Sixfolds of generalized Kummer type and K3 surfaces

We prove that any hyper-Kähler sixfold $K$ of generalized Kummer type has a naturally associated manifold $Y_K$ of $\mathrm {K}3^{[3]}$ type. It is obtained as crepant resolution of the quotient of $K$ by a group of symplectic involutions acting trivially on its second cohomology. When $K$ is projective, the variety $Y_K$ is birational to a moduli space of stable sheaves on a uniquely determined projective $\mathrm {K}3$ surface $S_K$. As an application of this construction we show that the Kuga–Satake correspondence is algebraic for the K3 surfaces $S_K$, producing infinitely many new families of $\mathrm {K}3$ surfaces of general Picard rank $16$ satisfying the Kuga–Satake Hodge conjecture.


Introduction
Together with manifolds of K3 [n] -type, deformations of generalized Kummer varieties constitute the most well studied hyper-Kähler manifolds.We refer to this deformation type in dimension 2n as to the Kum n -type.After Beauville [4] gave the first examples of such hyper-Kähler manifolds, many more have been constructed from moduli spaces of stable sheaves on abelian surfaces, see [55].However, the varieties so obtained always have Picard rank at least 2, and our understanding of a general projective variety of Kum n -type remains poor.In fact, for the time being, no construction of such variety is known (but see [44] for some recent ideas).
In the present article we partially remedy this for hyper-Kähler sixfolds of generalized Kummer type, by associating to any K of Kum 3 -type a hyper-Kähler manifold Y K of K3 [3] -type.They are related by a dominant rational map K Y K of degree 2 5 , described as follows.
Any K of Kum 3 -type admits an action of the group (Z/4Z) 4 ⋊ Z/2Z by symplectic automorphisms, where Z/2Z acts on the first factor as −1.In fact, this is the group Aut 0 (K) of automorphisms of K which act trivially on its second cohomology, which is deformation invariant by [19].It has been computed in [7] for the generalized Kummer variety associated to an abelian surface.We let G ⊂ Aut 0 (K) be the subgroup generated by the automorphisms whose fixed locus contains a 4-dimensional component.We will show in Lemma 2.4 that Theorem 1.1.-Let K be a manifold of Kum 3 -type.The quotient K/G admits a resolution Y K → K/G with Y K a manifold of K3 [3] -type.
The resolution Y K → K/G is obtained via a single blow-up of the reduced singular locus.It can also be described as the quotient by G of the blow-up of K at the union of the fixed loci of all non-trivial automorphisms in G.We show that there is an isometry of transcendental Hodge structures where on the right hand side the form is multiplied by 2. This parallels the classical construction of the Kummer K3 surface Km(A) associated to an abelian surface A, where H 2 tr (Km(A), Z) is Hodge isometric to H 2 tr (A, Z) (2).The study of the integral transcendental lattices of the manifolds Y K will be the subject of future work.
As a consequence, we obtain a well-defined K3 surface associated to any projective sixfold of generalized Kummer type.
Theorem 1.2.-Let K be a projective variety of Kum 3 -type.There exists a unique (up to isomorphism) projective K3 surface S K such that the variety Y K of K3 [3] -type given by Theorem 1.1 is birational to a moduli space M SK ,H (v) of stable sheaves on S K , for some primitive Mukai vector v and a v-generic polarization H.
We call S K the K3 surface associated to the sixfold K.It is characterized by the existence of a Hodge isometry H 2 tr (S K , Z) ∼ − − → H 2 tr (Y K , Z).These surfaces come in countably many 4-dimensional families of general Picard rank 16; up to isogeny, they are the K3 surfaces S admitting an isometric embedding H 2 tr (S, Q) ֒→ Λ Kum 3 (2) ⊗ Z Q of rational quadratic spaces.Here, Λ Kum 3 is the lattice which is the second cohomology of manifolds of Kum 3 -type; it was computed in [4] that Λ Kum 3 = U ⊕3 ⊕ −8 , where we denote by U the hyperbolic plane.
He uses Verbitsky's theory of hyperholomorphic sheaves [50] to produce very interesting algebraic cycles on general projective hyper-Kähler varieties via deformation.
Through our construction, we are able to deduce more cases of the Hodge conjecture from his results.
Our main application is to the Kuga-Satake Hodge conjecture for K3 surfaces [16].
The Kuga-Satake construction [11] associates via Hodge theory an abelian variety KS(S) to any projective K3 surface S, and the conjecture predicts the existence of an algebraic cycle inducing an embedding of the transcendental Hodge structure of the surface into the second cohomology of KS(S) × KS(S).It is known to hold in many cases for K3 surfaces of Picard number at least 17 ([37]), but it is wide open otherwise.There are a couple of families of K3 surfaces of general Picard rank 16 for which the conjecture is known, namely, the family of double covers of the plane branched at 6 lines [46] and the family of K3 surfaces with 15 nodes in P 4 [24].Theorem 1.3 below gives infinitely many new families of K3 surfaces of general Picard rank 16 for which the Kuga-Satake Hodge conjecture holds true.
Via Theorem 1.2, we deduce this statement from the validity of the Kuga-Satake Hodge conjecture for varieties of Kum n -type, established by Voisin [52] as a consequence of results of O'Grady [43] and Markman [30].Once Theorem 1.3 is proven, a result of Varesco [49] implies that the Hodge conjecture holds for all powers of any K3 surface satisfying the condition of the above theorem, see Corollary 5.8.
In the recent preprint [31], Markman proves that any rational Hodge isometry -type is algebraic.He expects that an extension of his argument will lead to the analogous result for varieties of Kum ntype.In dimension 6, we can obtain it from the K3 [3] case via Theorem 1.1.
Theorem 1.4.-Let K, K ′ be projective varieties of deformation type Kum 3 .Let Overview of the contents.-We sketch the proof of Theorem 1.1.First, we calculate the fixed loci K g of all the automorphims g ∈ G: we show that g =1∈G K g is the union of 16 hyper-Kähler manifolds of K3 [2] -type, and we determine the various intersections of these components.Thanks to the deformation invariance of G, it is in fact enough to calculate these loci in the special case of the generalized Kummer sixfold on an abelian surface.
We then describe explicitly the singularities of K/G.It turns out that these are of a particular simple nature, modeled on products of ordinary double points on surfaces.A resolution Y K of K/G is obtained via a single blow-up of the singular locus.The quotient K/G is a primitive symplectic orbifold as studied by Fujiki [14] and Menet [33], and we use a criterion due to Fujiki to prove that Y K is a hyper-Kähler manifold.Moreover, the resolution can be performed in families, so that the Y K are deformation equivalent to each other.To complete the proof of Theorem 1.1 it is therefore sufficient to find a single K of Kum 3 -type such that Y K is of K3 [3] -type.
The specific example we study is a Beauville-Mukai system K J (v 3 ) on a general principally polarized abelian surface Θ ⊂ J.It admits a Lagrangian fibration to the linear system |2Θ| = P 3 , whose general fibres parametrize certain degree 3 line bundles supported on curves in the linear system.We show that the norm map for line bundles induces a dominant rational map of degree 2 5 from K J (v 3 ) onto a Beauville-Mukai system M Km(J) (w 3 ), a moduli space of sheaves on the Kummer K3 surface Km(J) associated to J; the hyper-Kähler variety M Km(J) (w 3 ) is birational to Km(J) [3] .We next prove that the norm map descends to a birational map It follows that the hyper-Kähler manifold Y KJ (v3) is birational to Km(J) [3] , and hence Y KJ (v3) is of K3 [3] -type by Huybrechts' theorem [20] that birational hyper-Kähler manifolds are deformation equivalent.
Once Theorem 1.1 is proven, the other results follow from it.When K is projective, we define the associated K3 surface S K as the unique (up to isomorphism) K3 surface whose transcendental lattice is Hodge isometric to H 2 tr (Y K , Z).This is justified by the fact that H 2 tr (Y K , Z) has rank at most 6 and hence it appears as transcendental lattice of some K3 surface of Picard rank at least 16.Moreover, for such Picard numbers, two K3 surfaces with Hodge isometric transcendental lattices are isomorphic, and hence S K is determined up to isomorphism.To prove our applications, we exploit the algebraic cycle in K × Y K given by the rational map K Y K .
The organization of this text is as follows.In Section 2 we calculate the fixed locus of the automorphisms of G.In Section 3 we study in detail the case of K J (v 3 ) as outlined above.In Section 4 we prove Theorems 1. conversations related to these topics, and the anonymous referee for his/her comments.

