * (X) and CH * (X).Note that the Mukai pairing is bilinear but in general not symmetric, hence we need to distinguish between the notions of left and right orthogonal complements.Recall that for a vector space V equipped with a bilinear form −, − , the left (resp.right) orthogonal complement of a subspace U is by definition When the bilinear form is non-degenerate, we define the orthogonal projection from V onto ⊥ U (resp.U ⊥ ) as the projection with respect to the decomposition Lemma 4.3.The cohomological (resp.numerical) Fourier-Mukai transform are respectively the orthogonal projections onto v(O), v(O(1)), v(O(2)) ⊥ , which is the right orthogonal complement of the linear subspace spanned by the cohomological (resp.numerical) Mukai vectors of O X , O X (1), and O X (2), with respect to the Mukai pairing.
Proof.We only show the statement for the cohomology.The proof for CH * is the same.We first show a general result : for a smooth projective variety X and an exceptional object E in D b (X), the cohomological action of the left mutation functor L E on H * (X) is the orthogonal projection onto the subspace [v(E)] ⊥ , with respect to the Mukai pairing.Indeed, the Fourier-Mukai kernel of L E , denoted by F ∈ D b (X × X), fits into the distinguished triangle: which is exactly the orthogonal projector to [v(E)] ⊥ , where we used in the last step the relation [Huy06,Lemma 5.41].Now back to the case of cubic fourfolds : since P L X is the composition of the kernels of three left mutations (4), applying the above general result three times, we see that the cohomological transform Definition 4.4.The cohomology and the Chow group modulo numerical equivalence of the Kuznetsov component A X are defined, respectively, as the vector spaces Unlike the Mukai pairing on H * (X) or CH * (X), the restriction of the Mukai pairing to the above spaces becomes symmetric.This holds essentially because the Serre functor S , pp.1891-1892].This can also be checked directly by applying the Mukai pairing to the projections of two vectors.Thus the Mukai pairing endows both H(A X ) and CH(A X ) with a non-degenerate quadratic form.
Definition 4.5.Let X be a smooth cubic fourfold with hyperplane class h X .The primitive cohomology and the primitive Chow group modulo numerical equivalence of X are defined, respectively, to be Here, h 2 X ⊥ denotes the orthogonal complement of h 2 X inside H 4 (X) with respect to the intersection product.We also have the following alternative description for the space of primitive classes as the right orthogonal complement of all powers of the hyperplane class : The restriction of the Mukai pairing on H 4 prim (X) and on CH 2 prim (X) endows those spaces with a non-degenerate quadratic form that coincides with the intersection pairing.(As can readily be observed from (5), the Mukai pairing and the intersection pairing already agree on H 4 (X) and on CH 2 (X).)Proposition 4.6.We have the inclusions : CH 2 prim (X) ⊂ CH(A X ).
Proof.We only prove (6) as the proof of (7) is similar.By Lemma 4.3, the right-hand side of (6) coincides with the right orthogonal complement of the Mukai vectors of O X , O X (1), and O X (2), with respect to the Mukai pairing on H * (X).Therefore, it suffices to check that H As the Mukai vector of the sheaf O X (i) and exp(c 1 (X)/2) are all polynomials in the hyperplane section class h X , we have that for any i there is some rational number The inclusion (6) is proved.
Remark 4.7.Over the complex numbers (K = C), following Addington-Thomas [AT14], define the Mukai lattice of A X as its topological K-theory : where −, − is the Mukai pairing on K top (X) given by v, v ′ := χ(v, v ′ ).A weight-2 Hodge structure on H(A X , Z) is induced from the isomorphism v : K top (X) ⊗ Q → H * (X, Q) given by the Mukai vector.The cohomological action of the projector P L X recovers the Mukai lattice rationally : with respect to (the restriction of ) the Mukai pairing (5).Moreover, the Z-lattice λ 1 (h X ), λ 2 (h X ) equipped with the Mukai pairing is an A 2 -lattice.
