A motivic global Torelli theorem for isogenous K3 surfaces

We prove that the Chow motives of twisted derived equivalent K3 surfaces are isomorphic, not only as Chow motives (due to Huybrechts), but also as Frobenius algebra objects. Combined with a recent result of Huybrechts, we conclude that two complex projective K3 surfaces are isogenous (i.e. their second rational cohomology groups are Hodge isometric) if and only if their Chow motives are isomorphic as Frobenius algebra objects; this can be regarded as a motivic Torelli-type theorem. We ask whether, more generally, twisted derived equivalent hyper-Kaehler varieties have isomorphic Chow motives as (Frobenius) algebra objects and in particular isomorphic graded rational cohomology algebras. In the appendix, we justify introducing the notion of"Frobenius algebra object"by showing the existence of an infinite family of K3 surfaces whose Chow motives are pairwise non-isomorphic as Frobenius algebra objects but isomorphic as algebra objects. In particular, K3 surfaces in that family are pairwise non-isogenous but have isomorphic rational Hodge algebras.


Introduction
Torelli theorems for K3 surfaces. A compact complex surface is called a K3 surface if it is simply connected and has trivial canonical bundle. The Hodge structure carried by the second singular cohomology contains essential information of the surface. Indeed, the global Torelli theorem says that the isomorphism class of a K3 surface S is determined by the Hodge structure H 2 (S, Z) together with the intersection pairing on it [48,11].
The following more flexible notion due to Mukai [40] turns out to be crucial in the study of their derived categories : two complex projective K3 surfaces S and S ′ are called isogenous if there exists a Hodge isometry ϕ : H 2 (S, Q) ∼ −→ H 2 (S ′ , Q), i.e. an isomorphism of rational Hodge structures compatible with the intersection pairing on both sides. Recently, Buskin [12] proved that such an isometry ϕ is induced by an algebraic correspondence, as had previously been conjectured by Shafarevich [53] as a particularly interesting case of the Hodge conjecture. Let us call any such representative a Shafarevich cycle for this isogeny. Shortly afterwards, Huybrechts [29] gave another proof and showed that in fact ϕ is induced by a chain of exact linear equivalences between derived categories of twisted K3 surfaces ; thereby establishing the twisted derived global Torelli theorem [29,Corollary 1.4] : (1) S and S ′ are isogenous ⇐⇒ S and S ′ are twisted derived equivalent.
Here, following Huybrechts [29], we say that two K3 surfaces S and S ′ are twisted derived equivalent if there exist K3 surfaces S = S 0 , S 1 , . . . , S n = S ′ and Brauer classes α = β 0 ∈ Br(S), α 1 , β 1 ∈ Br(S 1 ), . . . , α n−1 , β n−1 ∈ Br(S n−1 ) and α ′ = α n ∈ Br(S ′ ) and exact linear equivalences between bounded derived categories of twisted coherent sheaves Note that by [13] any exact linear equivalence between bounded derived categories of twisted coherent sheaves on smooth projective varieties is of Fourier-Mukai type, so that in (2), each ) (unique up to isomorphism). Combined with his previous work [28] generalized to the twisted case, Huybrechts deduced that isogenous complex projective K3 surfaces have isomorphic Chow motives 1 . However, the converse does not hold in general : there are K3 surfaces having isomorphic Chow motives (hence isomorphic rational Hodge structures) without being isogenous. Examples of such K3 surfaces were constructed geometrically in [7] ; see Remark A.3 of Appendix A. We provide in Theorem A.13 two further constructions of infinite families of pairwise non-isogenous K3 surfaces with isomorphic Chow motives. The motivation of the paper is to complete the picture by giving a motivic characterization of isogenous K3 surfaces (Corollary 2). Our main result is the following : Theorem 1. Let S and S ′ be two twisted derived equivalent K3 surfaces over a field k. Then the Chow motives of S and S ′ are isomorphic as algebra objects, in fact even as Frobenius algebra objects (Definition 2.1), in the category of rational Chow motives over k.
In concrete terms, there exists a correspondence Γ ∈ CH 2 (S × S ′ ) with Γ • δ S = δ S ′ • (Γ ⊗ Γ) ("algebra homomorphism") and such that Γ is invertible as a morphism between the Chow motives of S and S ′ with Γ −1 = t Γ ("orthogonality"), where t Γ denotes the transpose of Γ and where δ S is the small diagonal in S × S × S viewed as a correspondence between S × S and S. Equivalently, Γ is an isomorphism such that (Γ ⊗ Γ) * ∆ S = ∆ S ′ and (Γ ⊗ Γ ⊗ Γ) * δ S = δ S ′ . The notion of Frobenius algebra object provides a conceptual way to pack these conditions. Roughly speaking, a Frobenius algebra object in a rigid tensor category is an algebra object together with an isomorphism to its dual object 2 with some compatibility conditions. The Chow motive of a smooth projective variety carries a natural structure of Frobenius algebra object in the rigid tensor category of Chow motives. We refer to §2 for more details on Frobenius algebra objects. ∼ −→ D b (S ′ , α ′ ) between twisted smooth projective surfaces, then v 2 (E) induces an isomorphism h 2 tr (S) ∼ −→ h 2 tr (S ′ ) of the transcendental motives of S and S ′ with inverse given by v 2 (E ∨ ⊗ p * ω S ), where E ∨ denotes the derived dual of E and p : S ×S ′ → S is the natural projection. (In the case of K3 surfaces, we have v 2 (E ∨ ⊗p * ω S ) = t v 2 (E).) This is achieved by exploiting known cases of Murre's Conjecture 1.2(B), and we thereby give an alternative proof of Huybrechts' [28, Theorem 0.1], generalized to all surfaces : two twisted derived equivalent smooth projective surfaces have isomorphic Chow motives ; see Theorem 1.1. This confirms the two-dimensional case of the following conjecture due to Orlov : Conjecture 1 (Orlov [46]). Let X and Y be two derived equivalent smooth projective varieties. Then their Chow motives are isomorphic.
We illustrate also in §1.3 how the same techniques can be used to establish Orlov's Conjecture 1 in some new cases in dimension 3 and 4 ; see Proposition 1.6.
In view of Theorem 1, we naturally ask that under what circumstances one could expect a "multiplicative Orlov conjecture", namely whether two derived equivalent smooth projective varieties have isomorphic Chow motives as algebra objects, or even as Frobenius algebra objects. According to the celebrated theorem of Bondal-Orlov [8], this holds true for varieties with ample or anti-ample canonical bundle, since any two such derived equivalent varieties must be isomorphic. The situation gets more intriguing for varieties with trivial canonical bundle and we cannot expect in general that derived equivalent varieties have isomorphic Chow motives as Frobenius algebra objects : there exists counter-examples for Calabi-Yau threefolds and abelian varieties, where even the graded cohomology algebras of the two derived equivalent varieties are not isomorphic as Frobenius algebras (see Example 4.3 and Proposition 4.5 (ii)). We notice that, on the other hand, two derived equivalent abelian varieties are isogenous and have isomorphic Chow motives as algebra objects (see Proposition 4.5 (i)).
Although we do not provide much evidence beyond the case of K3 surfaces, we are tempted to ask, because of the (expected) similarities of the intersection product on hyper-Kähler varieties with that on abelian varieties (cf. Beauville's seminal [5], and also [54,20]), the following Question 1. Let X and Y be two twisted derived equivalent projective hyper-Kähler varieties. Are their Chow motives isomorphic as algebra objects or even as Frobenius algebra objects ? In particular, are their cohomology H * (−, Q) isomorphic as graded Q-algebras or even as Frobenius algebras ? Canonicity of the Shafarevich cycle. In [29], Huybrechts shows that the restriction to the transcendental cohomology of an isogeny ϕ : H 2 (S, Q) ∼ −→ H 2 (S ′ , Q) is induced by the cycle v 2 (E n−1 ) • · · · • v 2 (E 0 ) ∈ CH 2 (S × S ′ ), where E 0 , . . . , E n−1 are the Fourier-Mukai kernels in (2). This provides a Shafarevich cycle for the isogeny ϕ. In §5, we give some evidence for the above cycle to be canonical, that is, independent of the choice of a chain of twisted derived equivalence inducing the isogeny. This depends on extending a result of Huybrechts and Voisin (Theorem 5.1) to twisted equivalences. We do however prove unconditionally in Theorem 5.4 that the intersection of the second Chern classes of two objects E 1 and E 2 in D b (S × S ′ ) inducing an equivalence D b (S) ∼ −→ D b (S ′ ) is proportional to c 2 (S) × c 2 (S ′ ) in CH 2 (S × S ′ ). This suggests that the Mukai vectors of twisted derived equivalences between K3 surfaces can be added to the Beauville-Voisin ring ; see §5.
Notation and Conventions. We fix a base field k. By a derived equivalence between smooth projective k-varieties, we mean a k-linear exact equivalence of triangulated categories between their bounded derived categories of coherent sheaves. Chow groups will always be considered with rational coefficients. Concerning the category of Chow motives over k, we follow the notation and conventions of [3]. This category is a pseudo-abelian rigid tensor category, whose objects consist of triples (X, p, n), where X is a smooth projective variety of dimension d X over k, The tensor product of two motives is defined in the obvious way, while the dual of M = (X, p, n) is M ∨ = (X, t p, −n + d X ), where t p denotes the transpose of p. The Chow motive of a smooth projective variety X is defined as h(X) := (X, ∆ X , 0), where ∆ X denotes the class of the diagonal inside X × X, and the unit motive is denoted 1 := h(Spec(k)). In particular, we have CH l (X) = Hom(1(−l), h(X)). The Tate motive of weight −2i is the motive 1(i) := (Spec(k), ∆ Spec(k) , i). A motive is said to be of Tate type if it is isomorphic to a direct sum of Tate motives.

