Hodge similarities, algebraic classes, and Kuga–Satake varieties

We introduce in this paper the notion of Hodge similarities of transcendental lattices of hyperkähler manifolds and investigate the Hodge conjecture for these Hodge morphisms. Studying K3 surfaces with a symplectic automorphism, we prove the Hodge conjecture for the square of the general member of the first four-dimensional families of K3 surfaces with totally real multiplication of degree two. We then show the functoriality of the Kuga–Satake construction with respect to Hodge similarities. This implies that, if the Kuga–Satake Hodge conjecture holds for two hyperkähler manifolds, then every Hodge similarity between their transcendental lattices is algebraic after composing it with the Lefschetz isomorphism. In particular, we deduce that Hodge similarities of transcendental lattices of hyperkähler manifolds of generalized Kummer deformation type are algebraic.

Introduction 0.1.Hyperkähler manifolds and the Hodge conjecture.Let X be a hyperkähler manifold, and let T (X) ⊆ H 2 (X, Q) be its transcendental lattice, which is the orthogonal complement of the Néron-Severi group of X in H 2 (X, Q) with respect to the Beauville-Bogomolov quadratic form.The relevance of this notion in the context of the Hodge conjecture can be evinced from the following observation: let X and Y be hyperkähler manifolds.By Lefschetz (1, 1) theorem, a Hodge morphism H 2 (X, Q) → H 2 (Y, Q) is algebraic if and only if the induced Hodge morphism T (X) → T (Y ) is algebraic.Recall that a Hodge morphism H 2 (X, Q) → H 2 (Y, Q) is said to be algebraic if the corresponding Hodge class in H 2n,2n (X × Y, Q) is algebraic, where 2n is the dimension of X.
In general, it is not known whether Hodge morphisms of transcendental lattices are algebraic or not.However, there have been promising results for the class of Hodge isometries.Recall that a Hodge isomorphism T (X) → T (Y ) is called a Hodge isometry if it is an isometry with respect to the Beauville-Bogomolov quadratic forms on X and Y .A result by Buskin [3] reproved by Huybrechts [12] shows that Hodge isometries of transcendental lattices of projective K3 surfaces are algebraic.The same has been proven by Markman [16] for Hodge isometries of transcendental lattices of hyperkähler manifolds of K3 [n] -type.
In this paper, we introduce a natural generalization of Hodge isometries which we call Hodge similarities: a Hodge isomorphism is a Hodge similarity if it multiplies the quadratic form by a non-zero scalar called multiplier, see Definition 1.2.Note that Hodge isometries are Hodge similarities with multiplier one.There are two contexts where Hodge similarities naturally appear.The main instance is given by hyperkähler manifolds X whose endomorphism field E := End Hdg (T (X)) is a totally real field of degree two: indeed, every totally real field of degree two is isomorphic to Q( √ d) for some positive integer d.One then sees that √ d : T (X) → T (X) is a Hodge similarity.This follows immediately from the fact that, as E is totally real, the Rosati involution is the identity.Note that in this case E is generated by Hodge similarities.A second source of examples of Hodge similarities is the following: given a hyperkähler manifold X, there might exist another hyperkähler manifold Y with transcendental lattice Hodge isometric to T (X)(λ), for some λ ∈ Q >0 , where (λ) indicates that the quadratic form is multiplied by λ.The identity of T (X) then defines a natural Hodge morphism T (Y ) → T (X) which is a Hodge similarity.At the time of writing this paper, there are very few examples of Hodge similarities that are not isometries which can be proven to be algebraic.For example, in the case of K3 surfaces with totally real endomorphism field E = Q( √ d), the algebraicity of √ d has been proven only for some one-dimensional families of such K3 surfaces.This is a result by Schlickewei [19] which has then been extended in [20].Note that the proof in the references involves the study of the Hodge conjecture for Kuga-Satake variety of these K3 surfaces, and does not use the fact that E is in these cases generated by Hodge similarities.0.2.Hodge similarities of K3 surfaces and symplectic automorphisms.Recall that the Hodge conjecture for the product of two K3 surfaces X and Y to the algebraicity of the elements of Hom Hdg (T (X), T (Y )).This follows from the Künneth decomposition and the fact that the quadratic form q X identifies (T (X) ⊗ T (Y )) 2,2 ∩ (T (X) ⊗ T (Y )) with Hom Hdg (T (X), T (Y )).As mentioned above, Hodge isometries between the transcendental lattices of two K3 surfaces are known to be algebraic.In particular, the Hodge conjecture holds for X × Y whenever Hom Hdg (T (X), T (Y )) is generated by Hodge isometries.This is the case when T (X) and T (Y ) are Hodge isometric and Hom Hdg (T (X), T (Y )) is Q or a CM field.
The main result of Section 2 is the proof of the algebraicity of some Hodge similarities for some families of K3 surfaces with totally real multiplication of degree two: Theorem 0.1 (Theorem 2.1, 2.9, and 2.15).Let X be a K3 surface Hodge isometric to a K3 surface with a symplectic automorphism of order p with p = 2, 3. Assume furthermore that Q( √ p) is contained in the endomorphism field of X.Then, √ p : T (X) → T (X) is algebraic.
