DEFORMATION PRINCIPLE AND ANDRÉ MOTIVES OF PROJECTIVEHYPERKÄHLER MANIFOLDS

ANDREY SOLDATENKOV

Abstract. Let X1 and X2 be deformation equivalent projective hyperkähler manifolds. We prove thatthe André motive of X1 is abelian if and only if the André motive of X2 is abelian. Applying this tomanifolds of K3[n], generalized Kummer and OG6 deformation types, we deduce that their André motivesare abelian. As a consequence, we prove that all Hodge classes in arbitrary degree on such manifolds areabsolute. We discuss applications to the Mumford-Tate conjecture, showing in particular that it holdsfor even degree cohomology of such manifolds.

Contents

1. Introduction 21.1. André motives and hyperkähler manifolds 21.2. Absolute Hodge classes 31.3. The Mumford-Tate conjecture 41.4. Organization of the paper 52. Motivated cohomology classes and André motives 52.1. Motivated cohomology classes 52.2. André motives 63. Hyperkähler manifolds and the Kuga-Satake embedding 63.1. Hyperkähler manifolds 73.2. The Kuga-Satake construction 73.3. The Kuga-Satake construction in families 74. The manifolds of K3[n], generalized Kummer and OG6 deformation types 94.1. Hilbert schemes of points on K3 surfaces and generalized Kummer varieties 94.2. Manifolds of OG6 deformation type 105. Deformation principle 105.1. The setting 105.2. The statement and preliminary constructions 115.3. Proof of Proposition 5.1 146. Motives of hyperkähler manifolds 146.1. Constructing smooth families of hyperkähler manifolds 146.2. Proof of theorem 1.1 16Acknowledgements 17References 17

Date: January 8, 2021.2010 Mathematics Subject Classification. primary 14C30, 14J32; secondary 14F42.

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1. Introduction

In this paper, we study André motives of projective hyperkähler manifolds. By a hyperkähler manifoldwe mean a compact simply connected Kähler manifold X such that H0(X,Ω2

X) is spanned by a symplecticform. We generalize the results of [2] and [29], showing that for most of the known deformation types ofhyperkähler manifolds their André motives are abelian.

1.1. André motives and hyperkähler manifolds. André motives were introduced in [1] as a refine-ment of Deligne’s motives [10]. They form a semi-simple Tannakian category, whose construction webriefly recall in section 2. The motives of abelian varieties generate a full Tannakian subcategory whoseobjects are called abelian motives. The theory of André motives has found applications to the study ofvarious arithmetic and Hodge-theoretic questions about algebraic varieties. Recall the theorem of Deligne[10] stating that any Hodge cohomology class on an abelian variety is absolute Hodge. More generally, itwas shown in [1] that for a projective manifold X whose André motive is abelian, all Hodge classes on Xare absolute. This shows that one part of the Hodge conjecture holds for varieties with abelian motives.Another application is related to the Mumford-Tate conjecture, which predicts a relation between theMumford-Tate groups and the Galois group action on the cohomology of a projective variety. We reviewthe absolute Hodge classes and the Mumford-Tate conjecture in more detail in sections 1.2 and 1.3 below.We recommend [25] for a general overview of the recent developments in this area.

The main new tool used in this paper is the generalized Kuga-Satake construction for hyperkählermanifolds, which was introduced in [21]. For any projective hyperkähler manifold X, this constructiongives an embedding of the cohomology groups of X into the cohomology of an abelian variety, whichrespects the Hodge structures. Therefore, our main goal is to prove that the Kuga-Satake embedding liftsto the category of André motives. To do this, we need to show that the cohomology class defining theembedding is motivated in the sense of [1], see also section 2.

Our approach is based on the deformation principle for motivated cohomology classes [1, Theorem 0.5].More precisely, assume that

(1.1) π : X → B

is a smooth projective morphism, B a connected quasi-projective variety and the fibres of π are hyperkählermanifolds. Assume that for some point b0 ∈ B the André motive of Xb0 = π−1(b0) is abelian. This impliesthat the Kuga-Satake embedding for Xb0 is motivated, and the deformation principle [1, Theorem 0.5]implies that it is motivated for any fibre Xb1 , b1 ∈ B. Therefore, it suffices to prove that there is onehyperkähler manifold in each deformation class whose motive is abelian. This is usually possible to dousing some explicit geometric construction. For example, in the case of K3 surfaces one can assume thatXb0 is a Kummer surface. Other deformation types of hyperkähler manifolds are discussed in section 4.

The approach outlined above has one subtlety. Namely, assume that X1 and X2 are deformation equiv-alent projective hyperkähler manifolds. In the moduli space of all hyperkähler manifolds the projectiveones are parametrized by a countable collection of divisors. We show in section 6 that it is possible torealize X1 and X2 as fibres of a smooth family as in (1.1), but it is a priori not clear if one can make themorphism π projective. To resolve this issue, we prove in section 5 a generalization of the deformationprinciple, which applies to the case when all fibres of π in (1.1) are projective, but the morphism π isprojective only over a dense Zariski-open subset of the base. Our main result is the following statement.

Theorem 1.1. Let X1 and X2 be deformation equivalent projective hyperkähler manifolds. The Andrémotive of X1 is abelian if and only if the André motive of X2 is abelian.

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The proof is given in section 6.2.We apply this theorem to several known deformation types of hyperkähler manifolds. We leave out

only the OG10 type, which has recently been treated by Floccari, Fu and Zhang in [12].

Corollary 1.2. Let X be a projective hyperkähler manifold of K3[n], generalized Kummer, or OG6 de-formation type. Then the André motive of X is abelian.

Proof. By Theorem 1.1, it suffices to find one manifold with abelian motive in each deformation class. Forthe Hilbert schemes of points on K3 surfaces and generalized Kummer varieties, the motives are abelianby [6], [7] and [36]. For OG6 deformation type, one can find a manifold with abelian motive using theconstruction from [24]. We recall all these constructions in section 4.

The above Corollary recovers the results of [1], [2] and [29], where the case of K3 surfaces and, moregenerally, K3[n]-type varieties has been considered. Unlike [29], we do not use the results of Markman onthe structure of the cohomology ring, which are specific for K3[n]-type varieties. The approach using theKuga-Satake construction is more general, and therefore allows us to treat other deformation types.

Let us also mention that a substantial amount of recent research has been devoted to the study of Chowmotives of hyperkähler manifolds and related questions, see e.g. [13] and references therein. Proving thatChow motives of hyperkähler manifolds are abelian seems to be a more difficult problem, and the methodsof the present paper are not sufficiently strong to deal with it.

Let us next review the applications of our results to the absolute Hodge classes and the Mumford-Tateconjecture.

1.2. Absolute Hodge classes. Let X be a non-singular projective variety over C. Denote by Xan thecorresponding complex manifold. Recall that the de Rham cohomology H•dR(X) is the hypercohomologyof the algebraic de Rham complex Ω•X/C. For every k, the C-vector space Hk

dR(X) is endowed with adecreasing Hodge filtration F •. On the other hand, the singular cohomology Hk(Xan,C) is endowedwith the Q-structure given by the subspace Hk(Xan,Q). Comparison results between the algebraicand the analytic cohomology of coherent sheaves and the quasi-isomorphism Ω•Xan ' C induce naturalisomorphisms of the cohomology groups Hk

dR(X) ' Hk(Xan,C).Recall that an element α ∈ F pH2p

dR(X) is called a Hodge class, if its image in H2p(Xan,C) under theisomorphism described above is contained in the subspace (2πi)pH2p(Xan,Q). The Hodge conjectureimplies that the property of α being Hodge should be stable under automorphisms of the field of complexnumbers. More precisely, let σ ∈ Aut(C/Q) be an automorphism and Xσ be the variety obtained by basechange via σ. We have a chain of isomorphisms of Q-vector spaces:

(1.2) HkdR(X) ' Hk

dR(X)⊗C,σ C ' HkdR(Xσ) ' Hk(Xan

σ ,C).