Proof.The decomposition (9) is established in [AT14, Proposition 2.3].We sketch the proof of (10) for the convenience of the reader.We define the polynomials (see [Huy19,) We write λ i for λ i (h X ) in the sequel ; λ i clearly defines an algebraic cycle defined over K. Let us mention that, geometrically (after a finite base-change), λ i agrees with the Mukai vector of p L X (O l (i)), where l is any line contained in X.It is easy to compute that λ 2 1 = λ 2 2 = −2 and λ 1 , λ 2 = 1.Now for any element in CH(A X ), which is necessarily of the form v(E) for some 4.3.Kuznetsov components and primitive motives.Let X → B be the universal family of smooth cubic fourfolds.We may refine the relative Chow-Künneth decomposition (2) and define the relative idempotent correspondence X ,prim and the restriction of π 4 X ,prim to any fiber X defines an idempotent π 4 prim ∈ CH 4 (X × K X) which cohomologically defines the orthogonal projector on the primitive cohomology H 4 prim (X).Using the Franchetta property for X × X of Proposition 3.2, we can show that the Fourier-Mukai kernels P L X and P R X enjoy the following property relatively to the projector π 4 prim .For an object F ∈ D b (X × X), we denote by v(F) := ch(F) • td(X × X) its Mukai vector and v i (F) the component of v(F) in CH i (X × X), for all 0 ≤ i ≤ 8. Lemma 4.9.The following relations hold in CH 4 (X × X) : Proof.We only prove the relation involving P L X ; the proof of the relation involving P R X is similar.We have to show that the composition is the identity map.Observe that π 4 prim is defined family-wise (which is the reason for focusing on π 4 prim , rather than on π 4 tr , in this section) and the Fourier-Mukai kernel P L X is also defined familywise (Remark 4.2), by the Franchetta property for X × K X in Proposition 3.2, we are reduced to showing that the composition (11) is the identity map modulo homological (or numerical) equivalence.This follows directly from Proposition 4.6.
Remark 4.10.It is maybe possible to prove Lemma 4.9 by a direct but tedious computation without using the Franchetta property.We leave the details to the interested reader.

Rational and numerical equivalence on codimension-2 cycles on cubic fourfolds.
Recall that a universal domain is an algebraically closed field of infinite transcendence degree over its prime subfield.The following lemma applies in particular to cubic fourfolds : Lemma 5.1.Let X be a smooth projective variety over a field K and let Ω be a universal domain containing K. Assume that CH 0 (X Ω ) is supported on a curve and that H 3 (X K , Q ℓ ) = 0 for some prime ℓ = char K. Then rational and numerical equivalence agree on Z 2 (X), where Z 2 denotes the group of algebraic cycles of codimension 2 with rational coefficients.
Proof.By a push-pull argument, we may assume that K is algebraically closed.The proof is classical and goes back to [BS83].By [BS83, Proposition 1], there exists a positive integer N , a 1-dimensional closed subscheme C ⊆ X, a divisor D ⊂ X and cycles Γ 1 , Γ 2 in CH dim X Z (X × K X) with respective supports contained in C × X and X × D, such that where CH * Z denotes the Chow group with integral coefficients.Let D → D be an alteration, say of degree d, with D smooth over K.The multiplication by N d map on CH 2 Z (X) then factors as where the arrows are induced by correspondences with integral coefficients.Since numerical and algebraic equivalence agree for codimension-1 cycles on D, we find that numerical and algebraic equivalence agree on CH 2 Z (X).It remains to show that the group of algebraically trivial cycles CH 2 Z (X) alg is zero after tensoring with Q.For that purpose, we consider the diagram (12) restricted to algebraically trivial cycles.We obtain a commutative diagram where the composition of the horizontal arrows is given by multiplication by N d, and where the vertical arrows are Murre's algebraic representatives [Mur85] (these are regular homomorphisms to abelian varieties that are universal).A diagram chase shows that CH 2 Z (X) alg → Ab 2 X ( K) is