In particular, the Hodge conjecture for X × X holds if End Hdg (T (X)) ≃ Q( √ p).
The condition "X is Hodge isometric to a K3 surface with a symplectic automorphism of order p" is equivalent to . This is deduced in Proposition 2.5 and Proposition 2.11 from the classical result by Nikulin [17], van Geemen and Sarti [10], and Garbagnati and Sarti [7].Using these conditions on the transcendental lattice, we show that the families of K3 surfaces satisfying the hypotheses of Theorem 0.1 are at most four-dimensional for p = 2 and two-dimensional for p = 3.We then produce examples of such maximal-dimensional families in Proposition 2.6 and Proposition 2.12.In particular, Theorem 0.1 provides the first four-dimensional families of K3 surfaces with totally real multiplication of degree two for which the Hodge conjecture can be proven for the square of its general member and the first two-dimensional family of K3 surfaces with totally real multiplication of degree two for which the Hodge conjecture can be proven for the square of all its members.0.3.Kuga-Satake varieties and Hodge similarities.In Section 3, we prove that the functoriality of the Kuga-Satake construction with respect to Hodge isometries extends to Hodge similarities in the following sense: Proposition 0.2 (Proposition 3.1).Let ψ : (V, q) → (V ′ , q ′ ) be a Hodge similarity of polarized Hodge structures of K3-type.Then, there exists an isogeny of abelian varieties ψ KS : KS(V ) → KS(V ′ ) making the following diagram commute where the vertical arrows are the Kuga-Satake correspondence for V and V ′ .
In Section 3.1, we exploit the observation that a similarity of quadratic spaces induces an isomorphism of even Clifford algebras to extend the result by Kreutz, Shen, and Vial [13] which shows that de Rham-Betti isometries between the second de Rham-Betti cohomology of two hyperkähler manifolds defined over Q are motivated in the sense of André.We note in Proposition 3.8 that the same proof as in the reference can be used to show that de Rham-Betti similarities are motivated.
In Section 4, we use the functoriality property of the Kuga-Satake construction proven in Proposition 0.2 to deduce the following: Theorem 0.3 (Theorem 4.5).Let X ′ and X be two hyperkähler manifolds for which the Kuga-Satake Hodge conjecture holds.Then, for every Hodge similarity ψ : T (X ′ ) → T (X), the composition is algebraic, where 2n := dim X and h X is the cohomology class of an ample divisor on X.
By a result of Voisin [23] based on previous results by Markman [15] and O'Grady [18], the Kuga-Satake Hodge conjecture holds for hyperkähler manifolds of generalized Kummer type.This is the main source of examples of manifolds which satisfy the hypotheses of Theorem 0.3.As the Lefschetz standard conjecture in degree two for these manifolds is proved by Foster [6], Theorem 0.3 shows that Hodge similarities between the transcendental lattices of two hyperkähler manifolds of generalized Kummer type are algebraic.Using the fact that the endomorphism field of these varieties is always generated by Hodge similarities, we then conclude the following: Theorem 0.4 (Theorem 5.1).Let X and X ′ be hyperkähler manifolds of generalized Kummer type such that T (X) and T (X ′ ) are Hodge similar.Then, every Hodge morphism between T (X ′ ) and T (X) is algebraic.
Note that, opposed to the case of K3 surfaces and hyperkähler manifolds of K3 [n] -type, already the algebraicity of Hodge isometries was not known in the case of hyperkähler manifolds of generalized Kummer type.Furthermore, note that Theorem 0.4 also applies for hyperkähler manifolds of generalized Kummer type of different dimension.
In the case of K3 surfaces, the Lefschetz standard conjecture is trivially true.Hence, if the Kuga-Satake Hodge conjecture holds for two given K3 surfaces, Theorem 0.3 shows that every Hodge similarity between their transcendental lattices is algebraic.In particular, this provides a more direct proof of the Hodge conjecture for the square of the K3 surfaces in the one-dimensional families of K3 surfaces with totally real field of degree two studied in [19,20] that we mentioned above.
For hyperkähler manifolds of K3 [n] -type, the Kuga-Satake Hodge conjecture is known only for certain families: the paper [5] proves this conjecture for countably many four-dimensional families of K3 [3] -type hyperkähler manifolds.Recall that, for hyperkähler manifolds of K3 [n] -type, the Lefschetz standard conjecture has been proven by Charles and Markman [4].Therefore, we deduce the algebraicity of Hodge similarities for the hyperkähler manifolds of K3 [3] -type appearing in [5].
As a final remark, note that the manifolds X and X ′ as in Theorem 0.3 are neither assumed to be of the same deformation type nor of the same dimension.

Main definitions
In this paper, all varieties are assumed to be projective.Unless otherwise stated, the definition field of the varieties we consider is C.
A hyperkähler manifold is a simply connected, projective, compact, Kähler manifold X such that H 0 (X, Ω 2 X ) is generated by a nowhere degenerate symplectic form.Denote by q X the Beauville-Bogomolov quadratic form, which is a non-degenerate quadratic form on H 2 (X, Q).Recall that q X induces the following direct sum decomposition where NS(X) is the Néron-Severi group of X and T (X) is the transcendental lattice of X.When talking about the transcendental lattice of a hyperkähler manifold X we will always refer to the rational quadratic subspace T (X) of H 2 (X, Q).The pair (T (X), −q X ) gives an example of polarized Hodge structures of K3-type: Definition 1.1.A rational Hodge structure V of weight two is called of K3-type if dim C V 2,0 = 1, and V p,q = 0 for |p − q| > 2.