A cohomology class α ∈ F pH2pdR(X) is called absolute Hodge, if its image under the isomorphism (1.2)

lies in (2πi)pH2p(Xanσ ,Q) for any σ ∈ Aut(C/Q). If α is an algebraic class, i.e. it is contained it the

Q-subspace spanned by the classed of algebraic subvarieties, then it is absolute Hodge. According to theHodge conjecture, every Hodge class should be algebraic, therefore absolute Hodge.

According to Deligne [10], any Hodge class on an abelian variety is absolute. Using the results of [1]and Theorem 1.1, we deduce the following statement.

Corollary 1.3. Let X be a projective hyperkähler manifold of K3[n], generalized Kummer, or OG6 de-formation type. Then all Hodge classes on X are absolute.

Proof. By Corollary 1.2, the André motive of X is abelian, and we can apply [1, section 6].

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1.3. The Mumford-Tate conjecture. The purpose of this section is to explain how the results ofFloccari [11] combined with Theorem 1.1 lead to the proof of some cases of the Mumford-Tate conjecturefor hyperkähler manifolds.

Assume that X is a non-singular projective variety defined over a subfield k ⊂ C finitely generated overQ. Recall the comparison isomorphism between the `-adic and singular cohomologies of X:

(1.3) Hket(Xk,Q`) ' Hk(Xan,Q)⊗Q`.

Let us briefly denote by Hk` this Q`-vector space.

The left-hand side of (1.3) is a representation of the Galois group Gal(k/k). Denote by Gk` the Zariskiclosure of the image of Gal(k/k) in GL(Hk

` ). Let Gk,` be the connected component of the identity in Gk` .The right-hand side of (1.3) is naturally a representation of the Mumford-Tate group MTk(X)⊗Q`. TheMumford-Tate conjecture predicts that these two subgroups of GL(Hk

` ) are equal:

(1.4) Gk,` = MTk(X)⊗Q`.

This conjecture has been a subject of an active research. For an overview of the recent developments,see [25]. There is a number of recent works on the Mumford-Tate conjecture that use methods similarto ours. In [2], the Mumford-Tate conjecture in degree 2 was proven for hyperkähler manifolds. In [26]this result was generalized to a wider class of varieties with h2,0 = 1; the proof relies on the Kuga-Satake construction. In [5], it was shown that for varieties with abelian André motive the validity of theMumford-Tate conjecture does not depend on `. In [11], the Mumford-Tate conjecture in arbitrary degreefor K3[n]-type varieties was proven; the method of the proof relies on the results of Markman about thestructure of the cohomology algebra, similarly to [29].

Let us remark, that at present the Mumford-Tate conjecture is not known even for general abelianvarieties. On the other hand, we know from [2] that it holds in degree 2 for any hyperkähler manifold.Using Theorem 1.1 and the results of [11], we can deduce the Mumford-Tate conjecture in all even degreesfor the same type of varieties as in Corollary 1.2.

One special type of abelian varieties for which the Mumford-Tate conjecture is known are the varietiesof CM type. In this case, one can use the results of [33], see also [25, Theorem 3.3.2 and Corollary4.3.15]. We deduce analogous results for hyperkähler manifolds. We will say that a projective hyperkählermanifold X is of CM type, if the Mumford-Tate group of H2(X,Q) is abelian. In this case the Mumford-Tate groups of Hk(X,Q) are abelian for all k. This follows from the fact that the Hodge structures on allcohomology groups of X are induced by the natural Lie algebra action, as we recall in section 3.

We summarize our discussion in the following statement. The idea of its proof is entirely due to Floccari,and we follow his arguments from [11]. The only ingredient that was missing in [11] is the deformationprinciple, Theorem 1.1. In the case of K3[n]-type it was obtained in [11] independently.

Corollary 1.4. Let X be a projective hyperkähler manifold of K3[n], generalized Kummer, or OG6 de-formation type. Then the Mumford-Tate conjecture holds for the cohomology of X in all even degrees. IfX is moreover of CM type, then the Mumford-Tate conjecture holds for the cohomology in all degrees.

Proof. Let H+` = ⊕kH2k

` (Xan,Q`) and H2` = H2(Xan,Q`). Denote by G+,

` ⊂ GL(H+` ) and G2,

` ⊂GL(H2

` ) the connected algebraic groups obtained from the Galois representations as explained above. LetMT+ ⊂ GL(H+

` ) and MT2 ⊂ GL(H2` ) be the Mumford-Tate groups. Since the André motive of X is

abelian, by [25, formula (3.3)] we have an inclusion G+,` ⊂ MT+. By the work of André [2], the analogous

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inclusion in degree two is an isomorphism G2,` ' MT2. We have the following diagram of groups

G+,` MT+

G2,` MT2

'

'

where the vertical arrows are induced by the projection H+` → H2

` . The arrow on the right is anisomorphism because the Hodge structures on the cohomology of X are induced by the action of theorthogonal Lie algebra gtot (see section 3 and [22, (1.7)]).

It follows from the diagram above that the upper arrow is also surjective, which proves the first partof the corollary. The second part follows from [25, Theorem 3.3.2], see also [25, Corollary 4.3.15].

1.4. Organization of the paper. In section 2, we discuss the notions of a motivated cohomology classand André motive. We recall all necessary definitions and constructions from [1]. In section 3, we recallbasic results about the cohomology of hyperkähler manifolds and the Kuga-Satake construction from [21].The main results of this section are Proposition 3.2 and Corollary 3.3, which generalize the results of [21]to the relative setting. In section 4, we recall some known results about the Hilbert schemes of points,generalized Kummer varieties and O’Grady’s 6-dimensional varieties. We explain that in each of thesedeformation types one can find a variety with abelian motive. In section 5, we prove a generalization ofthe deformation principle for motivated cohomology classes, Proposition 5.1. In section 6, we discuss theconstruction of families of hyperkähler manifolds and in 6.2 prove the main result, Theorem 1.1.

2. Motivated cohomology classes and André motives

In this section, we briefly recall some results of [1]. Let (Var/C) be the category of non-singularcomplex projective varieties and their morphisms. For a variety X ∈ (Var/C), we will denote by Hk(X)the singular cohomology group Hk(Xan,C). Let (GrAlg/Q) be the the category of finite-dimensionalgraded Q-algebras.