Derived equivalent surfaces
The aim of this section is to provide an alternative proof to the following result of Huybrechts [28,29] : (Huybrechts). Let S and S ′ be two (twisted) derived equivalent smooth projective surfaces defined over a field k. Then S and S ′ have isomorphic Chow motives.
The reason for including such a proof is threefold : first it provides anyway all the prerequisites and notation for the proof of Theorem 1 which will be given in §3.1 ; second by avoiding Manin's identity principle 3 (as is employed in [28]) we obtain an explicit inverse to the isomorphism v 2 (E) : h 2 tr (S) ∼ −→ h 2 tr (S ′ ) which will be essential to the proof of Theorem 1 ; and third it provides somehow a link between Orlov's Conjecture 1 and Murre's Conjecture 1.2 which itself is intricately linked to the conjectures of Bloch and Beilinson (see [33]).

1.1.
Murre's conjectures. We fix a base field k and a Weil cohomology theory H * (−) for smooth projective varieties over k. Concretely, we think of H * (−) as Betti cohomology in case k ⊆ C, or as ℓ-adic cohomology when char(k) = ℓ. Conjecture 1.2 (Murre [42]). Let X be a smooth projective variety of dimension d over k.
The filtration defined in (C) is conjecturally the Bloch-Beilinson filtration [6] (see also [58,Chapter 11]). In fact, as shown by Jannsen [33], the conjecture of Murre holds for all smooth projective varieties if and only if the conjectures of Bloch-Beilinson hold. We refer to [33] for precise statements. Proposition 1.3. Let X and Y be smooth projective varieties over k. Assume that h(X) and h(Y ) admit a Chow-Künneth decomposition as in Conjecture 1.2(A). Then, with respect to the Proof. This is formal : we have In other words, Murre's conjecture (B) implies that a motive of pure weight does not admit any non-trivial morphism to a motive of pure larger weight. Theorem 1.4 (Murre [41]). Let X be a smooth projective irreducible variety of dimension d X over a field k.
(i) The Chow motive of X admits a decomposition such that H * (h i (X)) = H i (X) ; in particular, Conjecture 1.2(A) holds for curves and surfaces. Moreover, such a decomposition can be chosen such that 3) holds in case X and Y are varieties of dimension ≤ 2 endowed with a Chow-Künneth decomposition as in (i).
Proof. Item (i) in the case of surfaces is the main result of [41]. In fact, for any smooth projective variety X of any dimension, h 1 (X) can be constructed as a direct summand of the motive of a smooth projective curve. As for (ii), this was checked by Murre [43] in the case one of X and Y has dimension ≤ 1. Thanks to item (i) and Proposition 1.3, it only remains to check that CH l (h 2 (X) ⊗ h 2 (Y )) = 0 for l = 0, 1 for smooth projective surfaces X and Y . For that purpose, we simply observe that for any choice of a Chow-Künneth decomposition (if it exists) on the motive of a smooth projective variety Z we have CH 0 (Z) = CH 0 (h 0 (Z)) and CH 1 (Z) = CH 1 (h 2 (Z) ⊕ h 1 (Z)). (Indeed, denote π i Z the projectors corresponding to the Chow-Künneth decomposition of Z, then by definition π 2i Z acts as the identity on H 2i (Z) and hence on im(CH i (Z) → H 2i (Z)), and by Murre [41] π 1 Z acts as the identity on ker(CH 1 (Z) → H 2 (Z)). Therefore π 2 Z + π 1 Z , which is a projector, acts as the identity on CH 1 (Z).) The following terminology will be convenient for our purpose. We say that a Chow motive M is of curve type (or of pure weight 1 ) if it is isomorphic to a direct summand of a direct sum of motives of the form h 1 (C), where C is a smooth projective curve defined over k. Motives of curve type form a thick additive subcategory 4 and enjoy the following property, which is also shared by Tate motives : Proposition 1.5. The full subcategory of motives whose objects are of curve type is abelian semi-simple. Moreover, the realization functor M → H * (M ) is conservative.
Proof. The first statement follows from the fact that this full subcategory of motives of curve type is equivalent to the category of abelian varieties up to isogeny, via the Jacobian construction ; see [3,Proposition 4.3.4.1]. The second statement follows from the first one together with the fact that H * (h 1 (A)) is a 2g-dimensional vector space for an abelian variety A of dimension g.

1.2.
Proof of Theorem 1.1. First, we observe as in [29,Section 2] that for twisted equivalences the yoga of Fourier-Mukai kernels, their action on Chow groups induced by Mukai vectors, and how they behave under convolutions works as in the untwisted case. Therefore, for ease of notation, we will only give a proof of Theorem 1.1 in the untwisted case.
1.2.1. Derived equivalences and motives, following Orlov. In general, let Φ E : be an exact equivalence with Fourier-Mukai kernel E ∈ D b (X ×Y ) between the derived categories of two smooth projective k-varieties of dimension d. Its inverse can be described as Φ −1 E ≃ 4 It is however not stable under tensor products and it does not contain any Tate motives. The tensor category generated by the Tate motives and the motives of curve type consists of the so-called motives of abelian type.
, where E ∨ is the derived dual of E and p, q are the projections from X × Y to X and Y respectively. As observed by Orlov [46], the Mukai vector In fact, as observed by Orlov [46], the latter identity shows that v(E) seen as a morphism of ind- The refined decomposition of the motive of surfaces, following Kahn-Murre-Pedrini. Let S be a smooth projective surface over k. The motive h(S) admits a Chow-Künneth decomposition as in Murre's Theorem 1.4(i) ; in particular h 0 (X) = h 4 (X)(2) ∨ is the unit motive 1 and h 1 (X) = h 3 (X)(2) ∨ is of curve type. Following Kahn-Murre-Pedrini [34], the summand h 2 (S) admits a further decomposition h 2 (S) = h 2 alg (S) ⊕ h 2 tr (S) defined as follows. Let k s be a separable closure of k and let E 1 , . . . , E ρ be non-isotropic divisors in CH 1 (S k s ) whose images in NS(X k s ) Q form an orthogonal basis. Up to replacing each E i by E i − (π 1 S ) * E i , we can assume that (π 1 S ) * E i = 0 for all i. Consider then the idempotent correspondence Since π 2 alg,S is the intersection form on NS(X k s ) Q , it is Galois-invariant, and hence does define an idempotent in CH 2 (S×S). The motive (S, π 2 alg,S , 0) is clearly isomorphic, after base-change to k s , to the direct sum of ρ copies of the Tate motive 1(−1). Moreover, it is easy to check that π 2 alg,S is orthogonal to π i S for i = 2 (use (π 1 S ) * E i = 0). Equivalently, we have π 2 alg,S • π 2 S = π 2 S • π 2 alg,S , so that (S, π 2 alg,S , 0) does define a direct summand of h 2 (S), denoted by h 2 alg (S). We then define π 2 tr,S := π 2 S − π 2 alg,S and h 2 tr (S) := (S, π 2 tr,S , 0). It is then straightforward to check that such a decomposition satisfies Hom(1(−i), h 2 tr (S)) = 0 for all i = 2. We note that t π 2 alg,S = π 2 alg,S , and since t π 0 S = π 4 S and t π 1 S = π 3 S we also have t π 2 tr,S = π 2 tr,S . Moreover, although this won't be of any use to us, we mention for comparison to [28] that Hom(1(−2), h 2 tr (S)) coincides with the Albanese kernel.
More generally, the above refined decomposition can be performed for direct summand of motives of surfaces, i.e. for motives of the form (S, p, 0), where S is a smooth projective ksurface and p ∈ CH 2 (S × S) is an idempotent. This will be used in the proof of Proposition 1.6. Indeed, by the above together with Theorem 1.4(ii), we have a decomposition (4) h , where none of the direct summands admit a non-trivial morphism to another direct summand placed on its right. It follows that the morphism p, expressed with respect to the decomposition of weight ≤ 2. Since Hom(h 2 tr (S), 1(−1)) = Hom(1(1), h 2 tr (S) ∨ ) = Hom(1(−1), h 2 tr (S)) = 0, we see that h 2 tr (S ′ ) is the only direct summand of weight 2 in (5) that admits a possibly non-trivial morphism from h 2 tr (S). Likewise, the only direct summand of (5) of weight ≤ 2 that maps possibly non-trivially in h 2 . It follows that the restriction of v 2 (E) induces an isomorphism tr,S ′ ; this fact will be used in the proof of Theorem 1. In a similar fashion, thanks to Theorem 1.4(ii) and having in mind that h 1 (S) ≃ h 3 (S)(1) is the direct summand of the motive of a curve (and similarly for S ′ ), v(E) induces isomorphisms , while the second one yields, thanks to Theorem 1.4(i), together with the semi-simplicity statement of Proposition 1.5, isomorphisms h 1 (S) ≃ h 1 (S ′ ) and h 3 (S) ≃ h 3 (S ′ ). This finishes the proof of Theorem 1.1.