Moreover, we say that a pair (V, q) is a polarized Hodge structure of K3-type if q : V ⊗V → Q(−2) is a morphism of Hodge structures whose real extension is negative definite on (V 2,0 ⊕V 0,2 )∩V R and has signature (dim V − 2, 2).
Let E := End Hdg (T (X)) be the endomorphism algebra of the Hodge structure T (X).As T (X) is an irreducible Hodge structure, E is a field.As explained in [11,Thm. 3.3.7],E is either totally real or CM.Recall that a field extension E of Q is totally real if every embedding E ֒→ C has image contained in R, and it is CM if E = F (ρ), where F is a totally real field and ρ satisfies the following: These two cases can be distinguished by the action of the Rosati involution, which is the involution on E which sends an element e ∈ E to the element e ′ ∈ E such that q X (ev, w) = q X (v, e ′ w), ∀v, w ∈ T (X).
The Rosati involution is the identity if E is totally real, and it acts as complex conjugation if E is CM.
As mentioned in the introduction, we focus in this paper on the notion of Hodge similarities: Definition 1.2.Let (V, q V ) and (V ′ , q V ′ ) be polarized Hodge structures of K3-type, and let ψ : V → V ′ be a Hodge isomorphism.We say that ψ is a Hodge similarity if there exists a non-zero We call λ ψ the multiplier of ψ.A Hodge isometry is a Hodge similarity ψ of multiplier λ ψ = 1.
We say that two hyperkähler manifolds are Hodge similar (resp., Hodge isometric) if there exists a Hodge similarity (resp., a Hodge isometry) between their transcendental lattices.Note that the multiplier of a Hodge similarity is always a positive number.

Symplectic automorphisms and algebraic Hodge similarities
Let X be a K3 surface, and denote by q the polarization on T (X) given by the negative of the intersection form.Identifying T (X) with its dual via q, we see that This shows that proving the Hodge conjecture for X 2 is equivalent to showing that every element of End Hdg (T (X)) is algebraic.In this section, considering K3 surfaces with a symplectic automorphism, we produce examples of K3 surfaces X with Q( √ p) ⊆ End Hdg (T (X)) for which the Hodge similarity √ p can be shown to be algebraic.
The starting observation is the following: given a K3 surface X with a symplectic automorphism of order p, there exists a K3 surface Y and an algebraic Hodge similarity ϕ : T (Y ) → T (X) of multiplier p.To show this, recall that, by [11,Prop. 15.3.11], the prime p is at most 7, the fixed locus of σ p is a finite union of points, and the minimal resolution of X/σ p is a K3 surface Y .Moreover, Y can also be obtained as follows: after a finite sequence of blowups of X at the fixed locus of σ p , we get a variety X with a free action σ p and Y ≃ X/ σ p .I.e., there is a commutative diagram As π : X → Y is a finite map of degree p and β : X → X just contracts the exceptional divisors, we see that is a Hodge similarity of multiplier p.Note that ϕ is algebraic.From this construction, we deduce the following: Theorem 2.1.Let X be a K3 surface Hodge isometric to a K3 surface with a symplectic automorphism of prime order p. Assume furthermore that Q( √ p) ⊆ End Hdg (T (X)).Then, the Hodge similarity √ p is algebraic.
Proof.As Hodge isometries of K3 surfaces are algebraic by [3] and [12], we may assume that X admits a symplectic automorphism of order p.Let ψ be the Hodge similarity of multiplier p on T (X), which exists since Q( √ p) ⊆ End Hdg (T (X)) by assumption.As remarked above, denoting by Y the minimal resolution of the quotient X/σ p , the map ϕ is then a Hodge isometry.In particular, ϕ −1 • ψ is algebraic by [3] and [12].As ϕ is algebraic, we conclude that ψ = ϕ • (ϕ −1 • ψ) is algebraic.This concludes the proof.
Remark 2.2.The two conditions "X is isometric to a K3 surface with a symplectic automorphisms of order p" and "the endomorphisms field of X contains Q( √ p)" are not related.In fact, the general K3 surface with a symplectic automorphism of order p has endomorphism field equal to Q.Moreover, note that the requirement "the endomorphisms field of X contains Q( √ p)" is equivalent to the condition "X admits a Hodge similarity ψ of multiplier d which is fixed by the Rosati involution".Indeed, if ψ such a Hodge similarity, then Q(ψ) is a totally real subfield of the endomorphism field of X.Using the fact that totally real fields have no non-trivial isometry, we see that ψ 2 /d is the identity, i.e., that In the remainder of this section, we construct families of K3 surfaces satisfying the hypotheses of Theorem 2.1.To do this, we use the following result is adapted from [9, Sec.3], we give here a detailed proof for later use.Proposition 2.3.Let d ∈ Z be a positive integer which is not a square, and let (Λ, q) be a rational quadratic space of signature (2, Λ − 2) with dim Λ > 4. Let ψ be a similarity of Λ of multiplier d which is fixed by the Rosati involution.Then, Λ is even-dimensional, and the locus of Hodge structures of K3-type on Λ for which ψ defines a Hodge similarity is either empty or of dimension (dim Λ)/2 − 2.