2.1. Motivated cohomology classes. Assume that X ∈ (Var/C) and let L be an ample line bundleon X. Denote by h ∈ H2(X) the first Chern class of L. The Lefschetz operator Lh ∈ End(H•(X)) isdefined as the cup product with h, and it induces isomorphisms Lkh : Hn−k(X) ∼→ Hn+k(X) for everyk = 0, . . . , n, where n = dimC(X). The subspace of primitive elements Hk

pr(X) ⊂ Hk(X), k = 0, . . . , n, isby definition the kernel of Ln−k+1

h .We will denote by ∗h ∈ End(H•(X)) the Lefschetz involution associated with h. Recall that for

x ∈ Hkpr(X) and i = 0, . . . , n− k we have ∗h(Lihx) = Ln−k−ih x. This uniquely determines ∗h, since H•(X)

is spanned by the elements of the form Lihx with x primitive.For X,Y ∈ (Var/C) and two ample line bundles L1 ∈ Pic(X), L2 ∈ Pic(Y ), let pX , pY denote the two

projections from X × Y , and let h = c1(p∗XL1 ⊗ p∗Y L2) ∈ H2(X × Y ). For any two classes of algebraiccycles α, β ∈ H•alg(X × Y ), consider the class

(2.1) pX∗(α ∪ ∗hβ).

Let H•M (X) be the Q-subspace of H•(X) spanned by the classes (2.1) for all Y , L1, L2, α, β as above.Elements of H•M (X) will be called motivated cohomology classes (or motivated cycles, in the terminologyof [1]). Let us list a few properties of the motivated classes.

(1) H•M (X) is a graded Q-subalgebra of H•(X);5

(2) For f : X → Y , we have f∗H•M (Y ) ⊂ H•M (X), hence we have a functor

H•M : (Var/C)op → (GrAlg/Q);

(3) For f as above, we have f∗H•M (X) ⊂ H•−dim(X)M (Y );

(4) All classes in H•M (X) are absolute Hodge;(5) The Künneth components of the diagonal are contained in H•M (X ×X) for any X ∈ (Var/C).

Let us also recall the following deformation principle:

Theorem 2.1 ([1, Theorem 0.5]). Let π : X → B be a smooth projective morphism. Assume that thebase B is a connected quasi-projective variety. Let ν ∈ Γ(B,R2pπ∗C). Assume that for some b0 ∈ B theelement νb0 ∈ H2p(Xb0) is a motivated cohomology class. Then νb1 is a motivated class on Xb1 for anyb1 ∈ B.

We will generalize the deformation principle in section 5, relaxing the condition of projectivity for themorphism π, see Proposition 5.1.

2.2. André motives. The construction of André motives is similar to that of Chow motives. Westart by defining the spaces of motivated correspondences: for X,Y ∈ (Var/C) with X connected, letCorkM (X,Y ) = H

k+dim(X)M (X × Y ). The properties of motivated classes listed above imply that we can

define the composition of motivated correspondences in the usual way.Define a Q-linear category (MotA) whose objects are triples (X, p, n), where X ∈ (Var/C), p ∈

Cor0M (X,X), p p = p, and n ∈ Z. The space of morphisms from (X, p, n) to (Y, q,m) is by defini-

tion q Corm−nM (X,Y ) p ⊂ Corm−nM (X,Y ). The tensor product on (MotA) is defined using the Cartesianproduct of varieties. It is shown in [1], that (MotA) is a semi-simple graded neutral Tannakian category.We will denote the André motives by boldface letters: for X ∈ (Var/C) define

M(X)(n) = (X, [∆X ], n) ∈ (MotA),

where ∆X denotes the diagonal in X×X. More generally, since the Künneth components of the diagonalare motivated, we can define the motives representing the cohomology groups of X:

Hk(X)(n) = (X, δk, n) ∈ (MotA),

where δk is the k-th Künneth component of [∆X ]. Therefore, we have the direct sum decompositionM(X)(n) = ⊕kHk(X)(n).

The subcategory of abelian motives (MotabA ) is the minimal full Tannakian subcategory of (MotA)

containing M(A)(n) for all abelian varieties A and n ∈ Z. It is shown in [1] that for any X ∈ (Var/C), ifM(X) ∈ (Motab

A ), then all Hodge classes on X are motivated, in particular absolute Hodge.We will use the well-known fact that the class of varieties with abelian motives is stable under two basic

operations. Namely, let X,Y ∈ (Var/C) and let f : X → Y be a surjective generically finite morphism.Then f∗ : M(Y ) → M(X) is injective, and M(X) ∈ (Motab

A ) implies M(Y ) ∈ (MotabA ). The second

basic operation is the blow-up: if i : X → Y is a closed immersion and M(X),M(Y ) ∈ (MotabA ), then

M(BlXY ) ∈ (MotabA ). As an example, consider a complex abelian surface A and denote by A[2] its 2-torsion

points. Then S = (BlA[2]A)/± 1 is the corresponding Kummer K3 surface. We see that M(S) ∈ (MotabA ).

3. Hyperkähler manifolds and the Kuga-Satake embedding

In this section, we recall basic properties of hyperkähler manifolds. We refer to [32], [17], [3] for moredetails. We also recall the results of [21] and extend them to the relative setting.

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3.1. Hyperkähler manifolds. In this paper, a hyperkähler manifold is a compact simply connectedKähler manifold X such that H0(X,Ω2

X) is spanned by a symplectic form. The dimension of any sym-plectic manifold is even; we let dimC(X) = 2n.

Let VZ = H2(X,Z) and V = VZ ⊗ Q. Let q ∈ S2V ∗ denote the Beauville-Bogomolov-Fujiki (BBF)form. Recall that this form has the following property: there exists a constant cX ∈ Q such that for allh ∈ H2(X,Q) the equality q(h)n = cXh

2n holds. We can assume that q is integral and primitive on VZ,and q(h) > 0 for a Kähler class h. The signature of q is (3, b2(X)− 3). Recall also that V carries a Hodgestructure of K3 type.

There exists a natural action of the orthogonal Lie algebra on the total cohomology of X. Let us brieflyrecall how to define this action. An element h ∈ V has Lefschetz property, if the cup product with hk

induces an isomorphism H2n−k(X,Q) ' H2n+k(X,Q) for all k = 0, . . . , 2n. In this case one can considerthe corresponding Lefschetz sl2-triple. Let gtot ⊂ End(H•(X,Q)) be the Lie subalgebra generated by allsuch sl2-triples.

Let V = 〈e0〉 ⊕ V ⊕ 〈e4〉 be the graded Q-vector space with ei of degree i, and V in degree 2. Letq ∈ S2V ∗ be the quadratic form such that: q|V = q, e0 and e4 are isotropic and orthogonal to V and spana hyperbolic plane. It was shown in [34] and [22] that there exists an isomorphism of graded Lie algebrasgtot ' so(V , q).

One can show that Hodge structures on the cohomology groups of X are induced by the action of gtot.More precisely, let W be the Weil operator that induces the Hodge structure on V , i.e. it acts on V p,q

as multiplication by i(p − q). Then W ∈ so(V, q) ⊂ so(V , q) and W induces the Hodge structures onHk(X,Q) for all k (see [34]).

3.2. The Kuga-Satake construction. To the K3 type Hodge structure V we can associate a Hodgestructure of abelian type, which is called the Kuga-Satake Hodge structure. Let us briefly recall theconstruction. Let H = Cl(V, q) be the Clifford algebra. There exists a natural embedding V → H. DefineH0,−1 to be the right ideal V 2,0 ·HC (see [30, Lemma 3.3]), and H−1,0 = H0,−1. This defines a rationalHodge structure on H.

Note that H is canonically an so(V, q)-module, and the Hodge structure on it is induced by the actionof the Weil operator W (see e.g. [30]). Since the hyperkähler manifold X is projective, one can show thatH is polarizable. Moreover, the polarization can be chosen Spin(V, q)-invariant (see e.g. [21] or [14]).