1.3.
A slight generalization to Theorem 1.1. The content of this paragraph won't be used in the proof of Theorem 1. Recall that Theorem 1.1 fits more generally into the Orlov Conjecture 1.
The method of proof of Theorem 1.1 can be pushed through to establish the following : Proposition 1.6. Let X and Y be two smooth projective varieties of dimension 3 or 4 over a field k. Assume either of the following : • dim X = 3 and CH 0 (X) is representable ; • dim X = 4, CH 0 (X) and CH 0 (Y ) are both representable, and X and Y have same Picard rank.
Here, we say that a smooth projective k-variety X of dimension d has representable CH 0 if for a choice of universal domain (i.e., algebraically closed field of infinite transcendence degree over its prime subfield) Ω containing k, there exists a smooth projective Ω-curve C and a correspondence γ ∈ Hom(h(X Ω ), h(C)) such that γ * CH 0 (C) = CH 0 (X Ω ). Examples of such varieties include varieties with maximally rationally connected quotient of dimension ≤ 1, and in particular rationally connected varieties.
Proof. We start with the case of threefolds. By [21], X admits a Chow-Künneth decomposition, where the even-degree summands are of Tate type, while the odd-degree summands are Tate twists of motives of curve type. The arguments of §1.2.1 show that h(Y ) is a direct summand of similar property to that of X (and hence has representable CH 0 ). The arguments of §1.2.3 provide isomorphisms Since the even-degree summands are of Tate type, and since dim H 2i (X) = dim H 6−2i (X) by Poincaré duality (and similarly for Y ), we conclude that h 2i (X) ≃ h 2i (Y ) for all i. Now, a theorem of Popa-Schnell [49] says that two derived equivalent complex varieties have isogenous reduced Picard scheme (see [24,Appendix] for the case of k-varieties). It follows from Theorem 1.4(i) that h 1 (X) ≃ h 1 (Y ), and then by duality that h 5 (X) ≃ h 5 (Y ). Since all terms of (6) are of curve type, we deduce from the semi-simplicity statement of Proposition 1.5 that h 3 (X) ≃ h 3 (Y ). Alternately, from [2], two derived equivalent threefolds have degree-wise isomorphic cohomology groups (the isomorphisms being induced by some algebraic correspondences) ; it then follows from the description of the motives of X and Y together with Proposition 1.5 that X and Y have isomorphic motives.
In the case of fourfolds, we first note by [56,Theorem 3.11] that the motive of a fourfold X with representable CH 0 admits a decomposition of the form for some curve C and some surface S. It follows from the arguments of §1.2.2 that h(X) admits a Chow-Künneth decomposition such that h 2i+1 (X)(i) is of curve type for all i, h 2i (X) is of Tate type for all i = 2, and h 4 (X) further decomposes as h 4 alg (X) ⊕ h 4 tr (X), with the property that h 4 alg (X) becomes, after base-change to k s , a direct sum of Tate motives 1(−2) and h 4 tr (X)(1) is a direct summand of the motive of a surface with Hom(h 4 tr (X), 1(−i)) = 0 for i = 3. The arguments of §1.2.3 then provides isomorphisms h 0 (X)⊕h 2 (X)(1)⊕h 4 alg (X)(2)⊕h 6 (X)(3)⊕h 8 and h 4 tr (X) ≃ h 4 tr (Y ).
As in the case of threefolds, since the Picard numbers of X and Y agree, we conclude that h 2i (X) ≃ h 2i (Y ) for all i = 2 and that h 4 alg (X) ≃ h 4 alg (Y ), while by utilizing the Theorem of Popa-Schnell [49], we conclude from Theorem 1.4(i) that h 1 (X) ≃ h 1 (Y ) and then by duality that h 7 (X) ≃ h 7 (Y ). It follows by cancellation (Proposition 1. (1). Since there is a Lefschetz isomorphism H 3 (X) ≃ H 5 (X)(1) and similarly for Y , we conclude (again from Proposition 1.5) that h 3 (X) ≃ h 3 (Y ) and h 5 (X) ≃ h 5 (Y ).

Motives of varieties as Frobenius algebra objects
2.1. Algebras and Frobenius algebras. Let C be a symmetric monoidal category with tensor unit 1. An algebra object in C is an object M together with a unit morphism η : 1 → M and a multiplication morphism µ : M ⊗M → M satisfying the unit axiom µ•(id ⊗η) = id = µ•(η ⊗id) and the associativity axiom µ • (µ ⊗ id) = µ • (id ⊗ µ). It is called commutative if moreover µ = µ • c M,M is satisfied, where c M,M is the commutativity constraint of the category C. A morphism of algebra objects between two algebra objects M and N is a morphism φ : M → N in C that preserves the multiplication µ and the unit η. We note that an algebra structure on an object M of C induces naturally an algebra structure on the n-th tensor powers M ⊗n of M , and that a morphism φ : M → N of algebra objects induces naturally a morphism of algebra objects φ ⊗n : M ⊗n → N ⊗n which is an isomorphism if φ is.
If C is moreover rigid and possesses a ⊗-invertible object then we can speak of Frobenius algebra objects in C : Definition 2.1 (Frobenius algebra objects). Let (C, ⊗, ∨, 1) be a rigid symmetric monoidal category admitting a ⊗-invertible object denoted by 1(1). Let d be an integer. A degree-d Frobenius algebra object in C is the data of an object M ∈ C endowed with • η : 1 → M , a unit morphism ; , an isomorphism, called the Frobenius structure ; satisfying the following axioms: is defined by dualizing µ via the following commutative diagram : We define also the counit morphism ǫ : M → 1(−d) by dualizing η via the following diagram: We remark that ǫ and δ satisfy automatically the counit and coassociativity axioms.
A Frobenius algebra object M is called commutative if the underlying algebra object is commutative : µ • c M,M = µ. Commutativity is equivalent to the cocommutativity of δ. The morphism β = ǫ • µ : M ⊗ M → 1(−d), called the Frobenius pairing, is also sometimes used. It is a symmetric pairing if M is commutative.
Remark 2.2. In the case of Frobenius algebra objects of degree 0, the ⊗-invertible object 1(1) is not needed in the definition, and it is reduced to the usual notion of Frobenius algebra object in the literature (see for example [1], [36]). In this sense, Definition 2.1 generalizes the existing definition of Frobenius structure by allowing non-zero twists. We believe that our more flexible notion is necessary and adequate for more sophisticated tensor categories than that of vector spaces, such as the categories of Hodge structures, Galois representations, motives, etc.

Remark 2.3 (Morphisms)
. Morphisms of Frobenius algebra objects are defined in the natural way, that is, as morphisms φ : M → N such that all the natural diagrams involving the structural morphisms are commutative. In particular, in order to admit non-trivial morphisms, the degrees of the Frobenius algebra objects M and N must coincide and the following diagram is then commutative : As a result, all morphisms between Frobenius algebra objects are in fact invertible. It is an exercise to show that an isomorphism φ : M → N between two Frobenius algebra objects respects the Frobenius algebra structures if and only if it is compatible with the algebra structure (i.e. with µ) and the Frobenius structure (i.e. with λ). This is proved in Proposition 2.11 in the case of Chow motives of smooth projective varieties. In addition, φ ⊗n : M ⊗n → N ⊗n is naturally an isomorphism of Frobenius algebra objects, as is the dual t φ : We summarize the above discussion in the following The "if" part follows from the definition. The "only if" part is explained in Remark 2.3.

Now let us turn to important examples of Frobenius algebra objects.
Example 2.5 (Cohomology as a graded vector space). Let X be a connected compact orientable (real) manifold of dimension d. Then its cohomology group H * (X, Q) is naturally a Frobenius algebra object of degree d in the category of Z-graded Q-vector spaces (where morphisms are degree-preserving linear maps and the ⊗-invertible object is chosen to be Q [1], the 1-dimensional vector space sitting in degree −1). The unit morphism η : Q → H * (X, Q) is given by the fundamental class [X] ; the multiplication morphism µ : is the cup-product ; the Frobenius structure comes from the Poincaré duality The induced comultiplication morphism δ : is the Gysin map for the diagonal embedding X ֒→ X × X ; the counit morphism ǫ : is the integration X . The Frobenius condition is a classical exercise. Note that H * (X, Q) is commutative, because the commutativity constraint in the category of graded vector spaces is the super one. If instead we consider the cohomology group as merely an ungraded vector space, then it becomes a Frobenius algebra object of degree 0 (i.e. in the usual sense) ; this is one of the main examples in the literature. Example 2.6 (Hodge structures). A pure (rational) Hodge structure is a finite-dimensional Z-graded Q-vector space H = ⊕ n∈Z H (n) such that each H (n) is given a Hodge structure of weight n. A morphism between two Hodge structures is required to preserve the weights. The category of pure Hodge structures is naturally a rigid symmetric monoidal category. The ⊗invertible object is chosen to be Q(1), which is the 1-dimensional vector space (2πi) · Q with Hodge structure purely of type (−1, −1).
Let X be a compact Kähler manifold of (complex) dimension d. Then H * (X, Q) is naturally a commutative Frobenius algebra object of degree d in the category of pure Q-Hodge structures. The structural morphisms are the same as in Example 2.5 up to replacing [d] by (d). For instance, λ : Our main examples of Frobenius algebra objects are the Chow motives of smooth projective varieties.