Automorphisms trivial on the second cohomology
Let K be a manifold of Kum 3 -type and let Aut 0 (K) be the group of automorphisms of K which act trivially on H 2 (K, Z).We let K h denote the fixed locus of an automorphism h of K.
Definition 2.1.-We denote by G the subgroup of Aut 0 (K) generated by the automorphisms g ∈ Aut 0 (K) such that K g has a 4-dimensional component.
We will see in Lemma 2.4 below that G ∼ = (Z/2Z) 5 .The main result proven in this section is then the following.
Theorem 2.2.-Let Z ⊂ K be defined as Z := g =1∈G K g .Then Z is the union of 16 fourfolds Z j , each of which is a smooth hyper-Kähler manifold of K3 [2] -type.
Moreover, two distinct components intersect in a K3 surface, three distinct components intersect in 4 points, and four or more distinct components do not intersect.
In order to prove this, we will compute the fixed loci of all automorphisms in G.
Thanks to the deformation invariance of automorphisms trivial on the second cohomology it will be sufficient to treat the case of the generalized Kummer variety associated to an abelian surface A, i. e.K is the fibre over 0 of the composition of the Hilbert-Chow morphism with the summation map.We denote by A (4) 0 ⊂ A (4) the fibre over 0 of .The restriction ν : K → A (4) 0 is a crepant resolution.
In this case the group of automorphisms of K acting trivially on the second cohomology has the natural description Aut 0 (K) = A 4 ⋊ −1 , and the action on K and A [4] is induced by that on A, see [7].

Notation. -We let
A n be the group of points of order n of A. Via the isomorphism above, we write the elements of Aut 0 (K 3 (A)) as (ǫ, ±1) for ǫ ∈ A 4 .For τ ∈ A 2 , we denote by A 2,τ the set {a ∈ A | 2a = τ }, which consists of 16 points.
The quotient surface A/ (τ, −1) has 16 nodes corresponding to points in A 2,τ ; its minimal resolution Km τ (A) is isomorphic to the Kummer K3 surface associated to A. Proof.-Since automorphisms in Aut 0 (K) deform with K, their fixed loci deform as well.Therefore, it suffices to prove the Lemma for the generalized Kummer variety K = K 3 (A) on an abelian surface A. In this case we show that [45,Lemma 3.5], the fixed locus (K 3 (A)) h is a union of K3 surfaces if 2ǫ = 0 and consists of isolated point if 2ǫ = 0. Consider now h = (ǫ, −1).