Let d = 14m dimQ(H). The following theorem was proven in [21].

Theorem 3.1. There exists a structure of graded so(V , q)-module on Λ•H∗ that extends the so(V, q)-module structure. For some m > 0 there exists an embedding of so(V , q)-modules

H•+2n(X,Q) → Λ•+2d(H∗⊕m).

This induces embeddings of Hodge structures

νi : Hi+2n(X,Q(n)) → Λi+2d(H∗⊕m)(d),

where i = −2n, . . . , 2n.

3.3. The Kuga-Satake construction in families. Let us consider a smooth projective morphismϕ : X → B whose fibres are hyperkähler manifolds. Let us fix a base point b0 ∈ B and denote by X

the fibre Xb0 . We would like to apply Theorem 3.1 to the family ϕ and obtain an embedding of the corre-sponding variations of Hodge structures. In order to do so, we need to construct a family of Kuga-Satakeabelian varieties such that the embeddings νi from Theorem 3.1 are π1(B, b0)-equivariant. We will explain

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below, that it is possible to do this after we pass to a finite étale covering of B. The base B can be anarbitrary complex-analytic space.

Denote by AutP (X) ⊂ GL(H•(X,Q)) the subgroup of algebra automorphisms that fix the Pontryaginclasses of X. We will denote by AutP (X)Z the arithmetic subgroup of AutP (X)Q that consists of allelements preserving the integral cohomology lattice.

Recall that so(V, q) acts on H•(X,Q) by derivations, and this action induces a homomorphism ofalgebraic groups

α : Spin(V, q)→ AutP (X),

see section 3.1 in [31] and references therein. Denote by Aut+(X) the image of α. Let MC(X) =Diff(X)/Diff(X) be the mapping class group of X. Here Diff(X) is the group of diffeomorphisms of X,and Diff(X) is the subgroup of diffeomorphisms isotopic to the identity. The mapping class group actson the cohomology of X fixing Pontryagin classes, hence a group homomorphism

β : MC(X)→ AutP (X)Z.

The following proposition generalizes [31, Proposition 3.5]. Let us mention that an analogous resultwas independently obtained in [16, Theorem 4.16].

Proposition 3.2. There exists a subgroup of finite index MC′ ⊂ MC(X) and a group homomorphismβ′ : MC′ → Spin(V, q)Q such that α β′ = β and the image of β′ is contained in an arithmetic subgroupof Spin(V, q)Q.

Proof. Recall that the action of AutP (X) on V preserves the form q, thus we have a homomorphismAutP (X)→ O(V, q), see [35, Theorem 3.5(i)]. Let Γ ⊂ AutP (X)Z be a torsion-free arithmetic subgroup.Since the induced homomorphism Γ → O(V, q)Q has finite kernel by [35, Theorem 3.5(iv)], it is actuallyinjective. Analogously, let Γ′ ⊂ Spin(V, q)Q be another torsion-free arithmetic subgroup. Then the actionof Spin(V, q) on V induces a homomorphism Γ′ → O(V, q)Q. This homomorphism is injective because Γ′

is torsion-free and the kernel of the map Spin(V, q)Q → O(V, q)Q is finite.The construction of MC′ is summarized in the following commutative diagram:

MC′ MC′′ MC(X)

Γ ∩ Γ′ Γ AutP (X)Z

Γ′ O(V, q)Q

β

Here Γ ∩ Γ′ is of finite index in Γ, because both Γ and Γ′ are arithmetic subgroups of O(V, q)Q. By thedefinition of Γ, it has finite index in AutP (X)Z. We then let MC′ = β−1(Γ ∩ Γ′), and define β′ to be thecomposition of the two maps in the left column of the diagram and the embedding Γ′ ⊂ Spin(V, q)Q.

Corollary 3.3. Let ϕ : X → B be a smooth family of hyperkähler manifolds with B connected, b0 ∈ B abase point, X the fibre of ϕ over b0, and 2n = dimC(X). Let V = H2(X,Q) and denote by q the BBFform on V . Then the following statements hold:

(1) There exists a finite étale covering B′ → B such that the action of π1(B′, b′0) on H•(X,Q) factorsthrough a homomorphism ρ : π1(B′, b′0)→ Spin(V, q)Q;

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(2) There exists a smooth family of compact complex tori ψ : A → B′ and an embedding of the varia-tions of Hodge structures

νi : Ri+2nϕ′∗Q(n) → Ri+2dψ∗Q(d),

where i = −2n, . . . , 2n, 2d = dimC(Ab′0) and ϕ′ : X ′ = X ×B B′ → B′;(3) Assume that there exists a monodromy-invariant cohomology class h ∈ H2(X,Q)π1(B,b0) such that

q(h) > 0. Then one can find B′ as above such that ψ is a projective family of abelian varieties.

Proof. (1) The monodromy action on the cohomology is induced by a homomorphism ρ : π1(B, b0) →MC(X). Using the subgroup MC′ ⊂ MC(X) from Proposition 3.2, we get a finite index subgroupρ−1(MC′) ⊂ π1(B, b0) and the corresponding covering B′. It satisfies the required properties by thedefinition of MC′.

(2) Denote by B the universal covering of B. Let H be the variation of the Kuga-Satake Hodgestructures over B that can be fibrewise described as in section 3.2. The fact that the fibrewise constructionindeed defines a variation of Hodge structures was proved e.g. in [30, section 3.2, in particular Proposition3.6]. By Proposition 3.2, the image of π1(B′, b′0) under ρ is contained in an arithmetic subgroup ofSpin(V, q)Q. This implies that the action of the fundamental group of B′ preserves some lattice Λ ⊂ H.Taking the quotient of H/Λ by the action of π1(B′, b′0), we get a family of complex tori ψ : A → B′. Theembeddings of Hodge structures νi from Theorem 3.1 are monodromy-equivariant. This implies that theyinduce the embeddings νi of the corresponding variations of Hodge structures over B′, see [31, Proposition3.7].

(3) Under our assumptions, the Kuga-Satake Hodge structures admit a Spin(V, q)-invariant polariza-tion, see e.g [21]. We can therefore fix a polarization type for the Kuga-Satake abelian varieties. Choosingappropriate finite étale cover B′ → B, we obtain a map from B′ to an arithmetic quotient of the Siegelhalf-space Γ\H2d. For a suitable choice of Γ, there exists a projective universal family of abelian varietiesover Γ\H2d, see e.g. [28, Theorem 8.11]. We can then construct the family ψ as the pull-back of theuniversal family.

4. The manifolds of K3[n], generalized Kummer and OG6 deformation types

In this section, we recall the results of [6], [7], [36] and [24]. They provide the necessary geometricinput for the proof of Corollaries 1.2, 1.3 and 1.4, showing that in each of the K3[n], generalized Kummerand OG6 deformation types there exists at least one variety with abelian motive.

4.1. Hilbert schemes of points on K3 surfaces and generalized Kummer varieties. Let S be anon-singular complex projective surface. We will denote by S[n] the Hilbert scheme of length n subschemesof S and by S(n) the n-th symmetric power of S. Recall that S[n] is non-singular and there exists a Hilbert-Chow morphism χ : S[n] → S(n).