Proof.The first statement is immediate from the fact that odd-dimensional quadratic spaces do not admit any similarity of multiplier d if d is not a square.
Let us assume that Λ is even-dimensional.As in Remark 2.2, we see that, for every Hodge structure on Λ for which ψ is a Hodge morphism, Q(ψ) ≃ Q( √ d) is a totally real subfield of the endomorphism field of Λ.
Note that Λ can be viewed as a From the fact that ψ is fixed by the Rosati involution, we deduce that this decomposition is orthogonal with respect to the quadratic form q on Λ R .
Recall that giving a Hodge structure of K3-type on Λ is equivalent to giving an element ω in the period domain Note that ψ defines a morphism of Hodge structures if and only if ω is an eigenvector.Therefore, d is orthogonal with respect to q and (Λ 2,0 ⊕ Λ 0,2 ) ∩ Λ R has to be positive definite, there exists a Hodge structure for which ψ is a Hodge morphism if and only if Λ − √ d is negative definite and Λ √ d has signature (2, (dim Λ)/2 − 2) or vice versa.Let us assume that ψ satisfy this hypothesis.Then, up to changing the sign of ψ, we may assume that Λ √ d has signature (2, (dim Λ)/2 − 2).We conclude that ψ defines a Hodge automorphism if and only if the Hodge structure corresponds to an element in Therefore, the locus of Hodge structures on Λ for which ψ defines a Hodge morphism has dimension equal to dim We use Proposition 2.3 to show that the families of K3 surfaces which satisfy the hypotheses of Theorem 2.1 are at most four-dimensional for p = 2 and two-dimensional for p = 3.Moreover, we produce examples of such families with these maximal dimensions.As we will see, no K3 surface satisfies the hypotheses of Theorem 2.1 for higher values of p.
Let us start from the case p = 2. Following [10], we call a symplectic involution on a K3 surface a Nikulin involution.By [10, Prop.2.2, 2.3], a K3 surface X admits a Nikulin involution if and only if the lattice E 8 (−2) is primitively embedded in the Néron-Severi group of X.Note that, up to an automorphism of the K3-lattice, there exists a unique primitive embedding of E 8 (−2) in the K3-lattice.Therefore, we deduce from [10, Sec.1.3] that (E 8 (−2)) ⊥ ≃ U 3 ⊕ E 8 (−2).From this fact, we get the following criterion in terms of the transcendental lattice of X: Proposition 2.5.A K3 surface X is Hodge isometric to a K3 surface admitting a Nikulin involution if and only if Proof.Let us first prove the "only if" part.Let X be a K3 surface such that T (X) is Hodge isometric to T (X ′ ) for some K3 surface X ′ admitting a Nikulin involution.By [10, Prop.2.2, 2.3], the lattice For the "if" part, let us assume that there is an embedding of quadratic spaces Denote by H 2 (X, Z) tr the transcendental part of the second integral cohomology of X. Clearing the denominators, we find a positive integer λ ∈ Z such that the above embedding restricts to an embedding of lattices Fix a primitive embedding ι : Let T ′ be the saturation of the lattice (ι • j)(H 2 (X, Z) tr ) ⊆ H 2 (X, Z).For any K3 surface X ′ such that H 2 (X ′ , Z) tr ≃ T ′ we get an embedding This embedding is primitive, since E 8 (−2) is obtained as an orthogonal complement.Therefore, X ′ admits a Nikulin involution by [10, Prop.2.2, 2.3].Note that T (X) and T ′ Q are isometric quadratic spaces.Hence, by the surjectivity of the period map we can find a K3 surface X ′ with H 2 (X ′ , Z) tr ≃ T ′ such that T (X ′ ) is Hodge isometric to T (X).This concludes the proof since the K3 surface X ′ is Hodge isometric to X and admits a Nikulin involution as required.
In particular, we deduce that the transcendental lattice of a K3 surfaces which is Hodge isometric to a K3 surface with a Nikulin involution is at most 13-dimensional.Proposition 2.3 then shows that the families of K3 surfaces satisfying the hypotheses of Theorem 2.1 in the case p = 2 are at most four-dimensional.To prove the existence of such a four-dimensional family of K3 surfaces we consider a particular quadratic subspace of U 3 Q ⊕ E 8 (−2) Q , and we show that it admits a similarity of multiplier 2.
Proposition 2.6.The locus of Hodge structures of K3-type on Λ := U 2 Q ⊕ E 8 (−2) Q which admit a Hodge similarity of multiplier 2 which is fixed by the Rosati involution is non-empty and has a four-dimensional component.