We have Aut
The fixed locus of h = (ǫ, −1) on A (4) consists of (4) , and subvarieties of lower dimension.If ǫ ∈ A 2 , then W ǫ is contained in A  for any τ ∈ A 2 , we let We denote by W τ and V τ,θ the subvarieties of K obtained as strict transform of W τ and V τ,θ under the Hilbert-Chow morphism, respectively.
Note that none among the W τ and V τ,θ is contained in the exceptional locus of 0 , so that W τ and V τ,θ are well-defined.We will show in Lemma 2.9 that the W τ are hyper-Kähler varieties of K3 [2] -type and that the V τ,θ are K3 surfaces.
We recall a result of Kamenova-Mongardi-Oblomonkov which will be used in the sequel.
The next lemma describes the varieties V τ,θ and W τ introduced above.
Lemma 2.9.-(i) For τ = 0, θ ∈ A 2 , the rational map φ τ,θ defined by the commutative diagram where q is the natural map, extends to a regular morphism, which is an isomorphism onto its image V τ,θ .
(ii) For τ ∈ A 2 , the rational map φ τ defined by the commutative diagram where q is the composition of the Hilbert-Chow morphism with the natural map (Km τ (A)) (2) → (A/ (τ, −1) ) (2) , extends to a regular morphism, which is an isomorphism onto its image W τ .
The various intersections of the submanifolds W τ ⊂ K are as follows.
(iii) The intersection of 4 or more distinct submanifolds W τ is empty. Proof.
Again, we may assume a ′′ = a.Then we must have −a+ τ 3 = a+ τ 1 + τ 2 ; equivalently, This set consists of 4 distinct points.Indeed, G acts transitively on it and the subgroup of G generated by the (τ i , −1), i = 1, 2, 3 acts trivially.Moreover, it is easy to see that if g ∈ G is not an element of this subgroup then it has no fixed points in the above set.Hence, the intersection point of this set belongs to the exceptional locus of the Hilbert-Chow morphism, so 2.11.Fixed loci.-We calculate the fixed locus of the automorphisms g ∈ G. Our results confirm those of Oguiso [45] and Kamenova-Mongardi-Oblomkov [25].
Proposition 2.12.-Let g = 1 ∈ G. Then: if g = (τ, 1), with τ = 0 ∈ A 2 , the fixed locus K g is the disjoint union of the 8 K3 surfaces V τ,θ , for θ varying in A 2 ; if g = (τ, −1) with τ ∈ A 2 , the fixed locus K g is the union of the K3 [2] -type variety W τ and 140 isolated points.The isolated points are given by the set Proof.-Let g = (τ, 1) with τ = 0 ∈ A 2 .The action of g on ξ = (a, b, c, d) ∈ A (4) is given by ξ → (a + τ, b + τ, c + τ, d + τ ).We see immediately that the surfaces V τ,θ are fixed by g.Conversely, if ξ is fixed and a belongs to its support, then also a + τ must be in the support.Therefore, ξ = (a, a + τ, b, b + τ ); if ξ ∈ A (4) 0 , then 2a + 2b = 0, so that θ := a + b belongs to A 2 and ξ ∈ V τ,θ .By the description of V τ,θ given in Lemma 2.9.(i), the fixed locus of Assume now that g = (τ, −1).If ξ = (a, b, c, d) ∈ A (4) , the action of g is given by As ξ belongs to A (4) We conclude that the fixed locus of g acting on A (4) 0 consists of W τ and the isolated points in the set There are 140 isolated fixed points.Indeed, the total number of ordered sequences (ǫ 1 , ǫ 2 , ǫ 3 , ǫ 4 ) such that i ǫ i = 0 is 16 3 .There are 16 such sequences with support a single point, and 6 16  2 = 16 • 45 with support two distinct points.The number of ordered sequences as above supported on four distinct point is 16 3  Hence the number of isolated fixed points is 210•16 24 = 140.Therefore, K g consists of 140 isolated fixed points and the fixed locus of g acting on ν −1 (W τ ).To conclude we have to check that the latter coincides with W τ .Using the description of W τ given in the proof of Lemma 2.9.(ii) and the second assertion of Proposition 2.8 it is readily seen that ν One can see directly from Definition 2.6 that h maps W τ to W τ +2ǫ , for any τ ∈ A 2 .
We can finally give the proof of the main result of this section.
Proof of Theorem 2.2.-Let us first assume that K is the generalized Kummer variety associated to an abelian surface A.
Consider a component V τ,θ of K g .The surface V τ,θ is clearly contained in W θ , and hence the same holds for their strict transforms.Let now g = (τ, −1) ∈ G. Let P be an isolated fixed point in K g .Then we have P = (ǫ 1 , ǫ 2 , ǫ 3 , ǫ 4 ) for some pairwise distinct ǫ i ∈ A 2,τ summing up to 0 in A. The last condition implies that we can write P = (ǫ 1 , ǫ 2 , ǫ 1 + θ, ǫ 2 + θ) for some 0 = θ ∈ A 2 .This is the same as which lies in W τ +θ .By Lemma 2.12 we conclude that for a generalized Kummer variety the union of the fixed loci K g consists of the 16 varieties W τ of K3 [2] -type.
The intersections of these components were computed in Lemma 2.
) such that the action on U x of the stabilizer subgroup of x is induced by that on the fibre K f (x) .Therefore, for any τ ∈ A 2 the fixed locus K g of g = (τ, −1) contains a smooth family W τ → B of manifolds of K3 [2] -type as the unique component of K g with positive dimensional intersection with the fibres of f .We conclude that for any b ∈ B we have and that the components W τ |K b intersect as claimed.

A Lagrangian fibration
Throughout this section, J denotes a general principally polarized abelian surface.
We fix a symmetric theta divisor Θ ⊂ J, which is unique up to translation by a point of J 2 .For any integer d, we consider the Mukai vector v d := (0, 2Θ, d − 4) on J and the moduli space M J (v d ) of Θ-semistable sheaves with Mukai vector v d ( [23]).The Albanese fibration is isotrivial [55].We denote the fibre of alb over (O J , 0) by K J (v d ).For d even, K J (v d ) is singular and admits a crepant resolution of OG6-type, by [42] and [28].
When d is odd, K J (v d ) is smooth, and it is a hyper-Kähler sixfold of Kum 3 -type [55].
In this case, we consider the group The main result of this section is the following.Theorem 3.1.-The quotient K J (v 3 )/G is birational to Km(J) [3] .
We will in fact show that for any odd d the quotient K J (v d )/G is birational to a variety of K3 [3] -type.Before proceeding with the proof we need to fix some notation.

3.3.
Geometric set-up.-Let Θ ⊂ J be the general principally polarized abelian surface of Picard rank 1, with Θ a symmetric theta divisor.By Riemann-Roch H 0 (J, O J (2Θ)) = 4; we will identify the complete linear system |2Θ| with P 3 .It is classically known that this linear system is base-point free and induces an embedding of J/ ± 1 into P 3,∨ as a quartic surface with 16 nodes.Blowing-up the nodes of this quartic surface one obtains the Kummer K3 surface Km(J) associated to J. We denote by H the divisor on Km(J) obtained as pull-back of a hyperplane section of J/ ± 1 ⊂ P 3,∨ .The linear system |H| is naturally identified with |2Θ| = P 3 .
Consider the universal family of genus 5 curves C → |2Θ|.The involution −1 of J acts trivially on |2Θ|, and we consider the quotient family C → P 3 , which is identified with the hyperplane linear system on J/ ± 1 ⊂ P 3,∨ .There is a degree 2 morphism A detailed study of the linear system |2Θ| and the map π : C → C can be found in Verra's article [51].In particular, he shows that, for any b ∈ P 3 , the curve Cb is smooth if and only if C b is so, and in this case the Prym variety of the étale double cover π b : Cb → C b is the abelian surface J. Moreover C b is smooth if and only if it does not contain any node of J/ ± 1 and it does not lie on a tangent hyperplane. of semistable sheaves with Mukai vector w d , with respect to a fixed w d -generic polarization.It is a hyper-Kähler variety of K3 [3] -type by [41], [54].The moduli space M Km(J) (w d ) parametrizes pure dimension 1 sheaves on Km(J) which are pushforward of torsion free sheaves of rank 1 and degree d on curves in the linear system |H|.There is a morphism supp : Pic d (D/P 3 ) → P 3 which is a Lagrangian fibration, whose general fibres are the degree d Picard varieties of the smooth curves in |H|.The pull-back π * has been studied by Rapagnetta [47] and by Mongardi-Rapagnetta-Saccà [34].They show that it gives a degree 2 dominant rational map exhibiting a variety of OG6-type as quotient of a variety of K3 [3] -type by a birational symplectic involution.We consider instead the norm map Nm π .Consider the commutative diagram Pic d (D/P 3 ) Pic 2d (D/P 3 ). [2] Nmπ Nmπ [2] Both the horizontal maps are dominant and generically finite of the same degree 2 6 .Indeed, they preserve the fibres of the respective support morphisms, which, generically, are torsors under abelian threefolds.Hence, the degree of [2] is the number of points of order 2 on an abelian threefold.
It thus suffices to prove the lemma for Nm π : K J (v 2d ) Pic 2d (D/P 3 ).Since ) is the closure of the image of π * , the involution σ * is the identity on K J (v 2d ).
Hence, the rational map coincides with multiplication by [2]; this composition is therefore dominant and generically finite of degree 2 6 .Since π * : Pic 2d (D/P 3 ) K J (v 4d ) is generically finite of degree 2, it follows that Nm π is dominant and it has degree 2 5 .