injective after tensoring with Q.We conclude with [Mur85, Theorem 1.9] which gives the upper bound dim Ab 2 5.2.Refined Chow-Künneth decomposition.Fix a smooth cubic fourfold X over K.We are going to produce a refined Chow-Künneth decomposition for X that is similar to that for surfaces constructed in [KMP07, §7.2.2], and extending the construction in [BP20] to arbitrary base fields.Refining the primitive motive to the transcendental motive is an essential step towards the proof of Theorem 1 as it makes it possible to use the "weight argument" of Lemma 5.5 below.For that purpose, recall from Lemma 5.1 that CH 2 (X K ) = CH 2 (X K ).This way we can complete h 2 X ⊂ CH 2 (X) to an orthogonal basis {h 2 X , α 1 , . . ., α r } of CH 2 (X K ) with respect to the intersection product.The correspondence then defines an idempotent in CH 4 (X K × K X K ).On the one hand, the correspondence π 4 alg comes from CH 2 (X K ) ⊗ CH 2 (X K ) and is Galois-invariant as it defines the intersection pairing on CH 2 (X K ), and the latter is obviously Galois-invariant.Since we are working with rational coefficients, by [Ful98, Example 1.7.6] and the fact that any cycle is defined over a finite Galois extension of K, it follows that π 4 alg is defined over K, i.e., is in the image of CH 4 (X × K X) after base-change to K. On the other hand, π 4 alg commutes with π 4 X and π 4 prim and is cohomologically the orthogonal projector on the subspace Im CH 2 (X K ) → H 4 (X) spanned by K-algebraic classes.In addition, we have We then define It is an idempotent correspondence in CH 4 (X × K X) which cohomologically is the orthogonal projector on the transcendental cohomology H 4 tr (X), i.e., by definition of transcendental cohomology, the orthogonal projector on the orthogonal complement to the K-algebraic classes in H 4 (X).In addition, π 4 tr commutes with π 4 prim and we have Note that, while π 4 prim is defined family-wise for the universal cubic fourfold X → B, π 4 tr and π 4 alg are not.
As an immediate consequence of (14), we have the following consequence of Lemma 4.9.
Lemma 5.2.The following relations hold in CH 4 (X × K X) : In other words, the correspondences v 4 (P L X ) and v 4 (P R X ) act as the identity on the transcendental motive h 4 tr (X).

A weight argument.
One defines a notion of weight on the Chow motives appearing in the decomposition (15) in the following way: for any i ∈ Z, the Tate motive 1(−i) has weight 2i; h 4 tr (X)(−i) and h 4 alg (X)(−i) have weight 4 + 2i.As a first step towards our weight argument below (Lemma 5.5), we need the following property of the refined Chow-Künneth decomposition (15).Proposition 5.3.Let X and X ′ be two smooth cubic fourfolds over a field K.
Proof.By definition of an effective motive, there exists a smooth projective variety X and an idempotent r ∈ End M (h(X)) such that M ≃ (X, r, 0).By assumption, r acts as zero on H 0 (X), so that CH 0 (M ) := r * CH 0 (X) = 0. Further, we have CH 1 (M ) := r * CH 1 (X) = 0 since by assumption r acts as zero both on Im(CH 1 (X) → H 2 (X)) and on H 1 (X) (hence on Pic 0 X (K)).We will need the following simple observation, which is an abstraction of [FV21, §1.2.3].
Lemma 5.5 (Weight argument).Let S := {N i , i ∈ I} be a collection of Chow motives whose objects N i are all equipped with an integer k i called weight such that any morphism from an object of smaller weight to an object of larger weight is zero.For r = 0, . . ., n, let M r be a Chow motive isomorphic to a direct sum of objects in S. Suppose we have a chain of morphisms of Chow motives ) such that M and M ′ are both of (pure) weight k for some integer k, i.e., such that M and M ′ are direct sums of objects of S all of weight k.Then the composition of morphisms in (16) is equal to the following composition where M w=k i means the direct sum of the summands (in S) of M i of weight k.
Proof.The composition in ( 16) is clearly the sum of all compositions of the form Therefore the only non-zero contribution is given by the case where k i = k for all 1 ≤ i ≤ n − 1.