In [6], the natural stratification of S(n) was used to describe the motive of S[n]. We briefly recall theconstruction. Let ν = (ν1, . . . , νn) denote a partition of n, so that n = ν1 + 2ν2 + . . . + nνn and νi > 0.Let l(ν) =

∑i νi and denote by S(ν) the product

∏i S

(νi). Recall that the points of S(νi) are the 0-cyclesof length νi in S. Consider the morphism S(ν) → S(n) that sends a collection of cycles (x1, . . . , xn) to thecycle x1 + 2x2 + . . .+ nxn. Let Zν denote the product S(ν) ×S(n) S[n] with the reduced scheme structure.

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We get a commutative square:

Zν S[n]

S(ν) S(n)

qν

pν χ

By construction, dim(S(ν)) = 2l(ν) and one can show that dim(Zν) = n+l(ν). Note that the symmetricpowers of S have quotient singularities. Hence all the natural operations on Chow groups with rationalcoefficients are well-defined. Consider the morphisms qν∗p∗ν : CHk

Q(S(ν))→ CHk+n−l(ν)Q (S[n]). It is shown

in [6, Theorem 5.4.1] that the sum of these morphisms gives an isomorphism of Chow groups

CH•Q(S[n]) ' ⊕νCH•−n+l(ν)Q (S(ν)),

where the direct sum is taken over all partitions of n. One deduces from this an isomorphism of motives

M(S[n]) ' ⊕νM(S(ν))(n− l(ν)),

which actually holds on the level of Chow motives with rational coefficients, cf. [6, Theorem 6.2.1]. Themotive of S(ν) is by definition a submotive of ⊗iM(Sνi). The latter is abelian if the motive of S is abelian.It is shown in [1, Theorem 7.1] that for any complex projective K3 surface S the motive of S is abelian,hence M(S[n]) ∈ (Motab

A ) .Analogous arguments show that the motive of a generalized Kummer variety is abelian. Namely, let A

be a complex abelian surface. Consider the Albanese morphism a : A[n+1] → A which sends an (n+1)-tupleof points on A into their sum. The fibre KnA = a−1(0) is called the generalized Kummer variety.

To describe the motive of KnA one uses the construction described above, replacing the symmetricpower A(n) by the fibre of the Albanese morphism A(n) → A. The fibres of the morphisms A(νi) → A

are finite quotients of abelian varieties. Therefore, repeating the arguments of [6] or using more generalresults of [7], we find that M(KnA) ∈ (Motab

A ). This was also shown in [36, Theorem 1.1] using similarmethods.

4.2. Manifolds of OG6 deformation type. The manifolds of OG6 deformation type were discoveredby O’Grady, see [27]. These 6-dimensional manifolds have originally been constructed as desingularizationsof certain moduli spaces of sheaves on abelian surfaces. To produce one manifold of this deformation typewith abelian motive, one can use a construction from [24]. In [24, section 6] one finds a diagram of theform

Xf←− Y2

g2−→ Y1g1−→ Y.

In this diagram: X is an OG6-type manifold; Y is a K3[3]-type manifold; f is a surjective genericallyfinite morphism; g1 and g2 are blow-ups with centers Z1 and Z2, where Z1 is the disjoint union of 256projective spaces and Z2 is isomorphic to (Bl(A×A∨)[2](A×A∨))/± 1 for some abelian surface A (see theproof of [24, Proposition 6.1]). The motives of Y , Z1 and Z2 are abelian, hence the motive of Y2 is alsoabelian. By projection formula, the motive of X embeds into the motive of Y2, hence it is also abelian.

5. Deformation principle

5.1. The setting. In this section, we will assume that

π : X → B10

is a smooth proper morphism with connected fibres between complex-analytic spaces, the base B is aconnected quasi-projective variety and for any b ∈ B the fibre Xb is projective. Moreover, we will assumethat there exists a line bundle L ∈ Pic(X ) and a dense Zariski-open subset U ⊂ B such that L|Xb is amplefor any b ∈ U . Let XU = π−1(U). We will assume that XU is a quasi-projective variety, and that thebundle L|XU and the morphism π|XU are algebraic.

Let us emphasize that we do not assume the total space X to be an algebraic variety, and for thefamilies that we consider below (see Proposition 6.2) it will typically not be algebraic.

5.2. The statement and preliminary constructions. The following proposition is a version of thedeformation principle for motivated cohomology classes, see section 2 and [1, Theorem 0.5]. We remarkthat similar results about specialization of motivated cycles were obtained in [4, Corollary 4.7]. We cannot apply those results in our setting, because we do not assume the total space X of our family to be analgebraic variety.

Proposition 5.1. In the above setting 5.1, let b0, b1 ∈ B be two points and consider a section ξ ∈H0(B,R2pπ∗C). Let ξbi ∈ H2p(Xbi ,C), i = 0, 1 be the values of ξ at bi. Then ξb0 is motivated if and onlyif ξb1 is motivated.

The proof is given below in 5.3. It uses the same idea as in [1], but in our case the variety X is notalgebraic, so we start by explaining some preliminary constructions.

First note that since B is quasi-projective, we can connect any two points in it by a chain of integralcurves. Since U is dense in B, we can also make sure that each of those curves intersects U . Choosingintermediate points between b0 and b1 at the intersections of the curves in the chain, we reduce to thecase when B is an integral quasi-projective curve. We may pull back the family X to the normalizationof this curve, and then assume that B is smooth.

The case when both b0 and b1 lie in U reduces to [1, Theorem 0.5], since in this case we can shrink Band assume that L is π-ample. If both b0 and b1 lie in B \U , we can choose an intermediate point b3 ∈ U ,and hence we reduce to the case when one of the points from the statement of Proposition 5.1 lies in Uand the other in B \ U . To fix the notation, we will assume that b0 ∈ U , b1 ∈ B \ U . After shrinking B,we may moreover assume that U = B \ b1. We will denote by Xi = π−1(bi) the two fibres.

The curve B is quasi-projective, hence there exists a projective curve B that contains B as a Zariski-open subset. Let us denote the boundary B \ B = z1, . . . , zn, and let U = B \ b1. Our next goal isto find a suitable compactification of X . We use the fact that over a punctured neighbourhood of everypoint zi the morphism π is projective and we can extend it over the puncture. The details appear in thefollowing lemma.

Lemma 5.2. There exists a compact complex manifold X , a flat morphism π : X → B, a line bundleL ∈ Pic(X ) and an open embedding j : X → X that satisfy the following conditions:

(1) π|j(X ) = π and X \ j(X ) = ∪ni=1π−1(zi);

(2) The open subset XU = π−1(U) ⊂ X is a quasi-projective variety;(3) The restriction of π to XU is an algebraic morphism onto U and L|XU is π-ample algebraic bundle.

Proof. Recall that πU : XU → U is a smooth morphism of quasi-projective varieties with relatively ampleline bundle LU = L|XU . For a big enough integer k, consider the vector bundle E = πU∗(LkU ) over U . Wecan find such k that the canonical morphism π∗U (E)→ LU induces a closed embedding XU → P(E). Notethat E is an algebraic vector bundle over the quasi-projective curve U , so we can extend it to a vector

11

bundle E over U . Denote by Z the closure of XU in P(E). Then Z is a quasi-projective variety fibredover U . It may be singular, the singularities lying over the points zi. Denote by ρ : Z → Z the resolutionof singularities which is obtained by a sequence of blow-ups with centres lying in the fibres over the pointszi. By construction, Z contains XU as an open subset, and the morphism Z → U agrees with πU onthis subset. We get the manifold X by gluing Z with X along their common open subset XU . Then Xis embedded into X by construction, and we get a morphism π : X → B that has the claimed property(1). The morphism π is flat, because it is equidimensional and X , U are smooth manifolds (see [15, page114]).