3.7.
The action of G. -Assume now that d is odd, so that K J (v d ) is a smooth variety of Kum 3 -type and the support map gives a Lagrangian fibration on it.In this case the group Aut 0 (K J (v d )) has been explicitly identified by Kim in [26].Proof.-We first show that Pic 0 (J) 2 × −1 acts on K J (v d ) via automorphisms trivial on the second cohomology and which preserve the Lagrangian fibration given by the support morphism.
Since −1 : J → J acts trivially on the cohomology of J in even degrees, the pullback of sheaves defines an automorphism (−1) * of K J (v d ); this automorphism is trivial for d even but not for d odd.By [35,Lemma 2.34], the action of (−1) * is symplectic, i. e., the induced action on the transcendental cohomology H 2 tr (K J (v d ), Z) is the identity.Since all curves in |2Θ| are stable under −1, the automorphism (−1) * preserves all fibres of the support fibration.The Picard rank of K J (v d ) is 2, because J is a general abelian surface by assumption (see [55]).Hence, by Lemma 3.10 below, the induced action of (−1) * on NS(K J (v d )) is also the identity.Since H 2 (K J (v d ), Z) is torsion-free, we conclude that (−1) * acts trivially on the second cohomology.
Let now L ∈ Pic 0 (J) 2 .Let B ⊂ |2Θ| be the locus of smooth curves, and consider the universal curve j : CB ֒→ J × B. By [53, Lemma 6.9], the corresponding pullback j * : Pic 0 (J) × B → Pic 0 ( C/P 3 ) B is an injection of abelian schemes.Note that the covering involution σ for the cover π : CB → C B is the restriction of −1 × id B .
Hence, the induced involution σ * on Pic 0 ( C/P 3 ) B is the inverse map on j * b (Pic 0 (J)), for each b ∈ B. As K J (v 0 ) is the fixed locus of this involution, the 2-torsion line bundle L on J corresponds uniquely to a 2-torsion section s L of K J (v 0 ) B → B. For any d, the variety K J (v d ) B is a torsor under K J (v 0 ) B and hence F → F ⊗ L induces a birational automorphism g L of K J (v d ), which restricts to a translation on the smooth fibres of the support morphism.This implies that g L is symplectic.We deduce from Lemma 3.10 that g L extends to a regular automorphism of K J (v d ), whose induced action is trivial on NS(K J (v d )) as well.
According to [26,Theorem 5.1] we have described all automorphisms in Aut 0 (K J (v d )) preserving the fibration supp : K J (v d ) → P 3 .We claim now that an automorphism in Aut 0 (K J (v d )) which does not preserve the fibration cannot fix a component of dimension 4 on K J (v d ); this will show that G = Pic 0 (J) 2 × −1 .
Let D = supp * (O P n (1)) ∈ Pic(K J (v d )) be the line bundle inducing the fibration.
Since Aut 0 (K J (v d )) acts trivially on H 2 (K J (v d ), Z), it acts on |D| = P 3 = |2Θ|.We obtain an action of Aut 0 (K J (v d ))/(Pic 0 (J) 2 × −1 ) ∼ = (Z/2Z) 4 on |2Θ|, which, up to conjugation by an automorphism of P 3 , is identified with the action generated by (x, y, z, w) → (z, w, x, y), (x, y, z, w) → (y, x, w, z), fixes a fourfold but does not preserve the support fibration.Then (K J (v d )) h contains one of the varieties Z j of K3 [2] -type from Theorem 2.2, and the image of Z j via the fibration must be a line R ⊂ |2Θ| fixed by h.But, by a theorem of Matsushita [32], the K3 [2] -variety Z j does not admit any non-constant morphism to P 1 .
-By [51], the relative Prym variety P → B of the double cover π : CB → C B is an isotrivial family over B with fibre J. Since the involution σ * acts as the inverse on the image of the pull-back j * , the constant abelian scheme j * (Pic 0 (J) × B) is contained into, and hence equal to, the relative Prym variety.
Lemma 3.10.-Let f : X → P n be a Lagrangian fibration, where X is a 2ndimensional projective hyper-Kähler variety of Picard rank 2. Let g : X X be a birational automorphism such that f • g = f .Then g extends to a regular automorphism of X, and g * | NS(X) : NS(X) → NS(X) is the identity.
Proof.-The Néron-Severi group NS(X) is a rank 2 lattice of signature (1,1).Any birational automorphism g induces an isometry g * of H 2 (X, Z) which restricts to an isometry of NS(X).Denote by D ∈ NS(X) the class of the line bundle f * (O P n (1)), and pick E ∈ NS(X) such that D and E generate NS(X)⊗Q.Since f •g = f , we have Using that D is isotropic one sees that this this forces g * (E) = E as well, so that g * is the identity on NS(X); by [13], g extends to an automorphism of X.
We can now complete the proof of the main result of this section.
Proof of Theorem 3.1.-Using the idenfication of G given in Proposition 3.8 we will show that the norm map descends to a birational map K J (v d )/G Pic d (D/P 3 ), for any odd d.The varieties Pic d (D/P 3 ) are of K3 [3] -type, and Pic 3 (D/P 3 ) is birational to Km(J) [3] , see [6,Proposition 1.3].
For a smooth curve Cb ∈ |2Θ| the norm map is induced by the map of divisors which Let instead L ∈ Pic 0 (J) 2 , and let s L : P 3 K J (v 0 ) be the rational section defined by s L (b) = j * b (L), where j : CB ֒→ J × B denotes the natural inclusion.By Remark 3.9, the section s L is contained in the relative Prym variety; in particular, the composition Nm π • s L gives the zero section of Pic 0 (D/P 3 ) B → P 3 .
Therefore, for any g ∈ G and y ∈ K J (v d ) B we have Nm π (g(y)) = Nm π (y), which means that Nm π : K J (v d ) Pic d (D/P 3 ) descends to a rational map This map is in fact birational, because Nm π : ) is generically finite of degree 2 5 by Lemma 3.6, and G has also order 2 5 .