5.4.Main result.Let X and X ′ be two smooth cubic fourfolds over a field K. Assume that their Kuznetsov components A X and A X ′ are Fourier-Mukai equivalent, this means there exists , where p X and p X ′ are the natural projections from X × K X ′ to X and X ′ respectively.Note that by Li-Pertusi-Zhao [LPZ23a], over K = C, any equivalence of triangulated categories between A X and A X ′ is a Fourier-Mukai equivalence.
Adding the right adjoints, we get a diagram denotes the right adjoint of E. Since F is an equivalence by assumption, F R is in fact the inverse of F , hence we have Theorem 5.6.The correspondence Γ tr := π 4 tr ′ • v 4 (E) • π 4 tr in CH 4 (X × K X ′ ) defines an isomorphism with inverse given by its transpose.In other words, via Proposition 2.1, the transcendental motives h 4 tr (X) and h 4 tr (X ′ ) are isomorphic as quadratic space objects.Proof.From the isomorphism of Fourier-Mukai functors (17), it is not clear whether one can deduce an isomorphism between their Fourier-Mukai kernels in D b (X × X), i.e., whether one has an isomorphism P R X * E R * P R X ′ * P L X ′ * E * P L X ≃ P L X , where * stands for the convolution of Fourier-Mukai kernels.Nonetheless, by Canonaco-Stellari [CS12, Theorem 1.2], the two sides have the same cohomology sheaves, and hence have the same class in K 0 (X × X).By taking Mukai vectors, one obtains the following equality in CH * (X × K X) : The above equality implies that the composition is equal to the composition Here the ranges of the (finite) direct sums are not specified since they are irrelevant.By the "weight argument" Lemma 5.5, combined with Proposition 5.3(i), we obtain that the composition is equal to the composition h 4 tr (X) ֒→ h 4 (X) , which is the identity map of h 4 tr (X) by Lemma 5.2.Writing h 4 = h 4 tr ⊕ h 4 alg and using Proposition 5.3(ii ′ ), we deduce that each map in (20) factors through h 4 tr or h 4 tr ′ .In other words, we have the following equality: Similarly, from (18), together with the weight argument, we obtain The equalities (21) and ( 22) say nothing but that π 4 tr ′ • v 4 (E) • π 4 tr and π 4 tr • v 4 (E R ) • π 4 tr ′ define inverse isomorphisms between h 4 tr (X) and h 4 tr (X ′ ).It remains to show that We will actually show the following stronger equality To see that (24) indeed implies (23), it is enough to compose both sides of (24) on the left with π 4 tr and on the right with π 4 tr ′ , and then to use (14).Let us show (24).Denoting h X , h X ′ ∈ CH 1 (X × K X ′ ) the pull-backs of the hyperplane section classes on X and X ′ via the natural projections, we have (see [Huy06,Lemma 5.41]) This yields the identity Therefore, to establish (24), it suffices to show the following lemma.
Lemma 5.7.For any We only show the first vanishing; the second one can be proved similarly.Note that However, by applying the excess intersection formula [Ful98,Theorem 6.3] to the following cartesian diagram with excess normal bundle O X (3): / / P 5 × K P 5 , we obtain that (∆ X ) * (3h X ) = ∆ P 5 | X×X = i h i X × h 5−i X , where the latter equality uses the relation ∆ P 5 = 5 i=0 h i × h j in CH 5 (P 5 × P 5 ), where h is a hyperplane class of P 5 .We can conclude by noting that for any i, we have π 4 prim • (h i X × h 5−i X ) = 0 by construction of π 4 prim .With Lemma 5.7 being proved, the equality (24), hence also (23), is established.The proof of Theorem 5.6 is complete.