The open subset XU is by construction identified with Z, hence it is quasi-projective and (2) is satisfied.The first part of (3) is also satisfied because Z is a blow-up of the algebraic variety Z.

Next we prove the existence of the line bundle L. Consider the line bundle L′ = OP(E)/U (1)|Z . Thebundle L′ is relatively ample over U and L′|XU ' LkU The blow-up morphism ρ is projective, and if wedenote by Ej the exceptional divisors of ρ, the bundle L′′ = ρ∗L′(

∑ajEj) is relatively ample over U for

suitably chosen integers aj . We have L′′|XU ' LkU and Lk|XU ' LkU , and we define L by gluing L′′ andLk over the open subset XU . The second part of (3) is then satisfied because L|XU ' L

′′ is algebraic andrelatively ample over U .

In what follows we will implicitly identify X with its image in X under the embedding j from thelemma above.

Lemma 5.3. The manifold X from Lemma 5.2 is Moishezon. There exists a projective manifold X anda birational morphism r : X → X that is an isomorphism over X \X1 and such that r−1(X1) is a simplenormal crossing divisor.

Proof. We use the notation from the statement of Lemma 5.2. We refer to [15, chapter VII, §6] for thediscussion of Moishezon manifolds. Our proof is standard: we use the line bundle L (or rather its power)to produce a birational map from X to a projective variety.

Let πU = π|XU and E = πU∗(L|XU ). After possibly replacing the bundle L by its power, we mayassume that the canonical morphism π∗

U(E) → L|XU defines a closed embedding XU → P(E). Now

consider the vector bundle E′ = π∗(L) over B and note that E′|U ' E. Since we do not assume thatL|X1 is ample (or even globally generated) the morphism π∗(E′) → L only defines a meromorphic mapϕ : X 99K P(E′) whose restriction to XU coincides with the embedding XU → P(E). The indeterminacylocus of ϕ is contained in the fibre X1. We can resolve the indeterminacy of ϕ by a sequence of blow-upsr′ : X → X , so that ϕ lifts to a morphism ϕ : X → P(E′). The latter morphism is proper, hence its imageis an analytic subvariety that we denote by Y. Since Y is a subvariety of the projective manifold P(E′),it is also projective. This shows that X is Moishezon.

Next we show that there is a birational morphism r′′ : Y → X for some projective manifold Y. Weconsider the birational map ϕ−1 : Y 99K X . We resolve the indeterminacy of the latter map and thesingularities of Y by a sequence of blow-ups s : Y → Y. Then Y is a projective manifold and there is abirational morphism r′′ : Y → X . Note that the blow-ups occur only in the fibre over the point b1 ∈ B, soall the constructed birational morphisms are isomorphisms over XU . The preimage of b1 in Y is a divisor.By blowing up Y further, we make sure that this divisor has simple normal crossings. We define X to bethe result of this final sequence of blow-ups and r to be the induced map to X . By construction, X hasthe claimed properties.

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Consider the morphism r : X → X constructed in Lemma 5.3. Then D = (π r)−1(b1) is a simplenormal crossing divisor by the lemma, and r(D) = X1. Let us denote by X1 the irreducible componentof D that dominates X1. It is a non-singular projective variety. The restriction of r to X1 is a birationalmorphism r1 : X1 → X1. We denote by αi : Xi → X the closed immersions. Since r is an isomorphismover X \X1, the immersion α0 lifts to α0 : X0 → X . We summarize our constructions in a diagram:

(5.1)X0 XU X X1

X0 X X X1

α0

r

α1

r1

α0

α1

The projective manifold X can be used as the 0-th term of a simplicial resolution for X , which is anaugmented simplicial variety X• → X with Xi smooth projective and X0 = X . Such resolution can beconstructed as in [9, 6.2.5]. Note that both X and X are smooth manifolds of the same dimension, henceby projection formula [19, IX.7.3] the pull-back on cohomology r∗ is injective. Thus we have the followingexact sequence (cf. [9, Proposition 8.2.5]):

0 −→ H2p(X ,C) r∗−→ H2p(X ,C) δ∗0−δ

∗1−→ H2p(X1,C),

where δ0, δ1 : X1 ⇒ X0 = X are the face maps.

Definition 5.4. Define the motive H2p(X ) to be the kernel of the morphism

δ∗0 − δ∗1 : H2p(X )→ H2p(X1).

Let Gi = π1(B, bi). There exist canonical isomorphisms H0(B,R2pπ∗C) ' H2p(Xi,C)Gi .

Lemma 5.5. We have the following equalities:

(5.2) ker(α∗0 : H2p(X ,C)→ H2p(X0,C)) = ker(α∗1 : H2p(X ,C)→ H2p(X1,C)),

and

(5.3) im(α∗i : H2p(X ,C)→ H2p(Xi,C)) = H2p(Xi,C)Gi , i = 0, 1.

Proof. The kernels in (5.2) can be identified with the kernel of the composition

(5.4) H2p(X ,C)→ H2p(X ,C)→ H0(B,R2pπ∗C),

where the last morphism comes from the Leray spectral sequence. This proves (5.2).To prove (5.3), it is sufficient to check that H2p(X ,C) → H2p(X0,C)G0 is surjective, because this

would imply surjectivity of the composition (5.4). Since X0 ⊂ XU , we have the composition

(5.5) H2p(X ,C) β→ H2p(XU ,C) γ→ H2p(X0,C)G0 .

The line bundle L defines a polarization on the fibres over U , hence the Leray spectral sequencedegenerates at E2. Since the action of π1(U, b0) on the cohomology of X0 factors through G0, it followsthat the morphism γ is surjective. The cohomology of XU carries a mixed Hodge structure, and sincethe Hodge structure on H2p(X0,C) is pure, we deduce that the restriction of γ to W2pH

2p(XU ,C) is stillsurjective. On the other hand, X is a smooth Moishezon compactification of XU , and the arguments from[8, section 3.2] show that W2pH

2p(XU ,C) is the image of β. This completes the proof of (5.3).

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Definition 5.6. Define the following motives:

H2p(X0)G0 = im(α∗0 : H2p(X )→ H2p(X0)),

H2p(X1)G1 = im(r1∗ α∗1 : H2p(X )→ H2p(X1)).

Note that α∗0 r∗ = α∗0 and by projection formula r1∗ α∗1 r∗ = r1∗ r∗1 α∗1 = α∗1, see the diagram(5.1). Therefore, it follows from (5.3) that H2p(Xi)Gi are the motives representing H2p(Xi,C)Gi .

5.3. Proof of Proposition 5.1. Consider the motives H2p(Xi)Gi introduced in Definition 5.6. The twoquotient maps H2p(X )→ H2p(Xi)Gi have the same kernel by (5.2). Hence the quotients H2p(Xi)Gi arecanonically isomorphic, and the corresponding isomorphism in cohomology is induced by H2p(Xi,C)Gi 'H2p(B,R2pπ∗C), by construction. We conclude that the isomorphism H2p(X0,C)G0 ' H2p(X1,C)G1 liftsto the category of André motives. It follows that for any section ξ ∈ H0(B,R2pπ∗C) the cohomology classξb0 ∈ H2p(X0,C) is motivated if and only if ξb1 ∈ H2p(X1,C) is motivated. This finishes the proof.