2.2.
Frobenius algebra structure on the motives of varieties. The category of rational Chow motives over a field k is rigid and symmetric monoidal. We choose the ⊗-invertible object to be the Tate motive 1(1). Then for any smooth projective k-variety X of dimension d, its Chow motive h(X) is naturally a commutative Frobenius algebra object of degree d in the category of Chow motives. Let us explain the structural morphisms in detail.
Let δ X denote the class of the small diagonal {(x, x, x) : x ∈ X} in CH d (X × X × X). Note that for α and β in CH * (X), we have (δ X ) * (α × β) = α · β, so that δ X seen as an element of Hom(h(X) ⊗ h(X), h(X)) describes the intersection theory on the Chow ring of X, as well as the cup product of its cohomology ring. So it is natural to define the multiplication morphism to be the one given by the small diagonal δ X ∈ CH 2d (X × X × X) = Hom(h(X) ⊗ h(X), h(X)) ; it can be checked to be commutative and associative. The unit morphism η : 1 → h(X) is again given by the fundamental class of X. The unit axiom is very easy to check.
The Frobenius structure is defined as the following canonical isomorphism, called the motivic Poincaré duality, given by the class of diagonal ∆ One readily checks that the induced comultiplication morphism is given by the fundamental class.
The following lemma proves that, endowed with these structural morphisms, h(X) is indeed a Frobenius algebra object.
Lemma 2.7 (Frobenius condition). Notation is as above. We have an equality of endomorphisms of h(X) ⊗ h(X) : Proof. We only show δ • µ = (µ ⊗ id) • (id ⊗ δ), the other equality being similar. We have a commutative cartesian diagram without excess intersection : where ∆ : X → X × X denotes the diagonal embedding. The base-change formula yields on Chow groups, hence also for Chow motives by Manin's identity principle [3, §4.3.1]. Now it suffices to notice that ∆ * is the comultiplication δ and ∆ * is the multiplication µ.
Remark 2.8. In general, a tensor functor F : C → C ′ between two rigid symmetric monoidal categories sends a Frobenius algebra object in C to such an object in C ′ . Example 2.6 is obtained by applying the Betti realization functor from the category of Chow motives to that of pure Hodge structures ; Example 2.5 (for Kähler manifolds) is obtained by further applying the forgetful functor (Q(1) is sent to Q[2]).

2.3.
(Iso)morphisms of Chow motives as Frobenius algebra objects. The notion of morphisms between two algebra objects is the natural one. Let us spell it out for motives of varieties. A non-zero morphism Γ : h(X) → h(Y ) between the motives of two smooth projective varieties over a field k is said to preserve the algebra structures if the following diagram commutes 5 is also a Q-algebra homomorphism. We say that the Chow motives of X and Y are isomorphic as algebra objects if there exists an isomorphism Γ : h(X) → h(Y ) that preserve the algebra structures. The following lemma is a formal consequence of the definition.
Lemma 2.9 (Algebra morphisms). Let X and Y be connected smooth projective varieties and let Γ : h(X) → h(Y ) be a non-zero morphism that preserves the algebra structures.
(ii) Suppose X and Y have same dimension and define c to be the rational number such that In particular, Γ is an isomorphism if and only if c = 0, and in this case, due to Lieberman's formula 6 , the inverse of Γ is equal to 1 c t Γ.
Proof. (i) This is the analogue of the basic fact that a non-trivial homomorphism of unital algebras preserves the units. Concretely, the fundamental class of X provides a morphism 1 X : 1 → h(X), and similarly for Y , and we need to show that Γ • 1 X = 1 Y . First, for dimension reasons we have Γ • 1 X = λ · 1 Y for some λ ∈ Q. Compose then the diagram (7) with the morphism 1 X ⊗ 1 X ; one obtains λ 2 = λ. If λ = 0, then by composing diagram (7) with the morphism 1 X ⊗ id X , we find that Γ = 0. Hence λ = 1 and we are done.
(ii) The commutativity of (7) provides the identity Γ where c is the rational number such that Γ * [Y ] = c [X] and where the second equality is Lieberman's formula. Since we assume that Γ is invertible, we get that Γ −1 = 1 c t Γ.
As is alluded to in Lemma 2.4, the notion of orthogonality is highly relevant when considering morphisms between Frobenius algebras. Let us recast it in the context of motives : 5 As explained in Lemma 2.9, a non-zero morphism between algebra objects that preserves the multiplication morphisms must also preserve the unit morphisms, and hence is a morphism of algebra objects in the sense of §2.1. 6  Finally, we can unravel the meaning of being isomorphic as Frobenius algebra objects for the motives of two varieties (and the same holds for Hodge morphisms between the cohomology algebras of smooth projective varieties of same dimension).
Proposition 2.11 (Frobenius algebra isomorphisms). Let X and Y be two smooth projective varieties of the same dimension and Γ : h(X) → h(Y ) be a morphism between their motives. Then the following are equivalent: (i) Γ is an isomorphism of Frobenius algebra objects.
(ii) Γ is an algebra isomorphism and Γ is orthogonal : that is, Proof. The equivalence between (i) and (ii) is a special case of Lemma 2.4. The equivalence between (ii) and (iii) can be read off Lemma 2.9(ii). For the equivalence between (ii) and (iv), one only needs to see that an orthogonal isomorphism ( But this again follows from Lieberman's formula.

Derived equivalent K3 surfaces
The aim of this section is to prove Theorem 1, Corollaries 1 and 2.
3.1. Proof of Theorem 1. The proof relies crucially on the Beauville-Voisin description of the algebra structure on the motive of K3 surfaces : [4]). Let S be a K3 surface and let o S be the class of any point lying on a rational curve on S. Then, as cycle classes in CH 2 (S × S × S), we have (Of course, this is due originally to Beauville-Voisin [4].) According to Proposition 2.11, in order to establish Theorem 1, it is necessary and sufficient to produce a correspondence Γ : h(S) → h(S ′ ) which is invertible and such that By the Beauville-Voisin Theorem 3.1, it is sufficient (in fact also necessary by looking at the contravariant action of (ii) on ∆ S ′ ) to produce a correspondence Γ : h(S) → h(S ′ ) which is invertible and such that We now proceed to the proof of Theorem 1, i.e. to constructing an invertible correspondence satisfying (i) and (ii ′ ) above. Given a K3 surface S, we consider the refined Chow-Künneth decomposition of Kahn-Murre-Pedrini as described in §1.2.2 given by where π i S denote the projectors on the corresponding direct summands and where o S denotes the Beauville-Voisin zero-cycle as in Theorem 3.1. Moreover, the decomposition is such that t π 2 alg,S = π 2 alg,S and t π 2 tr,S = π 2 tr,S . Consider now two twisted derived equivalent K3 surfaces S and S ′ . As in the proof of Theorem 1.1, we only give a proof in the untwisted case, the twisted case being similar. We fix an exact linear equivalence Φ E : The proof will proceed in two steps. First, we will construct an invertible correspondence is the algebraic summand of the motive of S ; second, we will construct an invertible correspondence on the transcendental summands of the motives of S and S ′ : will then provide the desired isomorphism of Frobenius algebra objects.
First, the numerical Grothendieck group K num 0 equipped with the Euler pairing is clearly a derived invariant. Using the Chern character isomorphism, we obtain an isometry between the quadratic spaces NS(S k s ) Q and NS(S ′ k s ) Q , where NS is the extended Néron-Severi group equipped with the Mukai pairing, hence is isometric to the (orthogonal) direct sum of the Néron-Severi lattice (endowed with the intersection pairing) and a copy of the hyperbolic plane. By Witt's cancellation theorem, the Néron-Severi groups NS(S k s ) Q and NS(S ′ k s ) Q of two derived equivalent surfaces are isomorphic both as Gal(k)-representations and as quadratic spaces ; there exists therefore a correspondence M = π 2 alg,S ′ • M • π 2 alg,S in CH 2 (S × k S ′ ) inducing an isometry NS(S k s ) Q ≃ NS(S ′ k s ) Q . This means that M induces an isomorphism h 2 alg (S) ∼ −→ h 2 alg (S ′ ) with inverse given by its transpose. It follows that Γ alg := o S ×S ′ +M +S×o S ′ induces an isomorphism h alg (S) . Since K3 surfaces have trivial first Chern class and trivial canonical bundle, it follows that the inverse of v 2 (E) is in fact its transpose. In other words, Γ tr := π 2 tr,S ′ • v 2 (E) • π 2 tr,S induces an isomorphism of Chow motives h 2 tr (S) ∼ −→ h 2 tr (S ′ ) with inverse its transpose. Finally, we do have (π 2 tr ) * o S = 0 because of the orthogonality of π 2 tr,S with π 4 S . The required correspondences Γ alg and Γ tr have thus been constructed and this concludes the proof of Theorem 1.