Proof of Theorem 1
Proposition 6.1 below, in particular, upgrades the quadratic space object isomorphism of Theorem 5.6 to a quadratic space object isomorphism h(X) ≃ h(X ′ ).Proposition 6.1.Let X and X ′ be two smooth cubic fourfolds over a field K, whose Kuznetsov components are Fourier-Mukai equivalent.Then their Chow motives are isomorphic.More precisely, there exists a correspondence Γ ∈ CH 4 (X × K X ′ ) such that Γ * h i X = h i X ′ for all i ≥ 0 which in addition induces an isomorphism of Chow motives with inverse given by its transpose t Γ.
Proof.As a first step, we construct an isomorphism Γ 4 alg : h 4 alg (X) → h 4 alg (X ′ ) of quadratic space objects.Let Φ : A X → A X ′ be the Fourier-Mukai equivalence.It induces a homomorphism which is clearly an isometry with respect to the Mukai pairings ( ) and is equivariant with respect to the action of the absolute Galois group of K (since the Fourier-Mukai kernel is defined over K).Recall from Proposition 4.8 that we have an orthogonal decomposition prim (X K ) with respect to the Mukai pairing.Since the planes λ 1 (h X ), λ 2 (h X ) and λ 1 (h X ′ ), λ 2 (h X ′ ) consist of Galois-invariant elements and are isometric to one another, we obtain from Theorem A.2, which is an equivariant Witt theorem, a Galois-equivariant isometry ≃ (Note that Theorem A.2 is stated for finite groups, but it indeed applies here: all the numerical Chow groups involved are finitely generated, hence the Galois group action factors through the Galois group of some common finite extension K ′ /K.)Let then {α 1 , . . ., α r } be an orthogonal basis of CH 2 prim (X K ).Having in mind that the Mukai pairing agrees with the intersection pairing on CH 2 (X K ) and that CH 2 (X K ) = CH 2 (X K ), we see, together with the construction and definition of h 4 alg (see ( 13)), that the correspondence is defined over K and defines an isomorphism h 4 alg (X) ≃ −→ h 4 alg (X ′ ) with inverse given by its transpose t Γ 4 alg .Finally, combining Γ 4 alg with Γ tr of Theorem 5.6, the cycle induces an isomorphism between h(X) and h(X ′ ), and its inverse is t Γ.Furthermore, by construction, we have Γ * (h i X ) = h i X ′ for all i.Remark 6.2.In the case where K = C and H * is Betti cohomology, the construction of the isomorphism Γ 4 alg : h 4 alg (X) → h 4 alg (X ′ ) in the proof of Proposition 6.1 is somewhat simpler.As a consequence of Theorem 5.6, we have a Hodge isometry (This Hodge isometry can also be obtained by considering the transcendental part of [Huy17, Proposition 3.4].)Since H 4 (X, Q) and H 4 (X ′ , Q) are isometric for all smooth complex cubic fourfolds, there is by Witt's theorem an isometry . ., α r } be an orthogonal basis of H 4 alg (X, Q).The correspondence Γ 4 alg of (25) then provides an isomorphism from h 4 alg (X) to h 4 alg (X ′ ), whose inverse is given by its transpose t Γ 4 alg .Note that, by combining (26) and ( 27), we obtain a Hodge isometry H 4 (X, Q) ≃ H 4 (X ′ , Q).
Theorem 1 then follows from combining Proposition 6.1 with the following proposition.
where v denotes the Chow-theoretic Mukai vector map.Likewise, using [CS12, Theorem 6.4], or alternately by the uniqueness of the Fourier-Mukai kernel in the twisted version of Orlov's Theorem ([CS07, Theorem 1.1]), (29) implies that As in Section 5, we define a refined Chow-Künneth decomposition for S. The general case of a smooth projective surface over K is due to [KMP07, §7.2.2].Since for a K3 surface rational and numerical equivalence agree on CH 1 (S K ), we can in fact construct such a refined Chow-Künneth decomposition in a more direct way.First, choose any degree-1 zero-cycle o ∈ CH 0 (S), and define the Chow-Künneth decomposition π 0 S := o × S, π 4 S := S × o, and π 2 S := ∆ S − π 0 S − π 4 S .Let {β 1 , . . ., β s } be an orthogonal basis for CH 1 (S K ).The correspondence π 2 alg,S := then defines an idempotent in CH 2 (S K × K S K ) which descends to K, which commutes with π 2 S and which cohomologically is the orthogonal projector on the subspace Im CH 1 (S K ) → H 2 (S) spanned by K-algebraic classes in H 2 (S).In addition, we have π 2 alg,S • π 2 S = π 2 S • π 2 alg,S = π 2 alg,S .We then define π 2 tr,S := π 2 S − π 2 alg,S .It is an idempotent correspondence in CH 2 (S × K S) which cohomologically is the orthogonal projector on the transcendental cohomology H 2 tr (S), i.e., by definition of transcendental cohomology, the orthogonal projector on the orthogonal complement to the K-algebraic classes in H 2 (S).