The K3 surface associated to a Kum 3 -variety
In this section we give the proof of Theorems 1.1 and 1.2.We let K be a manifold of Kum 3 -type and consider the quotient K/G by the group G ∼ = (Z/2Z) 5 of Definition 2.1.We will show that the blow-up of the singular locus of K/G yields a hyper-Kähler manifold Y K of K3 [3] -type.
4.1.The singularities of K/G.-By Theorem 2.2, the locus Z = 1 =g∈G K g is the union of 16 irreducible components Z i , for i = 1, . . ., 16. Denote by X i ⊂ K/G, the image of Z i via the quotient map q : K → K/G.We introduce the following stratification of K/G into closed subspaces: -The subspace X j is empty for j 4. For j < 4, a point x ∈ X j \ X j+1 has a neighborhood U x ⊂ K/G analytically isomorphic to where ι is the involution (x, y) → (−x, −y).
Proof.-Theorem 2.2 implies immediately that X j is empty for j 4. The group G acts freely on q −1 (X 0 \ X 1 ), and hence X 0 \ X 1 is smooth.
If K → B is a smooth proper family of Kum 3 -manifolds, the quotient K/G → B is a locally trivial family (see [14,Lemma 3.10]): any point x ∈ K/G has a neighborhood . Therefore, to prove the proposition we may assume that K is the generalized Kummer sixfold on an abelian surface A.
In this case the components Z i are the explicit W τ , for τ ∈ A 2 (see Definition 2.6).
Recall that W τ ⊂ K is the unique positive dimensional component of the fixed locus of (τ, −1) ∈ G, and that the induced action of G/ (τ, −1) on W τ is faithful.For , where the first factor is the normal space.The action of (τ, −1) on T z (K) is (−1, 1).Let now x ∈ X 1 \ X 2 , and let z ∈ K be a preimage of x.By definition, z belongs to exactly one component W τ .The stabilizer is G z = (τ, −1) .By the above, there exists a neighborhood V z of z ∈ K such that g(V z ) ∩ V z is empty for any g / ∈ G z , and In the other cases we proceed similarly.Let x ∈ X 2 \X 3 , and let z ∈ K be one of its preimages.Then z belongs to two distinct components W τ1 and W τ2 .The stabilizer There is a decomposition , and the action of G z is generated by (−1, 1, 1) and (1, −1, 1).This implies that the image in K/G of a sufficiently small neighborhood of Finally, let x ∈ X 3 and z ∈ K a preimage of it.Then z belongs to exactly 3 components W τi , for i = 1, 2, 3.The stabilizer G z is the subgroup generated by the involutions (τ i , −1), for i = 1, 2, 3, and there is a decomposition T z (K) = 3 i=1 N Wτ i |K,z of the tangent space.The action of (τ i , −1) on T z (K) is −1 on the i-th summand and the identity on the complement.We conclude that there exists a neighborhood V z of The proposition will be reduced to the following elementary statement.Lemma 4.5.-Let ι : C 2 → C 2 be the involution (x, y) → (−x, −y), let j, k be positive integers.Then the blow-up of (C 2 /ι) j × (C 2 ) k along its singular locus is isomorphic to (Bl 0 (C 2 /ι)) j × (C 2 ) k .It is thus smooth, and identified with the quotient Proof.-Given noetherian schemes T 1 and T 2 and closed subschemes V 1 T 1 and It is easy to see this when T 1 and T 2 are affine schemes, using the definition of blowingup via the Proj construction ([18, II, §7]).The singular locus of (C 2 /ι) j × (C 2 ) k is the union of pr −1 i ({0}) for i from 1 to j, where pr i : ) is the projection onto the i-th factor.Via the above observation, the statement is reduced to the case j = 1, k = 0, which is the well-known minimal resolution of an isolated We will now use a criterion due to Fujiki [14] to show that Y K is symplectic, i. e.
that it admits a nowhere degenerate holomorphic 2-form.
Proof.-The singular space K/G is a primitively symplectic V-manifold in the sense of Fujiki.According to his [14, Proposition 2.9], to show that Y K admits a symplectic form it suffices to check that the following two conditions are satisfied: (ii) the restriction of p : where p ′ is the minimal resolution of A.
We have already shown in the course of proof of Propositions 4.2 and 4.4 that these conditions hold, with A a neighborhood of the ordinary node 0 ∈ C 2 /ι, whose minimal resolution is the blow-up of the singular point.
Hence, Y K is symplectic.From its description as the quotient Bl Z (K)/G we obtain h 2,0 (Y K ) = 1 (see also §4.7).Finally, Y K is simply connected by [14,Lemma 1.2].
We can now complete the proof of our main result.-type such that Y K is hyper-Kähler of K3 [3] -type.
Consider the sixfold K J (v 3 ) introduced in Section 3, which is constructed from a Beauville-Mukai system on a principally polarized abelian surface J.By Theorem 3.1, in this case Y KJ (v3) is birational to the Hilbert scheme Km(J) [3] on the Kummer K3 surface associated to J. By [20,Theorem 4.6], birational hyper-Kähler manifolds are deformation equivalent, and therefore Y KJ (v3) is of K3 [3] -type.