6. Motives of hyperkähler manifolds

In this section, we consider hyperkähler manifolds of a fixed deformation type. Hence we fix a lattice Λrepresenting the second integral cohomology of these manifolds and the Beauville-Bogomolov-Fujiki formq. We denote by D ⊂ P(Λ ⊗ C) the period domain and fix one connected component M of the modulispace of marked hyperkähler manifolds of our fixed deformation type. We denote the period map byρ : M→D. Given a non-zero element h ∈ Λ, we denote by Λh ⊂ Λ the orthogonal complement to h andlet Dh = D ∩ P(Λh ⊗ C), Mh = ρ−1(Dh). Recall that if q(h) > 0, then all the hyperkähler manifoldsparametrized by Mh are projective by [17, Theorem 3.11]. We refer to [18] for a discussion of perioddomains and moduli spaces of marked hyperkähler manifolds.

6.1. Constructing smooth families of hyperkähler manifolds. We will need the following lemmaabout filling in families of hyperkähler manifolds over the punctured disc. We denote the unit disc by ∆and the punctured disc by ∆∗. If π′ : X ′ → ∆∗ is a family of hyperkähler manifolds, its marking is anisomorphism ϕ′ : R2π′∗Z ' Λ, where Λ is the constant sheaf with fibre Λ. In particular, the monodromyaction on H2 is trivial for a marked family. The marking induces a period map γ′ : ∆∗ → D. We will saythat some condition is satisfied for a very general point t ∈ ∆, if there exists a countable subset Z ⊂ ∆such that the condition is satisfied for any t ∈ ∆ \ Z.

Lemma 6.1. Let π : X → ∆ be a flat projective morphism and π′ : X ′ → ∆∗ its restriction to ∆∗. Assumethat π′ is a smooth family of marked hyperkähler manifolds, and the period map γ′ : ∆∗ → D extends to amorphism γ : ∆→ D. Assume that a very general fibre of π has Picard rank one. Let (X,ϕ) be a markedhyperkähler manifold such that ρ(X,ϕ) = γ(0). Then there exists a finite ramified cover α : ∆→ ∆, and asmooth family of hyperkähler manifolds π : X → ∆ such that X0 ' X and α∗X|∆∗ ' X |∆∗ , after possiblyshrinking ∆. Any line bundle L′ ∈ Pic(X |∆∗) can be extended to a line bundle L ∈ Pic(X ).

Proof. The statement is essentially equivalent to [20, Theorem 0.8]. Following the argument from [20,section 3], we pull back the universal family ofX via γ and obtain a smooth family of hyperkähler manifoldsξ : Y → ∆ with central fibre X. There exists a finite covering α : ∆→ ∆, and a cycle Z ⊂ α∗X ×∆ α∗Ythat induces a birational isomorphism between α∗X and α∗Y over ∆. We define X = α∗Y.

The local system R2ξ∗Z is trivial and by parallel transport we identify its fibres with H2(Y0,Z) 'H2(X,Z). Using the marking ϕ we then obtain an isomorphism R2ξ∗Z ' Λ, i.e. Y becomes a family ofmarked hyperkähler manifolds. For t ∈ ∆∗ let us denote by (Xt, ϕt) and (Yt, ϕ′t) the fibres of the two

14

families X and Y with the induced markings. Thus we obtain two maps from ∆∗ toM whose compositionswith the period map ρ are equal, by construction. Recall that we denote byM one connected componentof the moduli space of marked hyperkähler manifolds, and that by the global Torelli theorem [18, Corollary5.9] ρ induces an isomorphism between the Hausdorff reduction ofM and D. It follows that for t ∈ ∆∗

the marked manifolds (Xt, ϕt) and (Yt, ϕ′t) represent either the same point inM or a pair of non-separablepoints. By our assumption, for a very general t ∈ ∆∗ the Picard group of Xt is generated by an ampleline bundle, hence the Kähler cone of Xt coincides with a connected component of the positive cone. Thisimplies thatM contains no non-separable points over γ(t), see [23, section 5.3 and Theorem 5.16]. Hencethe cycle Zs is the graph of an isomorphism between Xα(s) and Yα(s) for a very general s ∈ ∆. The subsetof s ∈ ∆ for which Zs defines an isomorphism of the fibres is Zariski-open. This implies that the familiesα∗X|∆∗ and X |∆∗ are isomorphic, after possibly shrinking ∆.

To prove the last claim of the lemma, note that we have the following isomorphism

Pic(X ) ' ker(H0(∆, R2π∗Z)→ H0(∆, R2π∗OX )).

This isomorphism follows from the exponential exact sequence using the fact that ∆ is a Stein manifoldand the fibres of π are simply connected. The local system (R2π∗Z)|∆∗ is trivial, and its section thatdefines L′ extends to a section of R2π∗Z. This extension still lies in the kernel on the right hand side ofthe above formula, because the sheaf R2π∗OX is locally free. Thus we get a line bundle L ∈ Pic(X ) thatextends L′.

Given a hyperkähler manifold X1, we can choose a marking ϕ1 : H2(X1,Z) ∼→ Λ such that (X1, ϕ1) ∈M. Assume that X1 is projective with a very ample line bundle L, and let h = ϕ1(c1(L)). Then(X1, ϕ1) ∈Mh.

Proposition 6.2. In the above setting, assume that we have another marked hyperkähler manifold(X2, ϕ2) ∈ Mh. Then there exists a connected quasi-projective curve C, a smooth family of hyperkählermanifolds π : X → C and two points x1, x2 ∈ C such that Xi ' π−1(xi) and π is a projective morphismof algebraic varieties over C \ x2. There exists a line bundle L ∈ Pic(X ) that is algebraic and π-ampleover C \ x2.

Proof. We embed X1 into PN = PH0(X1, L) and denote by P its Hilbert polynomial. Let H be theHilbert scheme HilbP (PN ) with the reduced scheme structure and let ψ : X → H be the restriction to Hof the universal family. Let U ⊂ H be the open subset over which the morphism ψ is smooth. If U isdisconnected, we replace it by the connected component containing [X1]. Let µ : U → U be the universalcovering, and let ψ′ : X ′ → U be the pull back of the family X to U . The local system R2ψ′∗Z is trivialand by parallel transport its fibres can be identified with H2(X1,Z). Using the marking ϕ1 we identifythe fibres of the latter local system with Λ, and the relative Hodge to de Rham spectral sequence inducesan embedding of vector bundles ρ : ψ′∗(Ω2

X ′/U ) → Λ ⊗ OU . The family X ′ is polarized, the class of thepolarization being h ∈ Λ, hence ρ factors via Λh⊗OU . So ψ′∗(Ω2

X ′/U ) is a rank one subbundle of Λh⊗OU ,and this gives us a period map from U to Dh. We can find a torsion-free arithmetic subgroup Γ ⊂ O(Λh, q)and a finite covering U ′ → U such that the period map descends to a morphism µ : U ′ → Dh/Γ of quasi-projective varieties. Note that the morphism µ is dominant (see e.g. the proof of [31, Lemma 4.5]).