3.2.
Proof of Corollary 1. Let S and S ′ be two twisted derived equivalent K3 surfaces. Then due to Theorem 1 their motives are isomorphic as Frobenius algebra objects. As is explained in §2.1, isomorphisms of Frobenius algebra objects behave well with respect to (tensor) products, hence it suffices to see that for any n ∈ Z >0 the Hilbert schemes of n points Hilb n (S) and Hilb n (S ′ ) have isomorphic Chow motives as Frobenius algebra objects. To this end, we use the result of Fu-Tian [18] that describes the algebra object h(Hilb n (S)) in terms of the algebra objects h(S m ) for m ≤ n, together with some explicit combinatorial rules. More precisely, by [18, Theorem 1.6 and Remark 1.7], for a K3 surface S, we have an isomorphism of algebra objects : (9) φ : h(Hilb n (S)) ≃ where S n is the symmetric group acting naturally on S n ; for a permutation g, O(g) is its set of orbits in {1, · · · , n}, S O(g) is canonically identified with the fixed locus (S n ) g , and finally ⋆ orb,dt is the orbifold product with discrete torsion (see [17,18]) defined as follows (let us omit the Tate twists for ease of notation) : it is compatible with the S n -grading, and for any g, h ∈ S n , h) is given by the pushforward via the diagonal inclusion h) ), by [18, Lemma 9.3] : it is now clear from the above precise description that the right-hand side of (9) for S and for S ′ are isomorphic algebra objects, and the isomorphism can be chosen orthogonal. As the morphism φ in (9) satisfies φ −1 = t φ, we have h(Hilb n (S)) and h(Hilb n (S ′ )) are isomorphic Frobenius algebra objects. This completes the proof.
Remark 3.2 (Chow rings vs. algebra objects). It turns out that we do not need Theorem 1 to show that two twisted derived equivalent K3 surfaces have isomorphic Chow rings. Indeed, Huybrechts' result [28] (generalized to the twisted case in [29]) provides a correspondence Γ ∈ CH 2 (S × S ′ ) that induces an isomorphism of graded Q-vector spaces Γ * : CH * (S) with the extra property of being isometric on the Néron-Severi spaces CH 1 (S) ∼ −→ CH 1 (S ′ ). Now thanks to the theorem of Beauville-Voisin [4] saying that the image of the intersection product of two divisors on a K3 surface is of dimension 1, this already implies that Γ * is actually an isomorphism of graded Q-algebras.
In contrast, in the situation of Corollary 1, a derived equivalence between D b (S) and D b (S ′ ) does give rise to a derived equivalence between their powers and Hilbert schemes, thanks to Bridgeland-King-Reid [10] and Haiman [22]. However, it is not at all clear for the authors how to produce an isomorphism of the Chow rings (or even the rational cohomology rings) of two derived equivalent holomorphic symplectic varieties starting from the Fourier-Mukai kernel ; see Conjecture 4.6.
3.3. Proof of Corollary 2. The equivalence of (i) and (ii) is due to Huybrechts [29, Corollary 1.4], while the implication (ii) ⇒ (iii) is Theorem 1. We now prove the implication (iii) ⇒ (i). Suppose that Γ : h(S) → h(S ′ ) is an isomorphism that preserves the algebra structures. Let c be the rational number such that Γ * [S ′ ] = c [S], or equivalently such that Γ −1 = 1 c t Γ by Lemma 2.9 ; then the following diagram is commutative : The commutativity of the left-hand square of (11) is implied directly by the assumption that Γ preserves the algebra structures, while the commutativity of the right-hand square follows from the Poincaré dual of the identity Γ * [S ′ ] = c [S]. If in addition Γ preserves the Frobenius algebra structure, then c = 1 by Proposition 2.11. This means that S and S ′ are isogenous.

3.4.
A motivic global Torelli theorem. The aim of this section is to show that Lemma 2.9 directly allows to upgrade motivically the global Torelli theorem, without utilizing the decomposition of the diagonal of Beauville-Voisin (Theorem 3.1). We denote h(X) Z the Chow motive of X with integral coefficients. Proof. The equivalence of items (i) and (ii) is the global Torelli theorem. The implication (i) ⇒ (iii) is obvious. It remains to check that (iii) ⇒ (ii). Once it is observed that Lemma 2.9 holds with integral coefficients, we obtain the following commutative diagram (with c ∈ Z), which is similar to (11) in the proof of Corollary 2 Therefore, there is an isometry of lattices between H 2 (S, Z) ⊗ c and H 2 (S ′ , Z), which implies that c = 1.

Beyond K3 surfaces
Orlov's conjecture 1 predicts that the Chow motives of two derived equivalent smooth projective varieties are isomorphic. Motivated by Theorem 1, we raised the following question in the introduction : Question 4.1. When can we expect more strongly that a derived equivalence between two smooth projective varieties implies an isomorphism between their rational Chow motives as Frobenius algebra objects ?
We make some remarks and speculations on this question in this section. In general, one cannot expect in general a positive answer to Question 4.1. In fact, if h(X) and h(Y ) are isomorphic as Frobenius algebra objects then by applying the Betti realization functor, their cohomology are isomorphic as Frobenius algebras, that is, due to Proposition 2.11, there is a (graded) isomorphism of Q-algebras H * (X, Q) → H * (Y, Q) sending the class of a point on X to the class of a point on Y . However, as we will see below, this is not the case in general for derived equivalent varieties.   Let us know try to define a graded Hodge algebra isomorphism ψ : H * (X, Q) → H * (Y, Q). Consider the following formulas with the numbers a, b to be determined later : Q). These formulas define an algebra isomorphism if and only if b 2 = a d s. This equation has nonzero rational solutions when d is odd or d is even and s is a square in Q. Item (ii) is therefore proved.
The proof of (i) goes similarly as for (ii) by replacing Q by R. Notice that the analogous assumption that s is a square in R is automatically satisfied. So it is enough to see that there are always non-zero real solutions to the equation b 2 = a d s = 1, where the last equality reflects the Frobenius condition.

Abelian varieties.
This yields that the isogeny f is of degree λ 2g . (c) =⇒ (a) If there is an isogeny f : B → A of degree m 2g , then for any 0 ≤ i ≤ 2g consider the morphism Γ i : Here we use the motivic decomposition of Deninger-Murre [16] for abelian varieties h(A) = ⊕ 2g i=0 h i (A), and π i is the projector corresponding to h i . One readily checks that Γ := i Γ i : h(A) → h(B) is an isomorphism of algebra objects. Moreover, as π i = t π 2g−i for all i, we have that Γ −1 i = t Γ 2g−i , hence Γ −1 = t Γ, that is, Γ respects the Frobenius structures. Notice that by construction, Γ respects the decomposition of Deninger-Murre. The proof of (iii) is similar to the last part of the proof of (ii). One only needs to notice that there is no obstruction to taking the 2g-th root of a positive number in R.
As a consequence, given two derived equivalent abelian varieties, in general there is no isomorphism of Frobenius algebra objects between their Chow motives (or their cohomology). Indeed, by Proposition 4.5(ii), the motives of two derived equivalent abelian varieties that cannot be related by an isogeny of degree the 2g-th power of some positive integer are not isomorphic as Frobenius algebra objects. For instance, if one considers an abelian variety A with Néron-Severi group generated by one ample line bundle L, then any isogeny between A and A ∨ is of degree χ(L) 2 m 4g for some m ∈ Z >0 . But in general, χ(L) is not a g-th power in Z. On the other hand, A and A ∨ are always derived equivalent by Mukai's classical result [39].

4.3.
Hyper-Kähler varieties. One particularly interesting class of varieties for which we expect a positive answer consists of (projective) hyper-Kähler varieties ; these constitute higherdimensional generalizations of K3 surfaces. Note that by Huybrechts-Nieper-Wißkirchen [30], any Fourier-Mukai partner of a hyper-Kähler variety remains hyper-Kähler.
Conjecture 4.6. Let X and Y be two projective hyper-Kähler varieties. If there is an exact equivalence between triangulated categories D b (X) and D b (Y ), then there exists an isomorphism of Chow motives h(X) and h(Y ), as Frobenius algebra objects in the categories of Chow motives. In particular, their Chow rings as well as cohomology rings are isomorphic.
The following result is known to the experts ; it answers the last part of Conjecture 4.6 for cohomology with complex coefficients. Proposition 4.7. Let X and Y be two derived equivalent projective hyper-Kähler varieties. Then their cohomology rings with complex coefficients are isomorphic as C-algebras.
Proof. 8 As any exact equivalence D b (X) ≃ D b (Y ) is given by Fourier-Mukai kernel, it lifts naturally to an equivalence of differential graded categories. Therefore we have an isomorphism of graded C-algebras between their Hochschild cohomology: By a result of Calaque-Van den Bergh [14], which was also previously announced by Kontsevich, the Hochschild-Kostant-Rosenberg isomorphism twisted by the square root of the Todd genus gives rise to an isomorphism of C-algebras Now the symplectic forms on X induce an isomorphism between T X and Ω X , which yields isomorphisms of C-algebras: We can conclude by combining these isomorphisms. 8 It is not clear to the authors whether the isomorphism constructed in the proof preserves the Frobenius structure.
There are not so many known examples of derived equivalent hyper-Kähler varieties. Let us test Conjecture 4.6 for the available ones.
Example 4.8. Let S and S ′ be two derived equivalent K3 surfaces. Then for any n ∈ N * , the n-th Hilbert schemes Hilb n (S) and Hilb n (S ′ ) are derived equivalent. Indeed, by combining the results of Bridgeland-King-Reid [10] and Haiman [22], we have exact linear equivalences of triangulated categories : Sn (S n ), and similarly for S ′ ; the Fourier-Mukai kernel E ⊠ · · · ⊠ E induces an equivalence is the original Fourier-Mukai kernel inducing the equivalence between D b (S) and D b (S ′ ). We showed in Corollary 1 that h(Hilb n (S)) and h(Hilb n (S ′ )) are isomorphic as Frobenius algebra objects.   [17] predicts that the orbifold motive of [M/G] endowed with the orbifold product is isomorphic to the motive of X as algebra objects. In this sense, forgetting the Frobenius structure, we can obtain some evidences for the orbifold analogue of Conjecture 4.6 : for example between a K3 orbifold and its minimal resolution by [19], between [ker(A n+1 + − → A)/S n ] and the n-th generalized Kummer variety associated to an abelian surface A by [17], and between [S n /S n ] and the n-th Hilbert scheme of a K3 surface S by [18]. In fact, the authors suspect that the motivic hyper-Kähler resolution conjecture can be stated more strongly as an isomorphism of Frobenius algebra objects with complex coefficients, and the proofs of our aforementioned results do confirm this stronger version.