4.7.
The associated K3 surface.-The following computation is entirely analogous to [12, §3].If Z is a compact Kähler manifold, we let H 2 tr (Z, Z) denote the smallest sub-Hodge structure of H 2 (Z, Z) whose complexification contains H 2,0 (Z).Let K be a manifold of Kum 3 -type, and denote by K the blow-up of K along 1 =g∈G K g .By Proposition 4.4 we have a commutative diagram where p, p ′ are blow-up maps and q, q ′ are the quotient maps for the action of G.
The pull-back gives an isomorphism p ′ * : H 2 tr (K, Z) → H 2 tr ( K, Z).Since G acts trivially on H 2 (K, Z), the pushforward q ′ * : H 2 tr ( K, Z) → H 2 tr (Y K , Z) is injective, and q ′ * q ′ * is multiplication by 2 5 on H 2 tr ( K, Z).It follows that becomes an isomorphism of Hodge structures after tensoring with Q. Denote by q K and q YK the Beauville-Bogomolov forms on H 2 (K, Z) and H 2 (Y K , Z) respectively.
Proof.-Let c K and c Y be the Fujiki constants of K and Y respectively ( [15]).This means that we have K x 6 = c K q K (x, x) 3 for any x ∈ H 2 (K, Z), and similarly for Y K .
We now prove that every projective K of Kum 3 -type has a naturally associated K3 surface.
Proof of Theorem 1.2.-Let K be a projective variety of Kum 3 -type and let Y K be the crepant resolution of K/G.By the above lemma, the transcendental lattice ) is an even lattice of signature (2, k), and rank at most 6.By [36, Corollary 2.10], there exists a K3 surface S K such that H 2 tr (S K , Z) is Hodge isometric to H 2 tr (Y K , Z).A criterion due to independently Mongardi-Wandel [35] and Addington [1, Proposition 4] ensures that Y K is birational to M SK ,H (v), for some primitive there exists an isometric embedding of H 2 tr (S, Q) into Λ Kum 3 (2) ⊗ Z Q; -there exist a projective variety K of Kum 3 -type with associated K3 surface S K and a rational Hodge isometry H 2 tr (S, Q) Proof.-The second assertion implies the first thanks to Lemma 4.8.Conversely, let Φ : H 2 tr (S, Q) ֒→ Λ Kum 3 (2) ⊗ Z Q be an isometric embedding.Choose a primitive sublattice T ⊂ Λ Kum 3 such that T (2)⊗ Z Q coincides with the image of Φ. Equip T with the Hodge structure induced by that on H 2 tr (S, Q) via Φ.By the surjectivity of the period map [20], there exists a manifold K of Kum  [20] implies that K is projective.

Applications to the Hodge conjecture
In this section we prove Theorems 1.3 and 1.4.Throughout, all cohomology groups are taken with rational coefficients, which are thus suppressed from the notation.
We start with a simple observation.Let X, Y be smooth projective varieties and The lemma follows because the second summand consists of algebraic classes.
Proof of Theorem 1.4.-Let K and K ′ be projective varieties of Kum 3 -type and assume that f : ) is a rational Hodge isometry.By the above lemma, it suffices to show that the component f tr : the varieties of K3 [3] -type given by Theorem 1.1 applied to K and K ′ respectively, and let r : K Y and r ′ : K ′ Y ′ be the corresponding rational maps.They induce isomorphisms of Hodge structures r * : The inverse of r * is r * /2 5 , and similarly for r ′ .It follows that f tr is algebraic if and only if ftr is so.By Lemma 4.8, the map ftr is a rational Hodge isometry, and hence it is algebraic by Markman's theorem in [31].
5.2.The Kuga-Satake correspondence.-The Kuga-Satake construction associates an abelian variety to any polarized Hodge structure of K3-type.We briefly recall this construction, referring to [16] and [21,Chapter 4] for more details.
The Clifford algebra C(V ) is defined as the quotient In [11], Deligne shows that the Hodge structure of V induces a Hodge structure of weight 1 on C + (V ) of type (1, 0), (0, 1), which admits a polarization and hence defines an abelian variety KS(V ) up to isogeny.This 2 rk(V )−2 -dimensional abelian variety is called the Kuga-Satake variety of V .
Upon fixing some v 0 ∈ V , the action of V on C(V ) via left multiplication induces an embedding of weight 0 rational Hodge structures which maps v to the endomorphism w → vwv 0 .
Remark 5.3.-If V ′ ⊂ V is a sub-Hodge structure such that V ′ ⊥ consists of Hodge classes, then KS(V ) is isogenous to a power of KS(V ′ ).Replacing the form q with a non-zero rational multiple results in isogenous Kuga-Satake varieties.See Let X be a projective hyper-Kähler variety of dimension 2n.The Kuga-Satake variety KS(X) of X is the abelian variety obtained from (H 2 tr (X), q X ) via the Kuga-Satake construction, where q X is the restriction of the Beauville-Bogomolov form.
Identifying H 1 (KS(X)) with its dual, there exists an embedding of Hodge structures of H 2 tr (X) into H 2 (KS(X) × KS(X)).According to the Hodge conjecture this embedding should be induced by an algebraic cycle.Proof.-By Lemma 4.10, there exists a projective variety K of Kum 3 -type with associated K3 surface S K and a rational Hodge isometry t 0 : H 2 tr (S) ∼ − − → H 2 tr (S K ).Denote by Y K the crepant resolution of K/G given by Theorem 1.1 and let v, H be given by Theorem 1.2, so that Y K is birational to the moduli space M SK,H (v).Denote by r : K Y K the natural rational map of degree 2 5 .
By [9], [22], the isometry t 0 is algebraic.Next, by [39], there exists a quasitautological sheaf U over S K × M SK,H (v), which means that there exists an integer ρ such that for any F ∈ M SK,H (v) the restriction of U to S K × {F } is F ⊕ρ .Consider the algebraic class where pr : S K × M SK,H (v) → S K is the projection.By [41], its Künneth component By [10], the standard conjectures holds for M SK ,H (v) (this also follows from [8] via the arguments of [3]).Hence, all Künneth components of γ are algebraic.is induced by an algebraic cycle on S × K.The Kuga-Satake Hodge conjecture holds for K by [52].By Lemma 5.6, Conjecture 5.4 holds for the K3 surface S as well.
For the general projective variety K of Kum 3 -type, H 2 tr (K, Q) is a rank 6 Hodge structure with Hodge numbers (1, 4, 1) and the explicit knowledge of the quadratic form allows one to show that the Kuga Satake variety is A 4 , where A is an abelian fourfold of Weil type (cf.[16,Theorem 9.2]).O'Grady moreover shows in [43] that H 1 (A, Q) ∼ = H 3 (K, Q).This is a crucial ingredient in the subsequent work of Markman [30] and Voisin [52].O'Grady's theorem further allows us to apply a result of Varesco [49] to prove the Hodge conjecture for all powers of the K3 surfaces appearing in Theorem 1.3. .By [22], it is sufficient to prove the Hodge conjecture for all powers of S 0 .
To this end, it will be enough to check that S → B satisfies the two assumptions of [49,Theorem 0.2].As explained in [loc.cit., Theorem 4.1], the first of these assumptions is that the Kuga-Satake variety of a general K3 surface in the family is isogenous to A 4 for an abelian fourfold A of Weil type with trivial discriminant.This holds by O'Grady's theorem in [43] as already mentioned.The other assumption is that the Kuga-Satake Hodge conjecture holds for the surfaces S b for all b ∈ B, which is the content of Theorem 5.7.
Remark 5.9.-The conclusions of Theorem 5.7 and Corollary 5.8 were known for the two 4-dimensional families of K3 surfaces studied in [46] and [24].The general fibre has trascendental lattice U ⊕2 ⊕ −2 ⊕2 in the first case and U ⊕2 ⊕ −6 ⊕ −2 in the second.The reader may check that the K3 surfaces studied in [46] satisfy the assumption of our Theorem 5.7, while those appearing in [24] do not.
hence (ǫ, −1) ∈ G.If ǫ / ∈ A 2 , then W ǫ has empty intersection with A isolated points and the surfaces {(α, b, −b − ǫ, −α + ǫ), b ∈ A}, where α ∈ A satisfies α = −α − ǫ.The general point of such a surface is supported on 4 distinct point.Therefore, the fixed locus (K 3 (A)) h consists of surfaces and isolated points, because it is a union of symplectic varieties and the fibres of ν have dimension at most 3.We conclude that G is generated by the 16 involutions (τ, −1) with τ ∈ A 2 , and hence G = A 2 × −1 .