Let p1 and p2 be the images of (X1, ϕ1) and (X2, ϕ2) in Dh/Γ. By construction, p1 ∈ im(µ). We canfind a smooth quasi-projective curve C1 ⊂ Dh/Γ such that p1, p2 ∈ C1. For a very general point of Dh, thecorresponding hyperkähler manifold has Picard group of rank one (generated by h), and we can assume

15

that the same is true for a very general point of C1. Since µ is dominant, there exists a curve C2 ⊂ U ′

that maps dominantly to C1. Taking the normalization of C1 in the function field of C2, we get a curveC3 and a finite morphism ν : C3 → C1. By construction, there exists a rational map from C3 to H. SinceH is a projective variety, this map extends to a morphism ξ : C3 → H. We obtain a projective familyX ′ = C3 ×H X over C3.

C3 C1

H U ′ Dh/Γ

ν

ξ

µ

Let q1, q2 ∈ C3 be two points with ν(qi) = pi. Note that the fibre of X ′ over q1 is isomorphic to X1

by construction. The fibre over q2 might be non-smooth or not isomorphic to X2. We use Lemma 6.1 tomodify the family X ′ over a disk around q2, and produce a new family with fibre X2. Let ∆ ⊂ C3 and∆′ ⊂ C1 be two small disks around q2 and p2 such that ν(∆) ⊂ ∆′ and the covering map Dh → Dh/Γsplits over ∆′. Let ∆′′ ⊂ Dh map isomorphically onto ∆′ under the covering map. We obtain a morphismγ : ∆ → ∆′′ ⊂ Dh. Let ∆∗ = ∆ \ q2 and note that γ|∆∗ is the period map for X ′|∆∗ for some choiceof the marking. We can now apply Lemma 6.1. After passing to some finite ramified cover α : C4 → C3,we obtain a smooth family Y → α−1(∆) with central fibre X2 such that its restriction to α−1(∆∗) isisomorphic to the restriction of X ′′ = C4 ×C3 X ′. We modify X ′′ over α−1(∆) by gluing in Y, and obtaina new family X → C4 that contains both X1 and X2 as fibres. We restrict to an open subset C ⊂ C4 overwhich this family is smooth. Since the family X ′′ is projective, there exists a relatively ample line bundleL′′ ∈ Pic(X ′′). Using the last claim in Lemma 6.1, after gluing in Y we get a line bundle L ∈ Pic(X )whose restriction to all fibres except possibly X2 is ample. This completes the proof.

Lemma 6.3. Assume that π : X → C is a smooth family of hyperkähler manifolds such that: C is a smoothquasi-projective curve; all fibres of π are projective; π is a projective morphism of algebraic varieties overa dense Zariski-open subset U ⊂ C; there exits a line bundle L ∈ Pic(X ) that is algebraic and π-ampleover U. Assume that the André motive of the fibre Xb0 is abelian for some b0 ∈ C. Then for any b1 ∈ Cthe André motive of Xb1 is abelian.

Proof. Using part (1) of Corollary 3.3 and possibly replacing C by a finite cover, we may assume thatπ1(C, b0) acts on H•(Xb0 ,Q) via a homomorphism ρ : π1(C, b0) → Spin(V, q)Q, where V = H2(Xb0 ,Q)and q is the Beauville-Bogomolov-Fujiki form. Since the morphism π is projective over an open subset ofC, there exists a line bundle defining the polarization. The first Chern class of this line bundle gives amonodromy-invariant element h ∈ V such that q(h) > 0. Hence we can use parts (2) and (3) of Corollary3.3 to obtain a projective family ψ : A → C of Kuga-Satake abelian varieties.

Consider the product Y = X ×C A and denote by ξ : Y → C the induced morphism. The embeddingsνi from Corollary 3.3 can be viewed as global sections of the local system R2n+2dξ∗Q(n + d). We needto prove that the corresponding cohomology class νi,b1 ∈ H2n+2d(Xb1 ×Ab1 ,Q(n + d)) is motivated. ByProposition 5.1, it is enough to prove that νi,b0 is motivated. But the fibre Yb0 ' Xb0 ×Ab0 has abelianAndré motive, hence any cohomology class on it is motivated by [1, section 6].

6.2. Proof of theorem 1.1. We choose two markings ϕi : H2(Xi,Z) ∼→ Λ, so that (Xi, ϕi) ∈ M. Wechoose very ample line bundles on Xi and denote by hi ∈ Λ their classes. We use Proposition 6.2 toconnect X1 and X2 by several smooth families of hyperkähler manifolds and then apply Lemma 6.3 to

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these families. We connect X1 to X2 in several steps, depending on the relative position of the divisorsDh1 and Dh2 inside D.

Case 1. Assume that ρ(X1, ϕ1) ∈ Dh1 ∩ Dh2 or ρ(X2, ϕ2) ∈ Dh1 ∩ Dh2 . In this case we can applyProposition 6.2 to construct a family connecting X1 and X2.

Case 2. Assume that sign(q|〈h1,h2〉) = (1, 1). This condition implies that Dh1 ∩ Dh2 6= ∅. By thesurjectivity of the period map ρ : M → D (see [18, Theorem 5.5]), we can pick (X3, ϕ3) ∈ M such thatρ(X3, ϕ3) ∈ Dh1 ∩ Dh2 and reduce to Case 1 above.

Case 3. Assume the q|〈h1,h2〉 is positive definite. In this case Dh1 ∩ Dh2 = ∅, but we will find h3 ∈ Λsuch that q(h3) > 0 and Dh1 ∩ Dh3 6= ∅, Dh2 ∩ Dh3 6= ∅, reducing to Case 2. Consider the set

V = v ∈ Λ⊗ R | q(v) > 0, sign(q|〈h1,v〉) = sign(q|〈h2,v〉) = (1, 1).

This set is an open cone in Λ⊗R, and it suffices to prove that V 6= ∅. Choose three vectors e1, e2, e3 ∈ Λ⊗Rsuch that: q(e1) = q(e2) = 1, q(e3) = −1; ei are pairwise orthogonal; h1 = ae1, h2 = be1 + ce2 for somea, b, c ∈ R. If b = 0, then v = e1 + e2 + de3 ∈ V for 1 < d2 < 2. If b 6= 0, then v = be1 + ce2 + de3 ∈ V forc2 < d2 < b2 + c2.

Case 4. Assume the q|〈h1,h2〉 is degenerate. Then we will find h3 ∈ Λ such that q(h3) > 0 and therestrictions q|〈h1,h3〉, q|〈h2,h3〉 are non-degenerate, reducing to the previous cases. Consider the set

V = v ∈ Λ⊗ R | q(v) > 0, q|〈h1,v〉 and q|〈h2,v〉 non-degenerate.

As above, V is an open cone and we need to check that V 6= ∅. The condition that q|〈hi,v〉 is degenerate isgiven by the vanishing of the determinant of the Gram matrix, hence it defines a hypersurface in Λ⊗ R.Since the set v ∈ Λ⊗R | q(v) > 0 is clearly open and non-empty, V is also non-empty. This completesCase 4, and the proof of the theorem.

Acknowledgements

Thanks to Ben Moonen for communicating to me the problem of lifting the Kuga-Satake embeddingto the category of André motives, to Salvatore Floccari for his interest in this work and for pointing outthe relevance of [11, Theorem 5.1] for Corollary 1.4, to Giovanni Mongardi for the useful discussion ofO’Grady’s manifolds.

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Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 BerlinE-mail address: soldatea@hu-berlin.de

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