Proof.As dim Λ = 12, Proposition 2.3 shows that the locus of Hodge structures of K3-type on Λ which admit a Hodge similarity of multiplier 2 which is fixed by the Rosati involution has a four-dimensional component if non-empty.By Remark 2.4, we just need to produce a similarity ψ of Λ of multiplier 2 fixed by the Rosati involution such that Λ √ 2 has signature (2,4).As the quadratic space As in [9,Exmp. 3.4], we restrict to finding a similarity ψ which preserves the decomposition of Λ as above.I.e., we look for matrices M i ∈ GL 2 (Q) which satisfy the following: Then, ψ := M 1 ⊕ . . .⊕ M 6 will be fixed by the Rosati involution by the first condition and will be a similarity of multiplier 2. A direct computation shows that the following matrices satisfy all the above conditions and that the signature of Λ √ 2 is (2, 4).Thus, ψ satisfies the required properties.
Example 2.7.By [10, Sec. 1 4], the family of elliptic K3 surfaces with a section and a twotorsion section provides an example of a ten-dimensional family of K3 surfaces with a Nikulin involution and general transcendental lattice Λ = U 2 Q ⊕ −2 8 .By Proposition 2.6, there exists a four-dimensional family of elliptic K3 surfaces with a two-torsion section with endomorphism field containing Q( √ 2).
Remark 2.8.One can produce other examples of quadratic subspace of Q which admit a similarity of multiplier 2 which is fixed by the Rosati involution.For example, if d > 1 is a square-free integer such that 2 is a quadratic residue modulo d, the space U 2 Q ⊕ −2 7 ⊕ −2d admits a similarity of multiplier 2 and is not isometric to U 2 Q ⊕ E 8 (−2) Q .This provides other four-dimensional families of K3 surfaces satisfying the hypotheses of Theorem 2.1 for p = 2.
To sum up, our discussion shows that Theorem 2.1 in the case of Nikulin involutions gives the following: Theorem 2.9.For every K3 surface in the four-dimensional families of K3 surfaces with endomorphism field containing Q( √ 2) which are Hodge isometric to a K3 surface with a Nikulin involution, the endomorphism √ 2 is algebraic.In particular, the Hodge conjecture holds for the square of the general such K3 surface.
Remark 2.10.The only case where Theorem 2.9 is not enough to prove the Hodge conjecture for the square of the K3 surfaces X as in the statement is when the endomorphism field E of X is totally real of degree four and T (X) is twelve-dimensional: this follows from the well known fact that, if the endomorphism field E is totally real, the dimension of T (X) as E-vector space is at least three.Recall that if E is a CM field, then the Hodge conjecture for X 2 follows from [3] and [12] using the fact that E is generated by Hodge isometries.Similarly to Proposition 2.3, one sees that the families of K3 surfaces as in Theorem 2.9 with totally real endomorphism field of degree four are one-dimensional.
Let us come to the case p = 3.Let X be a K3 surface with a symplectic automorphism of order 3, and let Y be the minimal resolution of the quotient.As above, we have an algebraic similarity T (Y ) → T (X) of multiplier 3 and T (Y ) is Hodge isometric to T (X)( 13 ).By [7, Thm.4.1], a K3 surface X admits a symplectic automorphism of order 3 if and only if K 12 (−2) is primitively embedded in NS(X), where K 12 (−2) denotes the Coxeter-Todd lattice with the bilinear form multiplied by −2.With a similar proof as in Proposition 2.5, we can reformulate this in terms of the transcendental lattice as follows: Proposition 2.11.A K3 surface X is Hodge isometric to a K3 surface admitting a symplectic automorphism of order 3 if and only if ) are at most two-dimensional.As in the case of Nikulin involutions, we consider a particular quadratic subspace of Q , and we show that it admits a similarity of multiplier 3: Proposition 2.12.The locus of Hodge structures of K3-type on Q which admit a Hodge similarity of multiplier 3 which is fixed by the Rosati involution is non-empty and has a two-dimensional component.
Proof.As in the proof of Proposition 2.6, we will construct an explicit similarity ψ of Γ of multiplier 3 fixed by the Rosati involution such that Γ √ 3 has signature (2, 2).Then, by Proposition 2.3 and Remark 2.4, the locus of Hodge structures on Γ for which ψ is a Hodge morphism has a two-dimensional component.
Diagonalizing the quadratic space (A 2 ) Q , we see that there is an isometry As one checks, setting defines a similarity of multiplier 3 of Γ satisfying all the requirements.
Example 2.13.By [7, Prop.4.2], the family of elliptic K3 surfaces with a section and a threetorsion section provides an example of a six-dimensional family of K3 surfaces with a symplectic automorphism of order 3 and general transcendental lattice isometric to Γ = U 2 Q ⊕ (A 2 ) 2 Q .By Proposition 2.12, there is a two-dimensional subfamily of K3 surfaces with endomorphism field containing Q( √ 3).
Remark 2.14.As in the case of Nikulin involutions, one can construct other eight-dimensional quadratic subspaces of Q admitting a similarity of multiplier 3 as Γ of Propostion 2.12.