Denote by h i (S), h 2 tr (S) and h 2 alg (S) the Chow motives (S, π i S ), (S, π 2 tr,S ), and (S, π 2 alg,S ) respectively.From the above, we get the following refined Chow-Künneth decomposition : h(S) = h 0 (S) ⊕ h 2 alg (S) ⊕ h 2 tr (S) ⊕ h 4 (S), where h 2i (X) ≃ 1(−i) for i = 0, 2 and the base-change to K of h 2 alg (S) is a direct sum of copies of 1(−1).Now, as in the case of two cubic fourfolds, we want to apply the weight argument (Lemma 5.5) to the equalities (30) and (31).To this end, we need the following complement to Proposition 5.3.
Proposition 7.1.Let X be a cubic fourfold and S a projective surface.Then for all l > 1, Hom h 4 tr (X), h 2 tr (S)(−l) = 0. Proof.As is pointed out in the proof of Proposition 5.3, h 4 tr (X)(1) is a direct summand of the motive of a surface.Then we can apply Lemma 5.4 to conclude to the vanishing.
By the weight argument (Lemma 5.5), combined with Proposition 5.3, [FV21, Theorem 1.4(ii)] and Proposition 7.1, we can deduce that if we restrict the domain to h 4 tr (X), then each step of (30) factors through h 4 tr (X) or h 2 tr (S)(−1).In other words, π 4 tr,X • v 4 (P By the same argument as in the proof of (23), using Lemma 5.7, we can moreover show that the two inverse isomorphisms in (35) are transpose to each other.To summarize, we have proven the following : Theorem 7.2.The correspondence Γ tr := π 2 tr,S • v 3 (E) • π 4 tr,X in CH 3 (X × S) induces an isomorphism Γ tr : h 4 tr (X)(2) h 2 tr (S)(1) ≃ whose inverse is its transpose t Γ tr .

Appendix A. An equivariant Witt theorem
Throughout the appendix, F is a field of characteristic different from 2 and all the vector spaces are finite dimensional over F .
Let us first recall the classical Witt theorem.Let V 1 , V 2 be vector spaces equipped with quadratic forms, whose associated bilinear symmetric pairings are denoted by −, − .Suppose that V 1 and V 2 are isometric and we have orthogonal decompositions such that U 1 and U 2 are isometric.Then W 1 and W 2 are also isometric.This is often referred to as Witt's cancellation theorem, which is clearly equivalent to the following Witt's extension theorem : Let V be a non-degenerate quadratic space and let f : U → U ′ be an isometry between two subspaces of V .Then f can be extended to an isometry of V .
The goal of this appendix is to establish an equivariant version of the Witt theorem, in case the quadratic spaces are endowed with a group action.For a quadratic space V with a G-action, we denote O G (V ) the group of G-equivariant isometries, i.e. automorphisms of V that preserve the pairing and commute with the action of G.
Lemma A.1.Let V be a non-degenerate quadratic space equipped with an isometric action of a finite group G. Suppose that |G| is invertible in F .Then (1) The restriction of the quadratic form to V G , the G-fixed space, is non-degenerate.
(2) For any x, y ∈ V G with x, x = y, y = 0, there exists a G-equivariant isometry φ ∈ O G (V ) sending x to y.