2. 5 .
Relevant subvarieties.-In what follows we let K = K 3 (A) be the generalized Kummer variety on an abelian surface A. We will calculate the fixed locus of automorphisms in G = A 2 × −1 acting on K.The next definition introduces the relevant subvarieties.We denote by ξ = (a, b, c, d) a point of A (4) 0 , and refer to {a, b, c, d} as the support of ξ.Definition 2.6.-We define the following subvarieties of A
10. Let now f : K → B be a smooth and proper family of Kum 3 -manifolds over a connected base B, with a fibre K 0 isomorphic to the generalized Kummer variety on an abelian surface A. By the result of Hassett-Tschinkel [19, Theorem 2.1], up to a finite étale base-change, there is a fibrewise action of G = A 2 × −1 on K.By [14, Lemma 3.10], this action is locally trivial with resepect to f , i. e. any x ∈ K

3. 2 .
Preliminaries. -Let C be a smooth projective curve of genus g.Its Picard variety Pic • (C) = d Pic d (C) is the group of line bundles modulo isomorphism.Tensor product gives multiplication maps m : Pic a (C) × Pic b (C) → Pic a+b (C).We denote by [n] : Pic d (C) → Pic nd (C) the n-th power map.The Jacobian of C is the abelian variety Pic 0 (C); for d = 0, the variety Pic d (C) is a torsor under Pic 0 (C).Assume now that π : C → C is an étale double cover, so that C is of genus 2g − 1.In this situation, for each integer d, we have: -the pull-back map π * : Pic d (C) → Pic 2d ( C); -the norm map Nm π : Pic d ( C) → Pic d (C); -the covering involution σ : C → C, inducing σ * : Pic d ( C) → Pic d ( C).The composition Nm π • π * : Pic d (C) → Pic 2d (C) coincides with multiplication by 2, while π * • Nm π : Pic d ( C) → Pic 2d ( C) is given by L → L ⊗ σ * (L).We refer to Mumford's paper[40] for the following.The étale double cover π is uniquely determined by a 2-torsion line bundle η on C, which is the only nontrivial element in ker(π * : Pic 0 (C) → Pic 0 ( C)).Moreover, the image of the pull-back map π * : Pic d (C) → Pic 2d ( C) is precisely the fixed locus of σ * .The involution σ * acts as the inverse L → L ∨ on the kernel of the norm map Nm π : Pic 0 ( C) → Pic 0 (C).This kernel has two connected components; the one containing the neutral element is the Prym variety P (π) of the cover, an abelian variety of dimension g − 1.The fibres of Nm π : Pic d ( C) → Pic d (C) are torsors under P (π) × Z/2Z.

π
: C → C over P 3 .We denote by D → |H| the universal family of curves over the linear system |H| on Km(J); clearly, C b = D b if and only if C b does not pass through any of the nodes of J/ ± 1.

3. 4 .
Beauville-Mukai systems.-Taking the Jacobian of the smooth curves in the families introduced above yields families of abelian varieties over a Zariski open subvariety of P 3 .The total space of these families can be compactified considering suitable moduli spaces of stable sheaves on J and Km(J).The construction leads to certain hyper-Kähler varieties equipped with Lagrangians fibrations, called Beauville-Mukai systems[5],[38].Consider the Mukai vector v d = (0, 2Θ, d − 4) on J, for an integer d.The moduli space M J (v d ) parametrizes pure dimension 1 sheaves on J which are push-forward of semistable and torsion free sheaves of rank 1 and degree d supported on curves algebraically equivalent to 2Θ.Mapping a sheaf to its support[27] gives a morphismM J (v d ) supp − −−− → Pic 0 (J) × P 3 ,We define the degree d relative compactified Jacobian as Pic d ( C/P 3 ) := supp −1 ({O J } × P 3 ).If b ∈ P 3 corresponds to a smooth curve in |2Θ|, the fibre over b of the support morphism supp : Pic d ( C/P 3 ) → P 3 is the degree d Picard variety of the curve.The Albanese morphism alb : Pic d ( C/P 3 ) → J is an isotrivial fibration with fibre K J (v d ), and it maps a sheaf F ∈ K J (v d ) to the sum c 2 (F ) ∈ J, see [53, §6].A similar construction can be done for the family of curves D on Km(J).Given an integer d, we consider the Mukai vector w d = (0, H, d − 2) on Km(J).The degree d relative compactified Jacobian of curves in |H| is the moduli space Pic d (D/P 3 ) := M Km(J) (w d ),

Lemma 3 . 6 .
-For each d, the restriction Nm π : K J (v d ) Pic d (D/P 3 ) is dominant and generically finite of degree 2 5 .Proof.-For each k, the k-th power map [k] : Pic d ( C/P 3 ) Pic kd ( C/P 3 ) restricts to a rational map [k] : K J (v d ) K J (v kd ).This can be easily checked at smooth curves Cb in the family using the description of the Albanese map L → c 2 (L).

Proposition 3 . 8 .
-Let d be odd.Then G = Pic 0 (J) 2 × −1 , where −1 acts on K J (v d ) via the pull-back F → (−1) * (F ) of sheaves, and the action of L ∈ Pic 0 (J) 2 is given by F → F ⊗ L. Any element of G preserves the fibres of the support fibration.