Our discussion shows that Theorem 2.1 in the case p = 3 gives the following: Theorem 2.15.For every K3 surface in the two-dimensional families of K3 surfaces with endomorphism field containing Q( √ 3) which are Hodge isometric to a K3 surface with a symplectic automorphism of order 3, the endomorphism √ 3 is algebraic.In particular, the Hodge conjecture holds for the square of every such K3 surface.Remark 2.16.Note that in this case, Theorem 2.15 proves the Hodge conjecture for the square of every K3 surfaces of these families.The reason for this lies in the fact that the transcendental lattice of these K3 surfaces is at most eight-dimensional.Therefore, by a similar argument as in Remark 2.10, we see that the endomorphism field of such K3 surface is either Q( √ 3) or a CM field.In the latter case, the Hodge conjecture for the square of the K3 surface follows from the fact that CM fields are generated by Hodge isometries.
In the case of symplectic automorphisms of order bigger than 3, the same procedure does not produce any K3 surface.In fact, the endomorphism of a K3 surface with a symplectic automorphism of order 5 or 7 is always Q or a CM field.This can be deduced from [7, Prop.1.1]: indeed, the transcendental lattice of a K3 surface admitting a symplectic automorphism of order 5 is of dimension at most five, and for K3 surfaces with a symplectic automorphism of order 7 its dimension is at most three.As in Remark 2.10, one sees that, in both cases, the endomorphism field of these K3 surfaces cannot be a totally real field different from Q.

Kuga-Satake varieties and Hodge similarities
By a construction due to Kuga and Satake [14], given a polarized Hodge structure of K3-type (V, q), there exists an abelian variety KS(V ), called the Kuga-Satake variety of (V, q), together with an embedding of Hodge structures κ : V ֒→ H 1 (KS(V ), Q) ⊗2 .We refer the reader for this construction to [11,Ch. 4], [8], and [21].In this section, we prove the functoriality of the Kuga-Satake construction with respect to Hodge similarities: Proposition 3.1.Let ψ : (V, q) → (V ′ , q ′ ) be a Hodge similarity of polarized Hodge structures of K3-type.Then, there exists an isogeny of abelian varieties ψ KS : KS(V ) → KS(V ′ ) making the following diagram commute where the vertical arrows are the Kuga-Satake correspondence for (V, q) and (V ′ , q ′ ).
In the remainder of this section we prove Proposition 3.1.
Let (V, q) and (V ′ , q ′ ) be polarized Hodge structures of K3-type, and let ψ : (V, q) → (V ′ , q ′ ) be a Hodge similarity of multiplier λ ψ .The next lemma shows that ψ induces an isomorphism between the even Clifford algebras Cl + (V ) and Cl + (V ′ ).Recall that Cl + (V ) is defined as the even degree part of * V /I V , where I V is the two-sided ideal generated by elements of the Lemma 3.2.The isomorphism of graded rings Proof.From the definition, it is immediate to see that the map ψ ⊗ is an isomorphism of graded rings.Given v ∈ V , we have the following where in the last step we used that ψ is a similarity of multiplier λ ψ .This equality shows that ψ ⊗ (v ⊗ v − q(v)) belongs to the ideal of ev V ′ generated by w ⊗ w − q ′ (w) for w ∈ V ′ .Hence, the isomorphism ψ ⊗ descends to an isomorphism In the construction of the Kuga-Satake variety associated to a polarized Hodge structure of K3-type, the complex Hodge structure on Cl + (V ) R is given by left multiplication by J := e 1 •e 2 , with {e 1 , e 2 } an orthogonal basis of V R ∩ (V 2,0 ⊕ V 0,2 ) satisfying q(e 1 ) = q(e 2 ) = −1.As one checks, this complex structure does not depend on the choice of the basis.
Proof.Let {e 1 , e 2 } be an orthogonal basis of V R ∩ (V 2,0 ⊕ V 0,2 ) with q(e i ) = −1, and define As ψ is a Hodge similarity of multiplier λ ψ , one sees that {e ′ 1 , e ′ 2 } is an orthogonal basis of . Hence, the equality proves that ψ Cl,R is a morphism of complex vector spaces.
The Kuga-Satake variety of (V, q) is defined as (the isogeny class) of the complex torus KS(V ) := Cl + (V ) R /Cl + (V ), where Cl + (V ) R is endowed with the complex structure we recalled above.Lemma 3.3 then shows that ψ Cl : Cl + (V ) → Cl + (V ′ ) induces an isogeny of complex tori Recall that Kuga-Satake varieties of polarized Hodge structures of K3-type are abelian varieties: let (f 1 , f 2 ) ∈ V × V be a pair of orthogonal elements of V with positive square, and consider the pairing where tr(x) denotes the trace of the endomorphism of Cl + (V ) given by left multiplication by x ∈ Cl + (V ) and v * denotes the image of v under the involution of Cl + (V ) induced by the involution Then, Q defines up to a sign a polarization for the weight-one Hodge structure Cl + (V ).Note that the pair (ψf 1 /λ ψ , ψf 2 ) ∈ V ′ ×V ′ satisfies the same hypotheses.Hence, it defines a polarization ) is an isogeny of abelian varieties.
Proof.To prove the lemma, we need to show that Note that, by definition of ψ Cl , the following holds We then need to prove that, for every x ∈ Cl + (V ), the left multiplication by x on Cl + (V ) has the same trace as the left multiplication by ψ Cl x on Cl + (V ′ ).This can be checked as follows: let {b i } i be a basis of Cl + (V ) with dual basis {b i } i .By definition, we have that where ψ ∨ Cl is the dual action of ψ Cl .The trace of the left multiplication by ψ Cl x is then . This concludes the proof.