Chern classes of Fourier-Mukai equivalences between K3 surfaces
The aim of this final section is to provide evidence for the fact that the Chern classes of Fourier-Mukai equivalences between two K3 surfaces S and S ′ define "distinguished" classes in the Chow ring of S × S ′ , in the sense that they can be added to the Beauville-Voisin ring of S × S ′ and the resulting ring would still inject into cohomology via the cycle class map. 5.1. The Beauville-Voisin ring, and generalizations. Let S be a K3 surface and define its Beauville-Voisin ring R * (S) to be the subring of CH * (S) generated by divisors and Chern classes of the tangent bundle. By Beauville-Voisin's Theorem 3.1, this ring has the property that it injects into cohomology via the cycle class map.
In [54,Proposition 8.14] it was observed that the decomposition of the small diagonal (8) is equivalent to the above Chow-Künneth decomposition being multiplicative, meaning that the multiplication morphism h(S)⊗h(S) → h(S) is compatible with the grading given by the Chow-Künneth decomposition.
The following (formal) facts about multiplicative Chow-Künneth decompositions will be used. Let X and Y be two smooth projective varieties, both having motive endowed with a multiplicative Chow-Künneth decomposition. Then [54,Theorem 8.6] the product Chow-Künneth decomposition h n (X ×Y ) = i+j=n h i (X)⊗h j (Y ) is multiplicative. Moreover, if p : X ×Y → X denotes the projection, then p * : h(X) → h(X × Y ) is graded (i.e. compatible with the Chow-Künneth decompositions) and p * : h(X × Y ) → h(X) shifts the gradings by −2 dim Y .
A Chow-Künneth decomposition on the motive of X induces a bigrading on the Chow groups of X given by which in case the Chow-Künneth decomposition is multiplicative satisfies Given smooth projective varieties endowed with multiplicative Chow-Künneth decompositions, the products of which are endowed with the product Chow-Künneth decompositions, we therefore see that CH * (−) (0) defines a subalgebra of CH * (−) that is stable under pushforwards and pullbacks along projections, and stable under composition of correspondences belonging to CH * (− × −) (0) .
Murre's conjecture 1.2(B) and (D) imply that CH i (X) (0) := CH i (h 2i (X)) injects in cohomology with image the Hodge classes for any choice of Chow-Künneth decomposition. (This is known unconditionally in the case i = 0 and i = dim X.) In particular, in the above situation of smooth projective varieties endowed with multiplicative Chow-Künneth decompositions, it is expected that the subalgebra CH * (−) (0) injects into cohomology with image the Hodge classes. In that sense, CH * (−) (0) is a maximal subalgebra of CH * (−) with the property that it injects into cohomology via the cycle class map.  In light of the discussion in §5.1, it is natural to ask whether a more general statement could be true, namely :  (2) , it is enough to show that v 2 (E) (2) = 0. Let γ be any cycle in CH 2 (S ×S ′ ). On the one hand, we have γ (2) = π 2 tr,S ′ •γ•π 4 S +π 0 S ′ •γ•π 2 tr,S . On the other hand, we have γ • π 4 S is a multiple of (p ′ ) * o S ′ and henceforth since (π 2 tr,S ′ ) * o S ′ = 0 that π 2 tr,S ′ • γ • π 4 S = 0. Likewise, we have π 0 S ′ • v 2 (E) • π 2 tr,S = 0, and the proposition is established.
We deduce from §5.1 the following

Some speculations concerning Chern classes of twisted derived equivalent K3
surfaces. It is natural to ask whether Theorem 5.1 extends to derived equivalences between twisted K3 surfaces : We note that if v(E) preserves the Beauville-Voisin ring, then the same argument as in the proof of Proposition 5.3 gives that v 2 (E) belongs to CH 2 (S × S ′ ) (0) .
Let us now define E * (S × S ′ ) to be the subalgebra of CH * (S × S ′ ) generated by divisors, p * c 2 (S), (p ′ ) * c 2 (S ′ ), and the Chern classes of E, where E runs through objects in D b (S ×S ′ , α −1 ⊠ α ′ ) inducing exact equivalences Φ E : D b (S, α) ∼ −→ D b (S ′ , α ′ ) for some Brauer classes α ∈ Br(S) and α ′ ∈ Br(S ′ ). We then defineẼ * (S × S ′ ) to be the subalgebra of CH * (S × S ′ ) generated by cycles of the form where γ i ∈ E * (S i × S i+1 ) for all i for some K3 surfaces S = S 0 , S 1 , . . . , S n = S ′ . According to the discussion in §5.1, a positive answer to Question 5.5 would suggest that the following question should have a positive answer.
Appendix A. Non-isogenous K3 surfaces with isomorphic Hodge structures Recall that two K3 surfaces S and S ′ are said to be isogenous if their second rational cohomology groups are Hodge isometric, that is, if there exists an isomorphism of Hodge structures f : H 2 (S, Q) ≃ −→ H 2 (S ′ , Q) making the following diagram commute : We provide in this appendix infinite families of pairwise non-isogenous K3 surfaces with isomorphic rational Hodge structures, that is, for any two K3 surfaces S and S ′ belonging to the same family, we have H 2 (S, Q) ≃ H 2 (S ′ , Q) as Q-Hodge structures but there does not exist any isomorphism of Q-Hodge structures H 2 (S, Q) → H 2 (S ′ , Q) that is compatible with the intersection pairings given by (α, β) → deg(α ∪ β). By Mukai [40], such K3 surfaces are not derived equivalent to each other, and actually not even through a chain of twisted derived equivalences (cf. Huybrechts [29, Theorem 0.1]).
In fact, our families have a stronger property : any two K3 surfaces S and S ′ in the same family have Hodge-isomorphic rational cohomology algebras, that is, for any two K3 surfaces S and S ′ of the family, we have a graded Hodge isomorphism g : H * (S, Q) ≃ −→ H * (S ′ , Q) that respects the cup-product, i.e. such that the following diagram commutes : We say that H * (S, Q) and H * (S ′ , Q) are Hodge algebra isomorphic.
A.1. Motivation and statements. The aim of this appendix is to show that, for K3 surfaces, the notion of isogeny is strictly more restrictive than the notion of Hodge algebra isomorphic. In §A.5 this is upgraded motivically : we show that, for Chow motives of K3 surfaces, the notion of being isomorphic as Frobenius algebra objects is strictly more restrictive than the notion of being isomorphic as algebra objects, thereby justifying the somewhat technical condition (iii) involving Frobenius algebra objects in the motivic Torelli statement of Corollary 2.
First, we provide an infinite family of K3 surfaces whose rational cohomology rings are all Hodge algebra isomorphic (and, a fortiori, Hodge isomorphic), but they are pairwise nonisogenous. Precisely we have Theorem A.1. There exists an infinite family {S i } i∈Z >0 of pairwise non-isogenous K3 surfaces such that, for all j, k ∈ Z >0 , H * (S j , Q) and H * (S k , Q) are Hodge algebra isomorphic. Moreover, such a family can be chosen to consist of K3 surfaces of maximal Picard rank 20.
We will present two proofs of this theorem, one in §A.3 and the other in §A.4. Furthermore, we will show in Theorem A.13 that the Chow motives of K3 surfaces belonging to the family of K3 surfaces of Theorem A.1 are all isomorphic as algebra objects.
Second, if the Picard number is maximal, a family of K3 surfaces as in Theorem A.1 must have non-isometric Néron-Severi spaces (Lemma A.8). In this sense, we can improve Theorem A.1 in order to have in addition isometric Q-quadratic forms on the Néron-Severi spaces : Theorem A.2. There exists an infinite family {S i } i∈Z >0 of pairwise non-isogenous K3 surfaces such that, for all j = k ∈ Z >0 , we have • H * (S j , Q) and H * (S k , Q) are Hodge algebra isomorphic.
Furthermore, we will show in Proposition A.14 that, assuming Conjecture A.9, the K3 surfaces in such a family all have isomorphic Chow motives and isomorphic Chow rings, but their Chow motives are pairwise non-isomorphic as Frobenius algebra objects. This gives evidence that one cannot characterize the isogeny class of a complex K3 surface with its Chow ring. Remark A.3. As mentioned to us by Chiara Camere, examples of a pair of non-isogenous K3 surfaces with isomorphic rational Hodge structures were constructed geometrically, via the so-called Inose isogenies 10 [32], by Boissier-Sarti-Veniani [7]. Precisely, let f be a symplectic automorphism of prime order p of a K3 surface S and let S ′ be the minimal resolution of the quotient S/ f ; the surface S ′ is a K3 surface and, by definition, the rational map S S ′ is a degree-p Inose isogeny between these two K3 surfaces. On the one hand, we have isomorphisms of Hodge structures H 2 tr (S, On the other hand, [7, Theorem 1.1 and Corollary 1.2] provide many situations where H 2 tr (S, Q) and H 2 tr (S ′ , Q) are not isometric and consequently H 2 (S, Q) and H 2 (S ′ , Q) are not Hodge isometric. Note that this approach only produces finitely many non-isogenous K3 surfaces with isomorphic Hodge structures. The aim of this appendix is to show, via the surjectivity of the period map, that in fact one can produce an infinite family of such pairwise non-isogenous K3 surfaces.
A.2. Hodge isomorphic vs. Hodge algebra isomorphic. All the examples of non-isogenous K3 surfaces that we will consider will consist of K3 surfaces S and S ′ whose respective transcendental lattices T and T ′ become Hodge isometric after some twist, i.e. such that T ≃ T ′ (m) as Hodge lattices for some integer m. The aim of this paragraph is to show that the cohomology algebra of any two such K3 surfaces are Hodge algebra isomorphic ; cf. Lemma A.5 below. This will reduce the proofs of Theorems A.1 and A.2 to showing that the K3 surfaces in the families have Hodge isomorphic transcendental cohomology groups.
First we state a general lemma based on the classical classification of quadratic forms over Q. This lemma will also be used in the proof of Theorem A.2.
Lemma A.4. Let Q be a non-degenerate Q-quadratic form of even rank.
Proof. Obviously Q and Q(m) have the same rank and signature. As the rank of Q is even, their discriminants are also the same (that is, only differ by a square). By the classification theory of quadratic forms over Q, we only need to check that for any prime number ℓ, their ε-invariants are equal at all places ℓ. Assume that Q is equivalent to the diagonal form a 1 , . . . , a r where r is the rank of Q and a i ∈ Q ; its discriminant is thus given by disc(Q) = r i=1 a i and we have where (a, b) ℓ is the Hilbert symbol of a, b for the local field Q ℓ and the last equality uses the identity (m, m) ℓ = (m, −1) ℓ . One concludes by using the fact that (m, −1) ℓ = 1 for all prime numbers ℓ when m ∈ N Q(i) × . Given a K3 surface S of Picard rank 20, the subspace H 2,0 (S) ⊕ H 0,2 (S) ⊂ H 2 (S, Q) ⊗ C is defined over Q, and consequently, up to multiplying by a real scalar, we can fix a generator σ of the 1-dimensional space H 2,0 (S) so that σ +σ is rational, i.e., lies in H 2 (S, Q).
Let {D 1 , . . . , D ρ } be an orthogonal basis of the Néron-Severi space NS(S) Q . Write (D j , D j ) = 2d j with d j ∈ Q. We then have the following Q-basis of H 2 (S, Q) : Let V be the lattice H 2 (S, Z) equipped with the intersection pairing. We aim, via the (surjective) period map, to construct new K3 surfaces by constructing new Hodge structures on the lattice V . To this end, define a Q-linear map φ from the transcendental part H 2 tr (S, Q) to V Q by the following formula (13) φ(σ) = aσ +bσ + ρ j=1 c j D j φ(σ) = bσ +āσ + ρ j=1c j D j with a, b, c j ∈ C. The condition that the image of φ lies in V Q is equivalent to saying that where Q( √ −α) is the imaginary quadratic extension of Q with basis 1 and i √ α.
Consider the following subspaces of V C : By the global Torelli theorem [48], they define a Hodge structure of a K3 surface S ′ if and only if (φ(σ), φ(σ)) = 0 and (φ(σ), φ(σ)) > 0, or equivalently, the following numerical conditions are satisfied : 11 By construction, there is an isomorphism between the Q-Hodge structures of the two K3 surfaces S and S ′ : where φ is given on H 2 tr (S, Q) by φ and on NS(S) Q by any isomorphism to NS(S ′ ) Q . Note that there is a priori no reason for φ to be an isometry. Our goal is actually to provide examples where no Hodge isometry exists. If there exists a Hodge isometry then as H 2,0 is 1-dimensional, there is a λ ∈ C × , such that Since ψ respects the Q-structures, we find that a λ , b λ , The condition that ψ is an isometry implies in particular that (ψ(σ), ψ(σ)) = (σ,σ), that is, To summarize, any solution a, b, c j ∈ Q( √ −α) of (15) gives rise to a K3 surface S ′ with H 2 (S, Q) isomorphic to H 2 (S, Q) as Q-Hodge structures and with H 2 (S, Q) ⊗ m isometric to H 2 (S ′ , Q) with m := (φ(σ), φ(σ))/v, but it would be isogenous to S only if (16) |a| 11 We observe that these conditions imply that a and b cannot be both zero. Indeed, by the Hodge index theorem, one of the di's is positive, say d1, while all the others are negative. The triangular inequality applied to c 2 1 d1 = ρ j=2 c 2 j (−dj) yields |c1| 2 d1 ≤ ρ j=2 |cj | 2 (−dj), i.e., ρ j=1 |cj | 2 dj ≤ 0.
Lemma A.6. Let m 1 := 1 and for any j, m j+1 := j l=1 (2m 4 l + 1). Then for any j, 2m 4 j + 1 / ∈ N Q(i) × and for any j = k, Proof. A standard argument of infinite descent shows that a positive integer n is not the sum of squares of two rational numbers, i.e. not in N Q(i) × , if and only if it is not the sum of squares of two integers. By the theorem of sum of two squares, the latter is equivalent to the condition that any prime divisor of n with odd adic valuation is not congruent to 3 modulo 4.