4. 3 .
The symplectic resolution of K/G.-We will now conclude the proof of Theorem 1.1 in several steps.First we show that the blow-up of the singular locus of K/G resolves the singularities.Proposition 4.4.-Let Y K := Bl X 1 (K/G) be the blow-up of the singular locus with reduced structure.Then Y K is a smooth manifold.It is identified with the quotient by G of the blow-up of K at Z = 1 =g∈G K g .

A 1 -
singularity of a surface.Proof of Proposition 4.4.-By Proposition 4.2, any point x ∈ K/G has a neighborhood of the form (C 2 /ι) j ×(C 2 ) k .Therefore, Lemma 4.5 immediately implies that the blow-up Y K of the singular locus is smooth.It also implies that the natural birational map Bl Z (K)/G Y K extends to an isomorphism.

Proof of Theorem 1 . 1 . 4 and 4 . 6 ,
-Let K → B be a smooth proper family of manifolds of Kum 3 -type.Up to a finite étale base-change, G acts fibrewise on K, and the fixed loci of automorphisms in G are smooth families over B. By Theorem 2.2, the union Z of K g for 1 = g ∈ G has 16 components Z i , each of which is a smooth family of manifolds of K3 [2] -type over B. The blow-up of K along Z is a smooth family K over B, and the action of G extends to a fibrewise action on K.By Propositions 4.the quotient Y = K/G is a smooth proper family of hyper-Kähler manifolds over B, with fibre over the point b ∈ B the manifold Y K b .It thus suffices to find a single K of Kum 3 Mukai vector v and a v-generic polarization H on S K .The surface S K is uniquely determined up to isomorphism, because two K3 surfaces of Picard rank at least 12 with Hodge isometric transcendental lattices are isomorphic, see [21, Chapter 16, Corollary 3.8] Remark 4.9.-The K3 surfaces S K come in countably many 4-dimensional families.A projective K 0 of Kum 3 -type with Picard rank 1 gives such a family, consisting of the K3 surfaces S such that H 2 tr (S, Z) is a sublattice of H 2 tr (S K0 , Z).Up to isogeny, the K3 surfaces obtained are easily characterized as follows.Recall that Λ Kum 3 = U ⊕3 ⊕ −8 is the lattice H 2 (K, Z) for K of Kum 3 -type.Lemma 4.10.-Let S be a projective K3 surface.The following are equivalent:

Conjecture 5 . 4 (
Kuga-Satake Hodge conjecture).-Let X be a projective hyper-Kähler variety.There exists an algebraic cycle ζ on X × KS(X) × KS(X) such that the associated correspondence induces an embedding of Hodge structuresζ * : H 2 tr (X) ֒→ H 2 (KS(X) × KS(X)).Remark 5.5.-For a K3 surface X, the above form of the conjecture is equivalent to [16, §10.2].To see this, let Mot be the category of Grothendieck motives over C, in which morphisms are given by algebraic cycles modulo homological equivalence, see[2].The motive h(X) ∈ Mot of X decomposes as the sum of its transcendental part h tr (X) and some motives of Hodge-Tate type.Since the standard conjectures hold for X and KS(X), the tensor subcategory of Mot generated by their motives is abelian and semisimple [3, Theorem 4.1].Therefore, Conjecture 5.4 and the formulation of van Geemen [16, §10.2] are both equivalent to h 2 tr (X) being a direct summand of h 2 (KS(X) × KS(X)) in the category Mot.We will use the following easy lemma.Lemma 5.6.-Let X and Z be projective hyper-Kähler varieties.Assume that there exists an algebraic cycle γ on Z × X which induces a rational Hodge isometryγ * : H 2 tr (Z) ∼ −−→ H 2 tr (X)(k),for some non-zero k ∈ Q.Then, if Conjecture 5.4 holds for X, it holds for Z as well.Proof.-ByRemark 5.3 there exists an isogeny φ : KS(X) → KS(Z) of Kuga-Satake varieties.It induces an isomorphismφ * : H 2 (KS(X) × KS(X)) ∼ − − → H 2 (KS(Z) × KS(Z))of rational Hodge structures.If Conjecture 5.4 holds for X, there exists an algebraic cycle ζ on X × KS(X) × KS(X) giving an embedding of Hodge structuresζ * : H 2 tr (X) ֒→ H 2 (KS(X) × KS(X)).It follows that the embedding of Hodge structure given by the compositionφ * • ζ * • γ * : H 2 tr (Z) ֒→ H 2 (KS(Z) × KS(Z))is induced by an algebraic cylce on Z × KS(Z) × KS(Z).For convenience of the reader we restate Theorem 1.3.Theorem 5.7.-Let S be a projective K3 surface such that there exists an isometric embedding of H 2 tr (S) into Λ Kum 3 (2) ⊗ Z Q.Then Conjecture 5.4 holds for S.

Corollary 5 . 8 .
-Let S be a projective K3 surface such that there exists an isometric embedding of H 2 tr (S) into Λ Kum 3 (2) ⊗ Z Q.Then the Hodge conjecture holds for all powers of S. Proof.-By Remark 4.9, there exists a 4-dimensional family S → B of projective K3 surfaces of general Picard rank 16 such that: for each b ∈ B the fibre S b is of the form S K b for some K b of Kum 3 -type and for some 0 ∈ B there exists a rational Hodge isometry H 2 tr (S) ∼ − − → H 2 tr (S 0 ) 1 and 1.2.Finally, in Section 5 we use our construction to prove Theorems 1.3 and 1.4.Aknowledgements.-It is a pleasure to thank Ben Moonen, Kieran O'Grady, Claudio Pedrini and Stefan Schreieder for their interest in this project.I am grateful to Lie Fu and Giovanni Mongardi for their comments on a first version of this article, and to Grégoire Menet for useful discussions.I thank Mauro Varesco for suggesting to me Corollary 5.8.I thank Giacomo Mezzedimi and Domenico Valloni for many and only if θ ′ ∈ {θ, θ + τ }, which can be seen considering a point (a, a + τ, a, a + τ ): it belongs to V τ,θ if and only if a ∈ A 2,θ or a ∈ A 2,θ+τ .
be a morphism of Hodge structures.
Let now f : M SK ,H (v) Y K be a birational map.Then the pushforward f H 2 tr (Y K ) is a Hodge isometry.By Lemma 4.8, a multiple of the composition r * • f * gives a Hodge isometry t 2 : H 2 tr (M SK,H (v)) * : H 2 tr (M SK ,H (v)) ∼ − − →