The last ingredient for the proof of Proposition 3.1 is the compatibility of the isomorphism ψ Cl with the embedding ϕ V : V ֒→ End(Cl + (V )) given by the Kuga-Satake construction.Recall that ϕ V is given as follows: let v 0 ∈ V be an element with q(v 0 ) = 0, then ϕ Lemma 3.5.With the previous notation, the following diagram commutes , where End(ψ Cl ) is the map Proof.By definition, the composition of ϕ V with End(ψ Cl ) is the map This shows that the above square is commutative.Indeed, ϕ V ′ is the map Proof of Propostion 3.1.Lemma 3.4 shows that there exists an isogeny of abelian varieties Recall that the Kuga-Satake embedding is the composition where the isomorphism is given by the polarization Q on Cl + (V ) which induces an isomorphism between Cl + (V ) and Cl + (V ) * .Note that the commutativity of the square in the theorem follows from the commutativity of the square of Lemma 4.4 by the compatibility of ψ Cl with the polarizations Q and Q ′ .
3.1.De Rham-Betti similarities.In [13, Thm.9.5], the authors prove that de Rham-Betti isometries between the second de Rham-Betti cohomology groups of two hyperkähler manifolds defined over Q are motivated in the sense of André using the fact that the Kuga-Satake correspondence is motivated.We note here that the observation that similarities between two quadratic spaces induce isomorphisms between the respective even Clifford algebras shows that the result in [13] can be extended to de Rham-Betti similarities.
Let us briefly recall the notions of de Rham-Betti morphism and of motivated cycles as presented in [13].To simplify the exposition, we avoid going into too much detail of the Tannakian formalism.Definition 3.6.Let X be a smooth projective variety over Q, and let Q(k) := (2πi) k Q ⊆ C. The de Rham-Betti cohomology groups of X are the triples , where f dR is Q-linear and f B is Q-linear, and their C-linear extensions are compatible with c X and c X ′ .Definition 3.7.Let X be a smooth projective variety over Q.A motivated cycle on X is an element of H 2r B (X C , Q(r)) of the form p X, * (α∪ * L β), where α and β are algebraic cycles on X × Q Y for some smooth projective variety Y over Q, * L is the (inverse of the) Lefschetz isomorphism, and p X, * is the first projection.Note that, given a motivated cycle α B in H 2k B (X, Q(k)), there exists a de Rham cohomology class α dR in H 2k dR (X/Q) such that c X (α dR ) = α B .In particular, we see that a motivated cycle on X × Q Y induces a de Rham-Betti morphism between the de Rham-Betti cohomologies of X and Y .One says that a de Rham-Betti morphism between cohomology groups of two smooth projective varieties X and Y over Q is motivated if it is induced by a motivated cycle on X × Q Y .For a complete introduction on this subject we refer the reader to [2].
Following [13], a variety over Q is called a hyperkähler manifold over Q if its base-change to C is a hyperkähler manifold with second Betti number at least three.This last assumption is needed to ensure that the Kuga-Satake correspondence is motivated as proved by André [1].The following proposition extends the result of [13, Thm.9.5].
Proposition 3.8.Let X and X ′ be hyperkähler manifolds over Q.Then, any de Rham-Betti similarity H 2 dRB (X, Q) The proof of this theorem is exactly the same as the one in the reference with the only addition that similarities (and not just isometries) induce isomorphisms between the even Clifford algebras.We give here just a sketch of the proof.

Let
. By [13, Lem.6.2, 6.17], to prove the theorem, it suffices to prove the same statement over Q. I.e., that every Q-de Rham-Betti similarity between H 2 dRB (X, Q) and H 2 dRB (X ′ , Q) is a Q-linear combination of motivated cycles on X × Q X ′ .Let T 2 dRB (X, Q) be the orthogonal complement of the subspace of H 2 dRB (X, Q) spanned by divisor classes, and similarly define T 2 dRB (X ′ , Q).As in the reference, one shows that, to prove the result, it suffices to show that every de Rham-Betti similarity between T 2 dRB (X, Q) and Consider the Q-linear category C Q−dRB whose objects are triples (M dR , M B , c M ), where M dR and M B are finite dimensional Q-vector spaces and As in the reference, one then shows that this induces a G Q−dRB -invariant isomorphism of algebras J : End(Cl(V ) ⊗ Q) → End(Cl(V ′ ) ⊗ Q).One then shows that J is Q-motivated.This in turn implies that ψ is Q-motivated using the fact that the Kuga-Satake correspondence is Q-motivated as proven in [13,Prop. 8.5].This concludes the proof.

Hodge similarities and algebraic classes
We now go back to the case of hyperkähler manifolds defined over C and study the consequences of the functoriality of the Kuga-Satake construction relative to Hodge similarities in the case where the Hodge structure (V, q) is geometrical.In other words, we assume that there is a hyperkähler manifold X for which V = T (X) or V = H 2 (X, Q) and q is the Beauville-Bogomolov quadratic form with the sign changed.