As the m j 's are all odd, 2m 4 j + 1 ≡ 3 mod 4, hence is not the sum of two squares. On the other hand, by construction, for any j = k, 2m 4 j + 1 and 2m 4 k + 1 are coprime to each other, therefore their product admits a prime divisor p congruent to 3 modulo 4 with odd adic valuation. Hence Remark A.7. The reason for requiring that c 1 and c 2 are not zero in the proof of Theorem A.1 is the following. Let α ∈ Q >0 , and consider a solution to (15) with c 2 = · · · = c ρ = 0, that is, a solution to vab , v ∈ Q >0 and d ∈ Z. We observe that vab + c 2 d = 0 implies that |a||b| = ∓|c| 2 d v is a non-negative rational number, where the sign depends on the sign of d. Recall from footnote 11 that a and b cannot be both zero ; if one of them is zero, we may assume without loss of generality that it is b. Since |a| 2 is a rational number, this implies that |b| = t|a| for some rational number t ∈ Q ≥0 . It is then immediate to check that In other words, the equation vab + c 2 d = 0 (with a and b not both zero) forces the positivity of |a| 2 + |b| 2 + 2|c| 2 d/v, but also implies that it belongs to N Q( √ −α) × . Therefore, the new K3 surface S ′ obtained, via the global Torelli theorem, by imposing c 2 = · · · = c ρ = 0 in (13)  Second proof of Theorem A.1. We now take ρ = 20 and let S be the Fermat quartic surface ; its transcendental lattice T is isomorphic to Z(8) ⊕ Z (8). In order to prove Theorem A.1, it is enough to construct an infinite sequence of positive integers {m j } ∞ j=1 , such that the lattices T (m j ) ≃ Z(8m j ) ⊕ Z(8m j ) are pairwise non-isometric over Q. However, it is easy to see that for m, m ′ ∈ Z >0 , the two Q-quadratic forms Q(8m) ⊕ Q(8m) and Q(8m ′ ) ⊕ Q(8m ′ ) are isometric if and only if mm ′ belongs to {x 2 + y 2 | x, y ∈ Z}, which in turn is equivalent to the condition that any prime factor of mm ′ congruent to 3 modulo 4 has even exponent, by the theorem of the sum of two squares. Therefore a desired sequence is easy to construct, for example, one can take m j to be the j-th prime number congruent to 3 modulo 4. Proof of Theorem A.2. Given any even lattice T of signature (2, 2), (2,4), (2,6) or (2, 8) whose discriminant is a square, by Lemma A.4, the Q-quadratic forms T (m)⊗Q and T ⊗Q are isometric for any integer m ∈ Z >0 which is the sum of two squares. On the other hand, a generic choice of an isotropic element σ ∈ T ⊗ C gives rise to an irreducible Q-Hodge structure on T , hence on all its twists T (m), with minimal endomorphism algebra ; i.e. End HS (T Q ) ≃ Q. So for all m ∈ Z >0 which is the sum of two squares, the twists T (m) ⊗ Q are all Hodge isomorphic and isometric. However, since for any such integers m and m ′ , Hom HS (T (m) ⊗ Q, T (m ′ ) ⊗ Q) = Q, we see that T (m) ⊗ Q and T (m ′ ) ⊗ Q are Hodge isometric if and only if mm ′ is a square.
In order to realize the twists T (m) as transcendental lattices of K3 surfaces, we use Nikulin's embedding theorem [44, Theorem 1.14.4] to get a primitive embedding of T (m) into the K3 lattice Λ. For each m ∈ Z >0 , we can therefore construct a Hodge structure on Λ by declaring that T (m) carries the Hodge structure on T and T (m) ⊥ is of type (1, 1). By the surjectivity of the period map, for any m ∈ Z >0 , there exists a K3 surface S m with H 2 tr (S m , Q) Hodge isometric to T (m). Now, thanks to Lemma A.5, it remains to construct an infinite sequence of positive integers {m j } ∞ j=1 which are sums of two squares, such that the product of any two different terms is not a square. This is easily achieved : for example, one can take m j to be the j-th prime number congruent to 1 modulo 4.
A.5.1. The general expectations. The following conjecture is a combination of the Hodge conjecture and of the conservativity conjecture (which itself is a consequence of the Kimura-O'Sullivan finite-dimensionality conjecture or of the Bloch-Beilinson conjectures). The implication ⇒ holds unconditionally and is simply attained by applying the Betti realization functor. Regarding the implication ⇐, the Hodge conjecture predicts that an isomorphism H * (X, Q) ≃ H * (Y, Q) of graded Hodge structures and its inverse are induced by the action of a correspondence. Hence the homological motives of X and Y are isomorphic. By conservativity, such an isomorphism lifts to rational equivalence, i.e. lifts to an isomorphism between the Chow motives of X and Y .
Obviously, if h(X) ≃ h(Y ) as (Frobenius) algebra objects in the category of Chow motives, then by realization H * (X, Q) and H * (Y, Q) are Hodge (Frobenius) algebra isomorphic. We would like to discuss to which extent the converse statement could be true. In general, this is not the case : consider for instance a complex K3 surface S ; then the blow-up S 1 of S at a point lying on a rational curve and the blow-up S 2 of S at a very general point have Hodge isomorphic cohomology Frobenius algebras, but due to the Beauville-Voisin Theorem 3.1 we have 1 = rk CH 1 (S 1 ) ⊗ CH 1 (S 1 ) → CH 2 (S 1 ) = rk CH 1 (S 2 ) ⊗ CH 1 (S 2 ) → CH 2 (S 2 ) = 2, in particular their Chow motives are not isomorphic as algebra objects.
However, in the case of hyper-Kähler varieties, one can expect : Conjecture A.10. Two smooth projective hyper-Kähler varieties X and Y have isomorphic Chow motives as (Frobenius) algebra objects if and only if their rational cohomology rings are Hodge (Frobenius) algebra isomorphic : h(X) ≃ h(Y ) as (Frobenius) algebra objects ⇐⇒ H * (X, Q) and H * (Y, Q) are Hodge (Frobenius) algebra isomorphic.
We note that Corollary 2 establishes this conjecture in the case of K3 surfaces and Frobenius algebra structures. In general, Conjecture A.10 is implied by the combination of the Hodge conjecture and of the "distinguished marking conjecture" for hyper-Kähler varieties [20, Conjecture 2].
Proposition A.11. Let X and Y be hyper-Kähler varieties of same dimension d. Assume : • The Hodge conjecture in codimension d for X × Y ; • X and Y satisfy the "distinguished marking conjecture" [20, Conjecture 2]. Then Conjecture A.10 holds for X and Y .
Proof. By [20, §3], the distinguished marking conjecture for X and Y provides for all nonnegative integers n and m a section to the graded algebra epimorphism CH * (X n × Y m ) → CH * (X n ×Y m ) in such a way that these are compatible with push-forwards and pull-backs along projections. In addition, the images of the sections corresponding to CH * (X 2 ) → CH * (X 2 ) and CH * (Y 2 ) → CH * (Y 2 ) contain the diagonals ∆ X and ∆ Y , respectively. Here, CH * (−) denotes the Chow ring modulo numerical equivalence. In fact, since numerical and homological equivalence agree for abelian varieties, the same holds for X and Y (via their markings). As before, the Hodge conjecture predicts that a Hodge isomorphism H * (X, Q) ≃ H * (Y, Q) and its inverse are induced by the action of an algebraic correspondence. We fix the isomorphism φ : h(X) ∼ −→ h(Y ) to be the correspondence that is the image of the Hodge class under the section to CH * (X × Y ) → CH * (X × Y ) inducing the Hodge isomorphism of (Frobenius) algebras H * (X, Q) ∼ −→ H * (Y, Q). Since the (Frobenius) algebra structure on the motives of varieties is simply described in terms of the rational equivalence class of the diagonal and of the small diagonal, the isomorphism φ provides, thanks to the compatibilities of the sections on the product of various powers of X and Y , a morphism compatible with the (Frobenius) algebra structures.
Although we do not know how to establish Conjecture A.10 in general for K3 surfaces and algebra structures, we can still say something for K3 surfaces with a Shioda-Inose structure.
Recall that a Shioda-Inose structure on a K3 surface S consists of a Nikulin involution (that is, a symplectic involution) with rational quotient map π : S Y such that Y is a Kummer surface and π * induces a Hodge isometry T S (2) ≃ T Y , where T S and T Y denote the transcendental lattices of S and Y . If S admits a Shioda-Inose structure, let f : A → Y be the quotient morphism from the complex abelian surface whose Kummer surface is Y . By [38, §6], there is a Hodge isometry of transcendental lattices T S ≃ T A , and f * π * induces an isomorphism Proof. Let S and S ′ be two K3 surfaces with a Shioda-Inose structure. In a similar vein to Proposition A.11, the proposition is a combination of the validity of the Hodge conjecture for S × S ′ , together with the fact [20, Proposition 5.12] that S and S ′ satisfy the distinguished marking conjecture of [20,Conjecture 2]. The fact that the Hodge conjecture holds for S × S ′ reduces, via the correspondence-induced isomorphism H 2 tr (S, Q) ∼ −→ H 2 tr (A, Q) described above, to the fact that the Hodge conjecture holds for the product of any two abelian surfaces. The latter is proven in [50].
A.5.2. Non-isogenous K3 surfaces with Chow motives isomorphic as algebra objects. By combining Theorem A.1 with the fact [38, Corollary 6.4] that K3 surfaces of maximal Picard rank admit a Shioda-Inose structure, we can establish : Theorem A.13. There exists an infinite family of K3 surfaces such that • they are pairwise non-isogenous ; • their Chow motives are pairwise non-isomorphic as Frobenius algebra objects ; • their Chow motives are all isomorphic as algebra objects.
Proof. Let {S i } i∈Z >0 be a family of pairwise non-isogenous K3 surfaces of maximal Picard rank such that H * (S j , Q) and H * (S k , Q) are Hodge algebra isomorphic for all j, k ∈ Z. Such a family of K3 surfaces exist thanks to Theorem A.1. By Corollary 2 the Chow motives of these K3 surfaces are pairwise non-isomorphic as Frobenius algebra objects. The fact that the Chow motives of any two surfaces in the family are isomorphic as algebra objects is Proposition A.12.
Finally, the following proposition gives evidence that the notion of "isogeny" for K3 surfaces is strictly more restrictive than the notion of "isomorphic Chow rings" (see Remark 3.2) : Proposition A.14. Assume that Conjecture A.9 holds for K3 surfaces. Then there exists an infinite family {S i } i∈Z >0 of pairwise non-isogenous K3 surfaces with the property that, for all j, k ∈ Z >0 , there exists an isomorphism h(S j ) ∼ −→ h(S k ) of Chow motives inducing a ring isomorphism CH * (S j ) ∼ −→ CH * (S k ) such that the distinguished class o S j is mapped to the distinguished class o S k .
Proof. We consider the infinite family constructed in Theorem A.2. For any j = k, S j and S k are not isogenous. On the other hand, Conjecture A.9 implies that the Chow motives of S j and S k are isomorphic ; in particular, by the same weight argument as in §1.2.3, there exists an isomorphism between their transcendental motives : Γ tr : h 2 tr (S j ) ≃ h 2 tr (S k ). As NS(S j ) Q and NS(S k ) Q are isometric by construction, there is an isomorphism between the algebraic part of their weight-2 motives Γ 2 alg : h 2 alg (S j ) → h 2 alg (S k ) which induces the isometry between the Néron-Severi spaces. Combining them together, Γ := o S j × S k + Γ 2 alg + Γ tr + S j × o S k yields an isomorphism between their Chow motives : , with the extra property that it induces an isometry between the Q-quadratic spaces CH 1 (S j ) and CH 1 (S k ). Now as in Remark 3.2, according to the Beauville-Voisin theorem [4], the image of the intersection product CH 1 (S j )⊗CH 1 (S j ) → CH 2 (S j ) is 1-dimensional and similarly for S k . This implies that Γ induces an isomorphism of Chow rings with Γ * o S j = o S k .
Remark A. 15. In Proposition A.14, if one assumes Conjecture A.10 for K3 surfaces instead of Conjecture A.9, then the K3 surfaces of the family have isomorphic Chow motives as algebra objects.