Remark 4.1.In Section 3, we studied the Kuga-Satake construction for polarized Hodge structures of K3-type.The same construction also works for the second cohomology group of a hyperkähler manifold X even though it is not polarized by the Beauville-Bogomolov quadratic form.Indeed, using the direct sum decomposition H 2 (X, Q) ≃ T (X) ⊕ NS(X) Q , one sees that the even Clifford algebra of H 2 (X, Q) is a power of the even Clifford algebra of T (X).Thus, the Kuga-Satake variety KS(H 2 (X, Q)) is an abelian variety isogenous to a power of KS(T (X)).
Let KS(X) be the Kuga-Satake variety of H 2 (X, Q).The Kuga-Satake correspondence gives an embedding of Hodge structures: The Hodge conjecture predicts that κ X is algebraic: Conjecture 4.2 (Kuga-Satake Hodge conjecture).Let X be a hyperkähler manifold, then, the Kuga-Satake correspondence κ X is algebraic.Remark 4.3.Note that the Kuga-Satake correspondence depends on the choice of the three elements v 0 , f 1 , f 2 ∈ T (X) as in Section 3. Choosing a different v 0 ∈ T (X) changes the embedding by the automorphism of Cl + (H 2 (X, Q)) which sends w to w•v 0 • v 0 q(v 0 ) , and choosing a different pair f 1 , f 2 ∈ T (X) corresponds to changing the polarization on the complex torus KS(X).However, neither of these two operations affects the algebraicity of κ X .Hence, Conjecture 4.2 does not depend on the choices made in the definition of κ X .
Let 2n := dim X and N := dim KS(X).The transpose of κ ∨ X of κ X is the surjection κ ∨ X : H 4N −2 (KS(X) 2 , Q) ։ H 4n−2 (X, Q).Note that, as κ X and κ ∨ X are transpose of each other, κ X is algebraic if and only if κ ∨ X is algebraic.Let h X ∈ H 2 (X, Q) be the cohomology class of an ample divisor on X.By the strong Lefschetz theorem, the cup product with h 2n−1 X induces an isomorphism of Hodge structures where (2n − 2) denotes the Tate twist by Q(2n − 2).Let As mentioned in the introduction, Hodge similarities between transcendental lattices of hyperkähler manifolds appear naturally in two cases: as elements of totally real endomorphism fields of degree two, and as Hodge isomorphisms T (Y ) → T (X), where X and Y are hyperkähler manifolds with T (Y ) Hodge isometric to T (X)(λ) for some λ ∈ Q >0 .
Let us start from the case of hyperkähler manifolds of generalized Kummer type.For these varieties the Kuga-Satake Hodge conjecture is proven in [23] and the Lefschetz standard conjecture in degree two has been proven in [6].We thus get our main families of examples of varieties satisfying the hypotheses of Corollary 4.6, and we conclude that every Hodge similarity between the transcendental lattices of two hyperkähler manifolds of generalized Kummer type is algebraic.Note that the dimension of the transcendental lattice of a hyperkähler manifold of generalized Kummer type is at most six-dimensional.Therefore, its endomorphism field is either a CM field or a totally real field of degree one or two.In all cases, we see that it is always generated by Hodge similarities.We therefore deduce the following: Theorem 5.1.Let X and X ′ be hyperkähler manifolds of generalized Kummer type such that T (X) and T (X ′ ) are Hodge similar.Then, every Hodge morphism between T (X ′ ) and T (X) is algebraic.
Remark 5.2.Taking X = X ′ in Theorem 5.1, we see that every Hodge morphism in E := End Hdg (T (X)) is algebraic.Note that, Theorem 5.1 also covers the case where X and X ′ are hyperkähler manifolds of generalized Kummer type with Hodge similar transcendental lattice but of different dimension.Let us briefly recall why this happens: recall that the second cohomology group of a hyperkähler manifold X of generalized Kummer type of dimension 2n satisfies (H 2 (X, Q), q X ) ≃ U ⊕3 Q ⊕ Qδ n , where (δ n ) 2 = −2(n + 1).Let k be a positive integer.Using the fact that U Q is isometric with U Q (k), one sees that there is an isometry , for n ′ := k(n + 1) − 1.In other words, there is a similarity ) be a class satisfying (σ ′ ) 2 = 0 and (σ ′ , σ ′ ) > 0.Then, [σ ′ ] determines a Hodge structure on the quadratic space U ⊕3 Q ⊕ Qδ n ′ .Similarly the class [ψ(σ ′ )] satisfies the same hypotheses of σ ′ , hence, it defines a Hodge structure on U ⊕3 Q ⊕ Qδ n .By the surjectivity of the period map, we obtain a hyperkähler manifold X ′ of Kum n ′ -type whose symplectic form is given by σ ′ and a hyperkähler manifold X of Kum n -type whose symplectic form is given by ψ(σ).By construction, the morphism ψ defines a Hodge similarity between T (X ′ ) and T (X).Thus, X and X ′ satisfy the hypotheses for Theorem 5.1, and we conclude that any Hodge morphism T (X ′ ) → T (X) is algebraic.
Let us briefly comment on the application of our result to the case of K3 surfaces.In this case, the Lefschetz standard conjecture is trivially true.Hence, applying Corollary 4.6, we get the following: