On the motive of O'Grady's ten-dimensional hyper-K\"ahler varieties

We investigate how the motive of hyper-K\"ahler varieties is controlled by weight-2 (or surface-like) motives via tensor operations. In the first part, we study the Voevodsky motive of singular moduli spaces of semistable sheaves on K3 and abelian surfaces as well as the Chow motive of their crepant resolutions, when they exist. We show that these motives are in the tensor subcategory generated by the motive of the surface, provided that a crepant resolution exists. This extends a recent result of B\"ulles to the O'Grady-10 situation. In the non-commutative setting, similar results are proved for the Chow motive of moduli spaces of stable objects of the K3 category of a cubic fourfold. As a consequence, we provide abundant examples of hyper-K\"ahler varieties of O'Grady-10 deformation type satisfying the standard conjectures. In the second part, we study the Andr\'{e} motive of projective hyper-K\"ahler varieties. We attach to any such variety its defect group, an algebraic group which acts on the cohomology and measures the difference between the full motive and its weight-2 part. When the second Betti number is not 3, we show that the defect group is a natural complement of the Mumford--Tate group inside the motivic Galois group, and that it is deformation invariant. We prove the triviality of this group for all known examples of projective hyper-K\"ahler varieties, so that in each case the full motive is controlled by its weight-2 part. As applications, we show that for any variety motivated by a product of known hyper-K\"ahler varieties, all Hodge and Tate classes are motivated, the motivated Mumford--Tate conjecture holds, and the Andr\'e motive is abelian. This last point completes a recent work of Soldatenkov and provides a different proof for some of his results.

fundamentally, at the level of motives 1 . The prototype of such interplay we have in mind is del Baño's result [26], which says that the Chow motive of the moduli space M r,d (C) of stable vector bundles of coprime rank and degree on a smooth projective curve C is a direct summand of the Chow motive of some power of the curve; in other words, the Chow motive of M r,d (C) is in the pseudo-abelian tensor subcategory generated by the Chow motive of C. In [26], a precise formula for the virtual motive of M r,d (C) in terms of the virtual motive of C was obtained, a result which has been recently lifted to the level of motives in a greater generality by Hoskins and Pepin-Lehalleur [37].
In the realm of compact hyper-Kähler varieties [12] [40], this philosophy plays an even more important role: it turns out that taking the moduli spaces of (complexes of) sheaves on Calabi-Yau surfaces or their non-commutative analogues provides the most general and almost exhaustive way for constructing examples, see [62] [66] [67] [70] [88] [10] [11] [9] and [8] etc. As the first important relationship between the K3 or abelian surface S and a moduli space M := M S (v) of stable (complexes of) sheaves on S with (non-isotropic) Mukai vector v, the second cohomology of M is identified, as a Hodge lattice, with the orthogonal complement of v in H(S, Z), the Mukai lattice of S [68] [70]. Regarding the aforementioned result of del Baño in the curve case, a relation between the motive of S and the motive of M is desired. The first breakthrough in this direction is the following result of Bülles [18] based on the work of Markman [58]. Analogously to Theorem 1.3, our third result establishes Conjecture 1.6 in the ten-dimensional situation studied in [52], where a crepant resolution of M exists (it is again of O'Grady-10 deformation type). Recall that H(Ku(Y )) contains (and it is equal to, if Y is very general) a canonical A 2 -lattice generated by λ 1 and λ 2 (see [9] for the notation). As a by-product, we deduce Grothendieck's standard conjectures [34] [44] for many hyper-Kähler varieties of O'Grady-10 deformation type, cf. [20].

Defect groups of hyper-Kähler varieties.
It is easy to see that a general projective deformation of (a crepant resolution of) a moduli space of semistable sheaves (or objects) on a Calabi-Yau surface is no longer of this form (even by deforming the surface). If we still want to understand the motive of such a hyper-Kähler variety X in terms of some tensor constructions of a "surface-like" (or rather "weight-2") motive, the right substitution of the surface motive would be the degree-2 motive of X itself. We are therefore interested in the following meta-conjecture.
Meta-conjecture 1.9. Let X be a projective hyper-Kähler variety and fix some rigid tensor category of motives. If the odd Betti numbers of X vanish, then its motive is in the tensor subcategory generated by its degree-2 motive. If X has some non-zero odd Betti number, then its motive lies in the tensor subcategory generated by the Kuga-Satake construction of its degree-2 motive. In any case, the motive of X is abelian.
We will see in Proposition 6.4 that the analogous statement holds at the level of Hodge structures. This is essentially a consequence of Verbitsky's results [81], related works are [45] and [77]. Unfortunately, staying within the category of Chow motives (or Voevodsky motives), we are confronted with several essential difficulties: • As an immediate obstruction, to speak of the degree-2 motive, we have to admit the algebraicity of the Künneth projector, which is part of the standard conjecture. • Even in the case where the standard conjecture is known (for example [20]), the construction of the degree-2 part of the Chow motive h(X), denoted by h 2 (X), is still conjectural in general (see [65] [75]) and even when a candidate construction is available, it seems too difficult to relate h(X) and h 2 (X) for a general X in the moduli space of hyper-Kähler varieties. Nevertheless, let us point out that Bülles' Theorem 1.1 and our extensions Theorems 1.3, 1.4 and 1.7 indeed give some evidence in this direction (see also Corollary 4.7). • The algebraicity of the Kuga-Satake construction is wide open.
The third purpose of the paper is to make precise sense of the meta-conjecture 1.9. To circumvent the aforementioned difficulties we leave the category of Chow motive and work within the category of André motives [4]. Essentially, this amounts to replace rational equivalence by homological equivalence and formally add the cycles predicted by the standard conjectures; the result is a semisimple abelian Q-linear tannakian category, see §2.3 for a quick introduction. Through the tannakian formalism, most properties of an André motive M are encoded in its motivic Galois group G mot (M ). Note that since the Hodge theoretic version of meta-conjecture 1.9 holds, its validity at the level of André motives is implied by Conjecture 2.3 which says that all Hodge classes are motivated.
Our main contribution in this direction is about a Q-algebraic group, which we call the defect group, associated with a projective hyper-Kähler variety. Let X be a projective hyper-Kähler variety and let H(X) be its André motive. We have the Künneth decomposition H(X) = (iii) H(X) is abelian. (iv) Conjecture 2.3 holds for H(X): MT(H * (X)) = G mot (H(X)).
Thanks to Corollary 1.11, Conjecture 2.3 for hyper-Kähler varieties and the meta-conjecture 1.9 for their André motives are all equivalent to the following conjecture.
Conjecture 1.12. The defect group of any projective hyper-Kähler variety is trivial. Remark 1.13 (Potential approaches). We are not able to prove Conjecture 1.12 in general so far, but only for all the known examples of hyper-Kähler varieties (Corollary 1.15 below). However, (i) we will show in Corollary 7.2 that Conjecture 1.12 is implied by the following conjecture: an André motive is of Tate type if and only if its Hodge realization is of Tate type; (ii) The defect group satisfies many constraints. For example, its action on the rational cohomology ring is compatible with the ring structure as well as the Looijenga-Lunts-Verbitsky Lie algebra action [53] [81], and most importantly, it is a deformation invariant.
Theorem 1.14 (Deformation invariance of defect groups). Let S be a smooth quasi-projective variety and X → S be a smooth proper morphism with fibers being projective hyper-Kähler manifolds with b 2 = 3. Then for any s, s ′ ∈ S, the defect groups P (X s ) and P (X s ′ ) are canonically isomorphic, and similarly for the even defect groups. (iv) if X is of Kum n -type, then H(X) ∈ H 1 (KS(X)) and H + (X) ∈ H 2 (X) .
The item (i) on the abelianity of André motive is proved for K3 [n] -type by Schlickewei [74], for Kum n -type and OG6-type in the recent work of Soldatenkov [76].
Second, we can prove the Mumford-Tate conjecture for all known hyper-Kähler varieties defined over a finitely generated field extension of Q; see §7.2 for the precise statement of the conjecture. For varieties of K3 [m] -type, it has been proven in [31]. In fact, what we obtain in the following Theorem 1.17 is a stronger result in two aspects: • we identify the Mumford-Tate group and the Zariski closure of the image of the Galois representation via a third group, namely the motivic Galois group. This is the so-called motivated Mumford-Tate conjecture 7.3; • we can treat products. In general, it is far from obvious to deduce the Mumford-Tate conjecture for a product of varieties from the conjecture for the factors. Thanks to the work of Commelin [21] this can be done when the André motives of the varieties involved are abelian.  [76]). The combination of Corollary 1.11 and Theorem 1.14 (plus the fact that two deformation equivalent hyper-Kähler varieties can be connected by algebraic families) implies that the abelianity of the André motive of hyper-Kähler varieties is a deformation invariant property (Corollary 7.2(i)). When finalizing the paper, we discovered the recent update of Soldatenkov's preprint [76], where he also obtained this result, as well as Corollary 1.16(i), except for the O'Grady-10 case. We attribute the overlap to him. The proofs and points of view are somewhat different: [76] makes a detailed study of the Kuga-Satake construction in families, while our argument does not involve the Kuga-Satake construction when the odd cohomology is trivial, but relies on André's theorem [3] on the abelianity of H 2 . As a bonus of emphasizing the usage of defect groups in our study, on the one hand, there seems to be some promising approaches mentioned in Remark 1.13 to show the abelianity of the André motive of hyper-Kähler varieties in general; on the other hand, even if Conjecture 1.12 (hence the abelianity) turned out to fail for some deformation family of hyper-Kähler varieties, our method can still control their André motives by its degree-2 part together with informations on the André motive of one given member in that family, see Corollary 7.2 (ii), (iii).
Convention: From §3 to §7.1, all varieties are defined over the field of complex numbers C if not otherwise stated. We work exclusively with rational coefficients for cohomology groups and Chow groups, as well as for the corresponding categories of motives. For simplicity, the notation CHM (resp. DM, AM) stands for the category of rational Chow motives (resp. rational geometric motives in the sense of Voevodsky, rational André motives) over a base field k, which are usually denoted by CHM(k) Q (resp. DM gm (k) Q , AM(k)) in the literature.

Acknowledgement:
We want to thank Chunyi Li for helpful discussions and Ben Moonen for his careful reading of the draft.

Generalities on motives
In this section, we recall various categories of motives that we will be using, gather some of their basic properties, and explain some relations between them. Most of the content is standard and well-documented, except Proposition 2.2 and results of §2.4 in the non-projective setting.
2.1. Chow motives. Let SmProj k be the category of smooth projective varieties over an arbitrary base field k. Let CHM be the category of Chow motives with rational coefficients, equipped with the functor h : SmProj op k → CHM . We follow the notation and conventions of [5]. CHM is a pseudo-abelian rigid symmetric tensor category, whose objects consist of triples (X, p, n), where X is a smooth projective variety of dimension d X over the base field k, p ∈ CH d X (X × k X) with p • p = p, and n ∈ Z. Morphisms The tensor product of two motives is defined in the obvious way by the fiber product over the base field, while the dual of M = (X, p, n) is M ∨ = (X, t p, −n + d X ), where t p denotes the transpose of p. The Chow motive of a smooth projective variety X is defined as h(X) := (X, ∆ X , 0), where ∆ X denotes the class of the diagonal inside X × k X, and the unit motive is denoted by 1 := h(Spec(k)). In particular, we have CH l (X) = Hom(1(−l), h(X)). The Tate motive of weight −2i is the motive 1(i) := (Spec(k), ∆ Spec(k) , i). A motive is said to be of Tate type if it is isomorphic to a direct sum of Tate motives (of various weights).
Given a Chow motive M ∈ CHM, the pseudo-abelian tensor subcategory of CHM generated by M is by definition the smallest full subcategory of CHM containing M that is stable under isomorphisms, direct sums, direct summands, tensor products and duality. We denote this subcategory by M CHM ; it is again a pseudo-abelian rigid tensor category. Note that if M = h(X) is the motive of a smooth projective variety X, then any divisor on X gives rise to a splitting injection 1(−1) → h(X); therefore when X has a non-zero divisor, h(X) CHM contains the Tate motives and hence it is also stable under Tate twists.
2.2. Mixed motives. Let Sch k be the category of separated schemes of finite type over a perfect base field k. Let DM be Voevodsky's triangulated category of geometric motives over k with rational coefficients [84]. There are two canonical functors M : Sch k → DM and M c : (Sch k , proper morphisms) → DM .
For any X ∈ Sch k , M (X) is called its (mixed) motive and M c (X) is called its motive with compact support (or rather its Borel-Moore motive). There is a canonical comparison morphism M (X) → M c (X), which is an isomorphism if X is proper over k. The category DM is a rigid tensor triangulated category, where the duality functor is determined by the so-called motivic Poincaré duality, which says that for any connected smooth k-variety X of dimension d, The Chow groups are interpreted as the corresponding Borel-Moore theory. More precisely, if X is an equi-dimensional quasi-projective k-variety, then for any i ∈ N, An important property we will use is the localization distinguished triangle [84]: let Z be a closed subscheme of X ∈ Sch k , then there is a distinguished triangle in DM: Given a mixed motive M ∈ DM, the tensor triangulated subcategory of DM generated by M , denoted by M DM is the smallest full subcategory of DM containing M that is stable under isomorphisms, direct sums, tensor products, duality and cones (hence also shifts and direct summands). By definition, M DM is a pseudo-abelian rigid tensor triangulated category. Again for a smooth projective variety X admitting a non-zero effective divisor, M (X) DM contains all Tate motives, and hence it is also stable by Tate twists.
By [84], there is a fully faithful tensor functor which sends the Chow motive h(X) of a smooth projective variety X to its mixed motive M (X) ≃ M c (X); for any i ∈ Z, the Tate object 1(−i) in CHM is sent to 1(i) [2i]. In this paper we identify CHM op with its essential image in DM.
Indeed, it suffices to show the first inclusion in the case n = 0. Given any object N of M v 0 DM , by the boundedness of v, N can be obtained by a finite sequence of successive extensions of objects of M DM with non-negative v-weight, which have non-negative w-weight by (4). Therefore N ∈ DM w 0 , and the claim is proved. Now we show that the inclusions in (5) are actually equalities. Given N ∈ DM w 0 ∩ M DM , we have Hom(N, N ′ ) = 0 for all N ′ ∈ M v −1 DM since, by the second inclusion in (5), N ′ ∈ DM w −1 . Therefore N ∈ M v 0 DM . The first equality is proved; the argument for the second one is similar. As a consequence, Obviously, the proof shows that the same result holds if we start with a subcategory of CHM instead of just an object. The case of the subcategory of abelian motives is exactly [85, Proposition 1.2], from which we borrowed the argument above.
Note that due to the abstract machinery of weight structures, the above proof does not give a constructive way to eliminate the usage of cones if a Chow motive is explicitly expressed in terms of a second one by tensor operations and cones.
2.3. André motives. Let the base field k be a subfield of the field of complex numbers C. Replacing the Chow group by the Q-vector space of algebraic cycles modulo homological equivalence (here we use the rational singular homology group of the associated complex analytic space) in the construction of Chow motives ( §2.1) one obtains the category of Grothendieck motives, denoted GRM, which comes with a canonical full functor CHM → GRM. The category of Grothendieck motives is conjectured to be semisimple and abelian; Jannsen [42] showed that it is the case if and only if numerical equivalence agrees with homological equivalence, which is one of Grothendieck's standard conjectures.
The standard conjectures being difficult, in [4] an unconditional theory was proposed by André, refining Deligne's category of absolute Hodge motives [29]. He replaced in the construction of Grothendieck motives the group of algebraic cycles up to homological equivalence by the group of motivated cycles, which are roughly speaking cohomology classes that can be obtained by using algebraic cycles and the Hodge * -operator. The resulting category of André motives is denoted by AM, and it is a semisimple abelian category. The canonical faithful functor GRM → AM is an isomorphism if the standard conjecture holds true for all smooth projective varieties.
The virtue of AM is that it works well with the tannakian formalism. There are natural functors: where H is the functor that associates to a variety its André motive, HS pol Q is the category of polarizable rational Hodge structures, r is the Hodge realization functor, and F is the forgetful functor. The composition of r • H is equal to the functor H attaching to a smooth projective variety its rational cohomology group. It is easy to see that the functors r and F are conservative.

Mumford-Tate group and motivic Galois group. It is well-known that HS pol
Q is a neutral tannakian semisimple abelian category with fiber functor F . Given a polarizable rational Hodge structure V , let V HS be the full tannakian subcategory of HS pol Q generated by V . The restriction of F to this subcategory is again a fiber functor. The Mumford-Tate group of V , denoted by MT(V ), is by definition the automorphism group of the tensor functor F | V HS and V HS is equivalent to the category of representations of MT(V ). Note that as V is assumed to be polarizable, MT(V ) is reductive. Mumford-Tate groups are known to be always connected. The MT(V )-invariants in a tensor construction on V are precisely the Hodge classes of type (0,0).
In a similar fashion, AM is also neutral tannakian with fiber functor F • r. Given an André motive M ∈ AM, the tannakian subcategory M AM is again neutral tannakian, with fiber functor F • r| M AM ; the tensor automorphisms group of this functor is denoted by G mot (M ) and called the motivic Galois group of M . The tannakian category M AM is then equivalent to the category of representations of this reductive group, and the G mot (M )-invariants in any tensor construction of r(M ) are precisely the motivated classes.   Proof. Let X be a smooth and proper (non-necessarily projective) algebraic variety defined over k. Consider its Nori motive H Nori (X) = i H i Nori (X). For each i ∈ N, H i Nori (X) carries a weight filtration W • , inducing the weight filtration on its Hodge realization [39, Theorem 10.2.5]. In particular, r Gr W l H i Nori (X) = Gr W l H i (X). However, the Hodge structure H i (X) is pure 5 , Gr W l H i (X) is zero for all l = 0. By the conservativity of r, the Nori motive Gr W l H i Nori (X) is also trivial for l = 0. In other words, H i Nori (X) is pure. We conclude by invoking Arapura's theorem [7] which says that the category of pure Nori motives is equivalent to the category of André motives.
The following result generalizes André's deformation principle for motivated cycles [4, Théorème 0.5] to the proper setting (but always with projective fibers). It has been obtained recently by Soldatenkov [76, Proposition 5.1]. We include here an alternative proof with the point being that André's original proof actually works, when combined with Lemma 2.5. Theorem 2.6 (André-Soldatenkov). Let S be a connected and reduced variety and let f : X → S be a proper smooth morphism with projective fibers. Let ξ ∈ H 0 (S, R 2i f * Q(i)), and assume that there exists s 0 ∈ S such that the restriction ξ s 0 ∈ H 2i (X s 0 , Q(i)) of ξ to the fibre over s 0 is motivated. Then, for all s ∈ S, the class ξ s ∈ H 2i (X s , Q(i)) is motivated.
Proof. As in [4], we can assume that S is a smooth affine curve. Choose a smooth compactification X of the total space X and let j s : X s → X be the inclusion morphism for all s ∈ S. The theorem of the fixed part [28, 4.1.1] ensures that the image of the morphism of Hodge structures j * s : H 2i (X , Q(i)) → H 2i (X s , Q(i)) coincides with the subspace of monodromy invariants. André's proof uses the morphism j * s induced on André motives, and conclude that the subspace of monodromy invariants at s ∈ S is a submotive which does not depend on the chosen point. Now, in our case X is not necessarily projective, but still has a well-defined André motive H(X ) = i H i (X ) by Lemma 2.5, and j * s is a morphism of André motives. Then we can conclude via the same argument as in André [4].
The following definition extends slightly the usual notion of families of André motives. Definition 2.7 (cf. [60,Definition 4.3.3]). Let S be a smooth connected quasi-projective variety. An André motive (resp. generalized André motive) over S is a triple (X /S, e, n) with • f : X → S a smooth projective (resp. proper) morphism with connected projective fibers, such that for some s ∈ S (or equivalently by Theorem 2.6, for any s ∈ S), the value e(s) ∈ H 2d (X s × X s , Q(d)) is a motivated projector.
These objects, with morphisms defined in the usual way, form a tannakian semisimple abelian category denoted by AM(S) (resp. AM(S)). Obviously, a generalized André motive over a point is nothing else but an André motive introduced before. There is a natural realization functor from the category of generalized André motives over S to the tannakian category of algebraic variations 6 of Q-Hodge structures in the sense of Deligne [28, Definition 4.2.4]: By construction, for any smooth proper morphism f : X → S with projective fibers and any integer i, we have a generalized André motive H i (X /S) whose realization is R i f * Q ∈ VHS a Q (S). Given a (generalized) André motive M/S ∈ AM(S), we aim to study the variation of motivic Galois groups G mot (M s ) and Mumford-Tate groups MT(r(M ) s ) when s varies in S. Consider the monodromy representation π 1 (S, s) → GL(r(M ) s ) associated to the local system underlying the realization of M/S. The algebraic monodromy group at a point s ∈ S, denoted by G mono (M/S) s , is defined as the Zariski closure in GL(r(M ) s ) of the image of the monodromy representation. It is not necessarily connected, but it becomes so after some finiteétale cover of S; Deligne [28, Theorem 4.2.6] proved that G mono (M/S) 0 s is a semisimple Q-algebraic group. The variation of these groups with s determines a local system of algebraic groups G mono (M/S).   [60]; hence, we get two local systems of algebraic groups over U with the properties above. The fundamental group of S is a quotient of that of U . Since (i) holds over U , we can extend the generic Mumford-Tate and motivic Galois groups which we have over U to local systems MT(r(M )/S) and G mot (M/S) over S. We prove that these local systems satisfy the desired properties. Note that (i) and (ii) are immediate since both conditions can be checked over U , where we already know they hold. (iii). We only give the proof for the generic motivic Galois group; the argument for the generic Mumford-Tate group is similar. Up to a base change of the family M/S by a finiteétale cover of S, we may assume that the algebraic monodromy group is connected. Let s 0 ∈ S be any point such that G mono (M/S) s 0 is contained in G mot (M s 0 ); this is the case for a very general point, by (i) and (ii). The monodromy group acts on G mot (M s 0 ) by conjugation, and this defines a local system of algebraic groups G mot (M s 0 /S) with fiber isomorphic to the motivic Galois group at the point s 0 . Consider any tensor construction T /S = (M/S) ⊗m ⊗ (M/S) ∨,⊗n , and let ξ s 0 be the cohomology class of a motivated cycle in r(T ) s 0 . The class ξ s 0 is monodromy invariant, and therefore it extends to a global section ξ of the local system underlying r(T )/S. By Theorem 2.6 the restriction ξ s is motivated for any s ∈ S. By the reductivity of the groups involved, we deduce that for any s ∈ S we have G mot (M s ) ⊂ G mot (M s 0 /S) s , and we conclude by (ii) that the latter must be equal to G mot (M/S) s . This proves (iii) and that

In particular, each of the inclusion in (iii) is an equality if and only if
Since both sides are reductive, we only need to compare their invariants on the tensor constructions T /S on M/S as above. If ξ s ∈ r(T ) s is invariant for the action of G mono (M/S) 0 s · G mot (M s ), then it is the class of a motivated cycle which is monodromy invariant. By Theorem 2.6, it extends to a global section ξ of r(T )/S such that ξ s ′ is motivated at any s ′ ∈ S. It follows that ξ s is invariant for G mot (M/S) s . The proof of the assertion regarding the Mumford-Tate group is similar.

2.5.
Relations. We summarize in the diagram below the natural functors relating the various categories of motives we discussed above. For the sake of completeness, we inserted in the diagram also Nori's category of mixed motives MM Nori , whose pure part is the abelian category of André motives by Arapura's result in [7], see also [39] for a recent account.
Here the comparison functor C is due to Harrer [36,Theorem 7.4.17].

Motives of the stable loci of moduli spaces
In this section, we generalize an argument of Bülles [18] to give a relationship between the motive of the (in general quasi-projective) moduli space of stable sheaves on a K3 or abelian surface and the motive of the surface.
Pointer to references. For the case of Gieseker-stable sheaves, [58, Theorem 1] states the result for the cohomology class, but the proof gives the equality in Chow groups. Indeed, [57,Theorem 8], the statement is for Chow groups. Moreover, the assumption on the existence of universal family can be dropped ([57, Proposition 24]): it suffices to replace in the formula the sheaves E and F by certain universal classes in the Grothendieck group K 0 (S × M st ) constructed in [57,Definition 26]. More recently, it is shown in [56] that the technique of Markman can be adapted to obtain the result in the full generality as stated.
In terms of mixed motives, one can reformulate Proposition 3.2 as follows.
for finitely many integers k i 's and e i 's.
Proof. It is enough to remark that by (1) and (2), for any j ∈ Z, the space CH j (M st × S k i ) is equal to the space as well as to the space Remark 3.4 (Hodge realization). In Proposition 3.2, if one denotes γ = ⊕γ i and δ = ⊕δ i , then we get the following morphisms of mixed Hodge structures.
where the composition is precisely the comparison morphism from the compact support cohomology to the usual cohomology.
Remark 3.5 (Challenge for Kummer moduli spaces). In the case that S is an abelian surface, the moduli space M st is isotrivially fibered over S × S (which is the Albanese fibration when M st is projective). We usually denote by K st := K st S,H (v) its fiber. The analogue of Theorem 3.1 seems to be unknown for K st .

Motive of O'Grady's moduli spaces and their resolutions
In this section, we study the motive of O'Grady's 10-dimensional hyper-Kähler varieties [66]. Those are symplectic resolutions of certain singular moduli spaces of sheaves on K3 or abelian surfaces. We first recall the construction. for the (singular) moduli space of σ-semistable objects with the same Mukai vector. In [66], O'Grady constructed a symplectic resolution M of M (see also [43]), which is a projective (irreducible if S is a K3 surface) holomorphic symplectic manifold of dimension 10, not deformation equivalent to the fifth Hilbert schemes of the surface S. We know that these hyper-Kähler varieties are all deformation equivalent [70].
Let us briefly recall the geometry of M. We follow the notations in [66], see also [50] and [59, §2]. The moduli space M admits a filtration is the singular locus of M, which consists of strictly σ-semistable objects; and is the singular locus of Σ, hence the diagonal in Sym 2 (M S,σ (v 0 , α)). Notice that M S,σ (v 0 , α) is a smooth projective holomorphic symplectic fourfold deformation equivalent to the Hilbert squares of S.
In [66], O'Grady produced a symplectic resolution M of M in following three steps. As the explicit geometry is used in the proof of our main result, we briefly recall his construction.
Step 1. We blow up M along Ω, resulting a space M with an exceptional divisor Ω. The only singularity of M is an A 1 -singularity along the strict transform Σ of Σ. In fact, Σ is smooth, satisfying Σ ∼ = Hilb 2 (M S,σ (v 0 , α)), with the morphism Σ → Σ being the corresponding Hilbert-Chow morphism, whose exceptional divisor is precisely the intersection of Ω and Σ in M.
Step 2. We blow up M along Σ to obtain a (non-crepant) resolution M of M. The exceptional divisor Σ is thus a P 1 -bundle over Σ. We denote by Ω the strict transform of Ω. Then M is a smooth projective compactification of M st , with boundary being the union of two smooth hypersurfaces which intersect transversally.
Step 3. Lastly, an extremal contraction of M contracts Ω as a P 2 -bundle to Ω, which is a 3-dimensional quadric bundle (more precisely, the relative Lagrangian Grassmannian fibration associated to the tangent bundle) over Ω. The space obtained is denoted by M, which is shown to be a symplectic resolution of M.
Remark 4.1. By the main result of Lehn-Sorger [50], O'Grady's symplectic resolution can also be obtained by a single blow-up of M along its (reduced) singular locus Σ. The exceptional divisor Σ is nothing else but the image of Σ under the contraction in the third step described above, which is singular along Ω, the preimage of Ω. If we blow up M along Ω, we will obtain again M, with the exceptional divisor being Ω and the strict transform of Σ being Σ. In short, the order of blow-ups can be "reversed"; see the following commutative diagram from [59, §2]:  Proof. We assume dim X = n. Let ∆ X ⊆ X × X be the diagonal, then by [55, §9], we have which is a direct summand of h (Bl ∆ X (X × X)), hence is contained in the desired subcategory.
Since Ω ∼ = M S,σ (v 0 , α), it follows by [18, Theorem 0.1] that h(Ω) is in the thick tensor subcategory of Chow motives generated by h(S), hence the same is true for h( Ω) and h( Ω).
Similarly, the intersection Σ ∩ Ω is a smooth conic bundle over Ω, again by [82, Remark 4.6], its motive is in the tensor subcategory generated by that of Ω. One concludes as for Ω.
Here comes the key step of the proof. We give two proofs with the same starting point, namely Proposition 3.2. The difference is that the first one is elementary by staying in the category of Chow motives and is geometric so that in principle it gives rise to an explicit expression of the Chow motive h( M) in terms of h(S); however the second one is quicker by using mixed motives and Proposition 2.2, but is hopeless to deduce any concrete relation between these two motives.
For each i, the cycles γ i and δ i can be viewed as morphisms of motives On the other hand, we denote by j Σ and j Ω the closed embedding of Σ and Ω in M respectively. Then we have morphisms of motives It follows by (7) that the sum of all the above compositions add up to the identity on h( M).
Combining it with Lemma 4.3, we finish the proof.
Second proof of Proposition 4.4. By a repeated use of the localization distinguished triangle (3), we see that for a variety together with a locally closed stratification, if the motive of compact support of each stratum is in some triangluated tensor subcategory of DM, the so is the motive with compact support of the ambiant space; conversely, if the motive with compact support of the ambiant scheme as well as those of all but one strata are in some triangulated tensor subcategory of DM, then so is the motive with compact support of the remaining stratum.
(iii) Y is a Gushel-Mukai variety [35] [63] [24] of even dimension n = 4 or 6, and A is the Kuznetsov component defined as the semi-orthogonal complement of the exceptional collection where U is the rank-2 vector bundle associated to the Gushel map Y → Gr(2, 5) and O Y (1) is the pull-back of the Plücker polarization, see [47]. (iv) Y is a smooth hyperplane section of Gr (3,10), called the Debarre-Voisin (Fano) variety [25], and A is the semi-orthogonal complement of the exceptional collection where B Y is the restriction of the exceptional collection B of length 12 in the rectangular Lefschetz decomposition of Gr(3, 10) constructed by Fonarev [32], see [54, §3.3].
Assume that the manifold of stability conditions on A is non-empty, which is expected for all the cases in Example 5.1 and is established and studied for K3 and abelian surfaces in [17] (see also [87]), for the Kuznetsov component of cubic fourfolds by [9], and for the Kuznetsov component of Gushel-Mukai fourfolds by [71]. We denote the distinguished connected component of the stability manifold by Stab † (A). One can now extend Theorem 3.1 and Proposition 3.2 to the non-commutative setting as follows.
Proposition 5.2. The notation and assumption are as above.
π 13 (π * 12 (E), π * 23 (F)) denotes the class of the complex Rπ 13, * (π * 12 (E) ∨ ⊗ L π * 23 (F)) in the Grothendieck Proof. The proof of (i) is the similar to the proof of Markman's theorem [58] or rather its extension in [56]. Their proof only uses standard properties for stable objects and the Serre duality for K3 surfaces, which both hold for A. The proof of (ii) is exactly the same as in Proposition 3.2 (Bülles' argument), by replacing S by Y everywhere.
We consider first the situation where the stability agrees the semi-stability. Then v must be primitive and σ is v-generic. In this case, M is a smooth and projective hyper-Kähler variety, if it is not empty. Once we have the decomposition of the diagonal in Proposition 5.2 (ii), the same proof as in [18] yields the following generalization of Theorem 1.1. As a non-commutative analogue of Conjecture 1.2, we formulate the following conjecture. For an evidence to Conjecture 5.4, we restrict to the case where Y is a very general cubic fourfold and A is its Kuznetsov component. Let λ 1 and λ 2 be the cohomological Mukai vectors of the projections into A of O Y (1) and O Y (2) respectively. Then the topological K-theory of A is an A 2 -lattice with basis {λ 1 , λ 2 }, equipped with a K3-type Hodge structure [1]. Then for a generic stability condition σ (see [9]), there is an O'Grady-type crepant resolution of the singular moduli space M A,σ (2λ 1 + 2λ 2 ), which is of O'Grady-10 deformation type, see [52]. Our result is that Conjecture 5.4 holds true in this case. See Theorem 1.7 in the introduction for the precise statement.
Proof of Theorem 1.7. The argument is more or less the same as in §4: the singular locus of the moduli space of semistable objects M(2v 0 ) is Sym 2 (M(v 0 )), whose singular locus is the diagonal M(v 0 ). By the same procedure of blow-ups as in §4.1, we get a smooth projective variety M together with a stratification such that the motive with compact support of all strata belong to the tensor triangulated subcategory generated by the motive of M(v 0 ), hence also to the subcategory generated by h(Y ), by Theorem 5.3. The rest of the proof is the same as §4.

Defect groups of hyper-Kähler varieties
In this section we study the André motives of projective hyper-Kähler varieties with b 2 = 3. For any such X, we construct the defect group P (X), and prove Theorem 1.10 and Corollary 1.11. In the next section we will apply these results to the known examples of hyper-Kähler varieties.
The starting point and a main tool of our study is the following general theorem due to André. We review the Lie algebra action constructed by Looijenga-Lunts [53] and Verbitsky [81] on cohomology groups of varieties, as well as its remarkable properties when applied to compact hyper-Kähler manifolds. This action is crucial for the proof of Theorem 1.10. To ease the notation, the coefficient Q in all cohomology groups are suppressed.
6.1. The Looijenga-Lunts-Verbitsky (LLV) Lie algebra. Let X be a 2m-dimensional compact hyper-Kähler variety. A cohomology class x ∈ H 2 (X) is said to satisfy the Lefschetz property if the maps given by cup-product L j x : H 2m−j (X) → H 2m+j (X) sending α to x j ∪ α, are isomorphisms for all j 0. The Lefschetz property for a class x in H 2 (X) is equivalent to the existence of a sl 2 -triple (L x , θ, Λ x ), where θ ∈ End H * (X) is the degree-0 endomorphism which acts as multiplication by k − 2m on H k (X) for all k ∈ N. Moreover, in this case Λ x is uniquely determined by L x and θ. Note that the first Chern class of an ample divisor on X has the Lefschetz property by the hard Lefschetz theorem.
The LLV-Lie algebra of X, denoted by g LLV (X), is defined as the Lie subalgebra of gl(H * (X)) generated by the sl 2 -triples (L x , θ, Λ x ) as above for all cohomology classes x ∈ H 2 (X) satisfying the Lefschetz property. It is shown in [53, §(1.9)] that g LLV (X) is a semisimple Q-Lie algebra, evenly graded by the adjoint action of θ. The construction does not depend on the complex structure; therefore, g LLV (X) is deformation invariant.
Let us denote by H the space H 2 (X) equipped with the Beauville-Bogomolov quadratic form [12]. Let H denote the orthogonal direct sum of H and a hyperbolic plane U = v, w equipped with the form v 2 = w 2 = 0 and vw = 1. We summarize the main properties of the Lie algebra g LLV (X).
Moreover, g 0 (X) ∼ = so(H)⊕Q·θ, is the centralizer of θ in g LLV (X). The abelian subalgebra g 2 (X) is the linear span of the endomorphisms L x , for x ∈ H 2 (X), and g −2 (X) is the span of the Λ x , for all x ∈ H 2 (X) with the Lefschetz property. The above theorem is proved in [81], and in [53,Proposition 4.5], see also the appendix of [45]. These proofs are carried out with real coefficients, but immediately imply the result with rational coefficients: since g LLV (X) is defined over Q, the equality g LLV (X) ⊗ R = so( H) ⊗ R of Lie subalgebras of gl( H) ⊗ R implies that the same equality already holds with rational coefficients. acts on H i (X) via multiplication by (−1) i . Note also that the action induced by ρ and ρ + is via algebra automorphisms, thanks to Theorem 6.2(ii).
Proposition 6.4. The notation is as above.
(i) The morphism π + 2 is an isomorphism. In particular, the Hodge structure H + (X) belongs to the tensor subcategory of HS pol Q generated by H 2 (X).
(ii) If X has non-vanishing odd cohomology, the morphism π 2 is an isogeny with kernel ι ≃ Z/2Z. Moreover, if A is any Kuga-Satake variety for H 2 (X) in the sense of Definition A.2, we have H * (X) HS = H 1 (A) HS .
The natural choice for A is the abelian variety obtained through the Kuga-Satake construction on H 2 (X) equipped with the Beauville-Bogomolov form, see §A.1; let us remark that also the construction of [45] yields a Kuga-Satake variety for H 2 (X) in our sense.
The proof of the proposition will be given after some preliminary results. Recall ( [27]) that the algebraic group CSpin (H)  to be the induced representations on the even cohomology and on H 2 (X) respectively.
is an isogeny of degree-2 onto its image. The natural projection pr + 2 : i GL(H 2i (X)) → GL(H 2 (X)) maps the image of σ + isomorphically onto the image of σ 2 . (ii) If X has non-vanishing odd cohomology, the representation σ : is faithful, and the projection pr 2 : i GL(H i (X)) → GL(H 2 (X)) induces a degree-2 isogeny between the image of σ and the image of σ 2 .
Proof. By Remark 6.3 and the explicit description of w, the kernels of σ + and σ 2 both coincide with the central subgroup of order 2 generated by (−1, 1) = (1, ι). This proves part (i). If X has non-vanishing odd cohomology, σ is faithful by Remark 6.3, and the second assertion follows.
Remark 6.6. Note that the twisted representation σ ′ = w ′ · ρ where w ′ (λ) acts on H i (X) via multiplication by λ i−2m is the representation obtained via integration of g 0 → i gl(H i (X)).
The point of introducing the above representation is that it controls the Mumford-Tate group. Proof. Let G = Im(σ). Since both MT(H * (X)) and G are reductive, by [29, Proposition 3.1] it suffices to check that for any tensor construction any element α of T that is invariant for G is also fixed by MT(H * (X)). Let α ∈ T be such an invariant for G. Then the image of α in T ⊗ C is annihilated by all elements of ρ(so(H)) ⊗ C. By Theorem 6.2(iii), α is annihilated by the Weil operator W . Therefore α is of type (p, p) for some integer p. However, since w(G m ) also acts trivially on α, we must have p = 0; hence α is a Hodge class of type (0, 0) and is thus fixed by the Mumford-Tate group.
Proof of Proposition 6.4. (i) Lemma 6.7 implies that MT(H + (X)) ⊂ Im(σ + ). The morphism π + 2 is the restriction of the natural projection pr + 2 : i GL(H 2i (X)) → GL(H 2 (X)). Lemma 6.5 implies in particular that the restriction of pr + 2 to Im(σ + ) is injective; hence the restriction to the small group MT(H + (X)) is also injective, i.e. π + 2 is injective and hence it is an isomorphism.
(ii) Assume now that the odd cohomology of X is non-trivial. Since MT(H * (X)) ⊂ Im(σ) by Lemma 6.7, we deduce as above that the kernel of the morphism π 2 : MT(H * (X)) → MT(H 2 (X)) is contained in the kernel of pr 2 : Im(σ) → Im(σ 2 ). By Lemma 6.5, this is an order 2 central subgroup of Im(σ), generated by w(−1) = ι. Clearly w(−1) is contained in MT(X), and it follows that π 2 is an isogeny of degree 2 whose kernel is generated by ι. We have just proven that π 2 is an isogeny of degree 2, and we know that the morphism π ev is also an isogeny of degree 2, see §A.2; we conclude that π ev 2 is an isomorphism and hence H * (X) ev HS = H 2 (X) HS .

Finally, let
The following observation will be used in the proof of Theorem 1.10.
Lemma 6.8. Let G be a group acting on H * (X) via graded algebra automorphisms. If G acts trivially on H 2 (X), then the G-action commutes with the action of the LLV Lie algebra ( §6.1).
Proof. Let g ∈ G. By assumption, g commutes with θ and L x , for any x ∈ H 2 (X). Moreover, if x has the Lefschetz property, then g commutes with Λ x as well: indeed, L x = gL x g −1 , θ = gθg −1 and gΛ x g −1 form an sl 2 -triple, and since Λ x is uniquely determined by the elements L x and θ, we must have gΛ x g −1 = Λ x . One can conclude since the various operators L x and Λ x , for x ∈ H 2 (X), generate the Lie algebra g LLV (X).
We now turn to the proof of the main result of this section.
Recall that P + (X) is defined as the kernel of the map π + 2,mot . We deduce that G mot (H + (X)) is the semidirect product of its subgroups P + (X) and MT(H + (X)), which intersect trivially. In order to show that G mot (H + (X)) = MT(H + (X)) × P + (X), it thus suffices to show that P + (X) and MT(H + (X)) commute. By Lemma 6.7, it suffices to show that P + (X) commutes with the image of the representation σ + . Since P + (X) preserves the grading on H + (X), its action clearly commutes with the weight cocharacter w. Note that every element of G mot (H + (X)) acts via algebra automorphisms, since the cup-product is given by an algebraic correspondence (namely, the small diagonal in X × X × X). Moreover, if p ∈ P + (X), then by definition p acts trivially on H 2 (X); hence, its action commutes with that of the LLV-Lie algebra thanks to Lemma 6.8. It follows that P + (X) commutes with the image of the representation ρ + , and therefore P + (X) commutes with σ + as desired.
Assume now that the odd cohomology of X does not vanish, and choose a Kuga-Satake variety A for H 2 (X), see Appendix A. By Lemma 6.9 below, the motive H 1 (A) belongs to H(X) AM .
We consider the commutative diagram The morphisms π A and i A are isomorphisms by Proposition 6.4(ii) and Theorem 2.4 respectively. Note that by Theorem A.4, the kernel P (X) of π A,mot does not depend on the choice of the Kuga-Satake abelian variety A; this group is by definition the defect group of X. As above, we deduce the existence of a section of π A,mot with image MT(H * (X)), and to conclude we need to show that P (X) and MT(H * (X)) commute. To this end, we consider the commutative diagram with exact rows The group Q(X) commutes with the action of g LLV , by Lemma 6.8, and it therefore commutes with the Mumford-Tate group, thanks to Lemma 6.7. Since P (X) is a subgroup of Q(X), it also commutes with MT(H * (X)), and we have G mot (H(X)) = P (X) × MT(H * (X)). Also note that we have Q(X) ∩ MT(H * (X)) = ι , and that Q(X) = P (X) × ι .
In the previous proof, we have used the following result. See Appendix A for the notation.  (H 1 (A)). The desired conclusion is equivalent to the inclusion ker(q) ⊂ ker(q A ). In fact, this precisely means that the tannakian category generated by H 1 (A) is contained in H(X) , which then implies that q is an isomorphism. We consider the analogous morphisms for the even parts Note that the analogous of Remark A.5 holds for André motives, and therefore ker(q) and ker(q A ) equal the connected component of the identity of the preimage in G mot (H(X) ⊕ H 1 (A)) of ker(q ev ) and ker(q ev A ) respectively. Hence, it suffices to show that ker(q ev ) ⊂ ker(q ev A ). To this end, consider the commutative diagram with short exact rows The rightmost vertical map is an isomorphism by assumption. The snake lemma now yields that ker(j) = ker(q ev ), which shows that ker(q ev ) ⊂ ker(q ev A ).
6.3. What does the defect group measure? With the structure result of the motivic Galois group being proved in Theorem 1.10, we can deduce that the defect group indeed grasps the essential difficulty of meta-conjecture 1.9 for André motives. See Corollary 1.11 for the precise statement.
Proof of Corollary 1.11. We first treat the even motive. It follows immediately from Theorem 1.10 that (i + ) and (iv + ) are equivalent. (i + ) implies (ii + ): By the definition of P + (X), if it is trivial, then the natural surjection G mot (H + (X)) → G mot (H 2 (X)) is an isomorphism. Then (ii + ) follows from the Tannaka duality. The implication from (ii + ) to (iii + ) follows from the fact that H 2 (X) is abelian, which is André's Theorem 6.1. Finally, (iii + ) implies (iv + ) thanks to André's Theorem 2.4.
In the presence of non-vanishing odd Betti numbers, the proof is similar to the even case: the equivalence of (i) and (iv) is immediate from Theorem 1.10. (ii) obviously implies (iii); (iii) implies (iv) by André's Theorem 2.4. Finally, let us show that (i) implies (ii): if P (X) is trivial then G mot (H(X)) → G mot (H 1 (A)) is an isomorphism, where A is any Kuga-Satake variety for H 2 (X) in the sense of Definition A.2. Therefore, H(X) is in H 1 (A) AM by Tannaka duality.
6.4. Deformation invariance. We have seen in the above proof that the action of the defect group commutes with the LLV-Lie algebra. We prove now that defect groups are deformation invariant in algebraic families, see Theorem 1.14 for the statement. The relevant notation and results are recalled in §2.4. Let f : X → S be a smooth and proper family over a non-singular quasi-projective variety S such that all fibres X s are projective hyper-Kähler varieties with b 2 = 3. We have naturally the following generalized André motives over S (Definition 2.7): H(X /S), H i (X /S) and H + (X /S). Up to replacing S by anétale cover, we can assume that the algebraic monodromy group G mono (H(X /S)) is connected.
Proof of Theorem 1.14. We prove the invariance of the even defect group; the same argument proves the invariance of the full defect group if the odd cohomology does not vanish. For any s ∈ S, we have G mot (H + (X s )) = MT(H + (X s )) × P + (X s ) by Theorem 1.10. Let s 0 ∈ S be a very general point. By Theorem 2.8 (i) and (ii), we have G mono (H + (X /S)) s 0 ⊂ MT(H + (X s 0 )). Hence, the monodromy acts trivially on P + (X s 0 ), which therefore extends to a constant local system P + (X /S) such that we have a splitting G mot (H + (X /S)) = MT(H + (X /S)) × P + (X /S) of local systems of algebraic groups over S. Moreover the inclusion of G mot (H + (X s )) into G mot (H + (X /S)) s is the direct product of inclusions MT(H + (X s )) ֒→ MT(H + (X /S)) s and P + (X s ) ֒→ P + (X /S) s , for any s ∈ S.
It is enough to show that for all s, the equality P + (X s ) = P + (X /S) s holds. By Theorem 2.8(iv), for all s ∈ S, we have G mono (H + (X /S)) s · G mot (H + (X s )) = G mot (H + (X /S)) s .
But we know that G mono (H + (X /S)) s is contained in and therefore we have which forces P + (X s ) = P + (X /S) s . 7. Applications 7.1. André motives of hyper-Kähler varieties. As we have seen in Theorem 1.14, the defect group does not change along smooth proper algebraic families. In fact, the defect group is invariant in the whole deformation class.
Proof. Pick two deformation equivalent projective hyper-Kähler varieties X and X ′ with b 2 = 3. It has been shown by Soldatenkov ([76,§6.2]) that there exists finitely many smooth proper algebraic families f i : Y i → S i , i = 0, 1, . . . , k over smooth quasi-projective curves S i and points a i , b i ∈ S i together with isomorphisms We therefore find a chain of smooth proper families with projective fibers connecting X and X ′ . The conclusion now follows via an iterated application of Theorem 1.14.
(ii). This follows via a reinterpretation of Theorem 1.10 in terms of a defect motive. Recall that we have G mot (H + (X)) = MT(H + (X)) × P + (X). The category Rep(P + (X)) can be seen as the subcategory of H + (X) AM on which MT(H + (X)) acts trivially, i.e., it consists of the motives in H + (X) whose realization is of Tate type. By [29, Proposition 3.1], the category Rep(P + (X)) is generated as a tannakian category by any faithful representation of P + (X); choosing one such representation determines a motive D + (X) ∈ H + (X) , such that inside AM, H + (X) = D + (X), H 2 (X) .
By Corollary 7.1, if X ′ is deformation equivalent to X, we have P + (X ′ ) ∼ = P + (X), and therefore we can choose D + (X ′ ) = D + (X). Hence, the motive D + = D + (X) has the desired property.
(iii). Same argument as above.
We can now prove that the defect group of any known projective hyper-Kähler variety is trivial.
Proof of Corollary 1.15. As the triviality of the defect group is a deformation invariant property by Corollary 7.1, it is enough to find in each of the known deformation classes a representative whose defect group is trivial, or equivalently, whose André motive is abelian.
, which is a continuous Q ℓ -representation of Gal(k/k). These two cohomology theories provide realization functors from AM(k), the category of André motives over Spec(k): Given a Galois representation σ : Gal(k/k) → GL(V ) on a Q ℓ -vector space V , we let G(V ) denote the Q ℓ -algebraic subgroup of GL(V ) which is the Zariski closure of the image of σ. This algebraic group is not necessarily connected, but becomes so after a finite field extension of k. It is not known to be reductive in general. The category of Q ℓ -Galois representations is a neutral tannakian abelian category, and the tannakian subcategory V is equivalent to the category of finite dimensional Q ℓ -representations of G(V ).
These different realizations are related via Artin's comparison theorem: for any M ∈ AM(k) there is a canonical isomorphism of Q ℓ -vector spaces γ : r B (M ) ⊗ Q ℓ ∼ = r ℓ (M ). This gives rise to an isomorphism of Q ℓ -algebraic groups γ : GL(r B (M )) ⊗ Q ℓ ∼ = GL(r ℓ (M )), under which G mot (M C ) ⊗ Q ℓ is identified with G mot,ℓ (M k ), where the latter is the motivic Galois group of the tannakian category M k AM(k) with fiber functor r l composed with the forgetful functor. The following conjecture is a motivic extension of the Mumford-Tate conjecture [64].
Remark 7.4. The first equality is the content of Conjecture 2.3 and the last equality is the analogous statement saying that all Tate classes are motivated. The original statement of the Mumford-Tate conjecture only predicts that under γ we have MT(H * B (X)) ⊗ Q ℓ = G(H * ℓ (X)) 0 , for any smooth and projective variety X over k.
Let us define a hyper-Kähler variety over k to be a smooth projective variety X over k such that X C is a hyper-Kähler variety. The following result confirms Conjecture 7.3 for the degree-2 part of the motive of X, see Moonen [61] for some generalizations.
Proposition 7.6. If P + (X) is finite (resp. trivial), then the Mumford-Tate conjecture (resp. the motivated Mumford-Tate conjecture) holds for the motive H + (X). If P (X) is finite (resp. trivial), then the Mumford-Tate conjecture (resp. the motivated Mumford-Tate conjecture) holds for the motive H(X) Proof. Let us identify G mot (M k ) ⊗ Q ℓ and G mot,ℓ (M k ) using Artin's comparison isomorphism. Consider the following commutative diagram The two horizontal morphisms on the bottom are isomorphisms due to Theorem 7.5, the vertical map on the left is an isomorphism thanks to Proposition 6.4 and the top left arrow is an isomorphism by Theorem 1.10 since P + (X) is finite by assumption. It follows that all arrows in the diagram are isomorphisms, and so G(H + ℓ (X)) 0 ∼ = G mot (H + (X k )) 0 ⊗ Q ℓ ∼ = MT(H + B (X)) ⊗ Q ℓ . If P + (X) is actually trivial, then G mot (H + (X k )) is connected, and we conclude that the motivated Mumford-Tate conjecture holds for H + (X) in this case.
If the odd cohomology of X is trivial, we are done. Otherwise, assume that P (X) is finite, which implies that also P + (X) is finite. We consider another commutative diagram The horizontal arrows on the bottom are isomorphism due to the above; the top left horizontal map is an isomorphism by Theorem 1.10, since P (X) is finite by assumption, while the leftmost vertical arrow is an isogeny due to Proposition 6.4. It follows that also the middle vertical arrow is an isogeny. We deduce that G mot (H(X k )) 0 ⊗ Q ℓ and G(H * ℓ (X)) 0 are connected algebraic groups of the same dimension over Q ℓ . Hence, the inclusion G(H * ℓ (X)) 0 ֒→ G mot (H(X k )) 0 ⊗ Q ℓ is an isomorphism. If P (X) is actually trivial, then G mot (H(X k )) is connected, and we conclude that the motivated Mumford-Tate conjecture holds for the full André motive H(X).
Definition 7.7. Let k ⊂ C be a finitely generated field. Define C k to be the tannakian subcategory of AM(k) generated by the motives of all hyper-Kähler varieties whose associated complex manifold is one of the four known deformation types.
Remark 7.8. Note that this category contains already the motive of cubic fourfolds, as they are motivated by their Fano varieties of lines (see for example [48]). Very likely, C k also contains the motive of some interesting Fano varieties whose cohomology is of K3-type, for instance, Gushel-Mukai varieties [35] [63], Debarre-Voisin Fano varieties [25] and many more [30].
Theorem 7.9. The motivated Mumford-Tate conjecture holds for any motive M ∈ C k . In particular, for any smooth projective variety motivated by a product of projective hyper-Kähler varieties of known deformation type, the Hodge conjecture and the Tate conjecture are equivalent.
Proof. By Commelin [21,Theorem 10.3], the subcategory of abelian André motives satisfying the Mumford-Tate conjecture 9 is a tannakian subcategory. Therefore, it suffices to check the abelianity and the Mumford-Tate conjecture for the generators of C k . By Corollary 1.15 the defect group of any hyper-Kähler variety X of known deformation type is trivial. Hence, the motive H(X) ∈ C k is abelian by Corollary 1.11, and the motivated Mumford-Tate conjecture holds for its André motive by Proposition 7.6.
Remark 7.10. Thanks to [21], we can put even more generators in the category C k to obtain new evidence for the Mumford-Tate conjecture. Since the conjecture is known to hold for (i) geometrically simple abelian varieties of prime dimension, by Tankeev [79], (ii) abelian varieties of dimension g with trivial endomorphism ring over k such that 2g is neither a k-th power for some odd k > 1 nor of the form 2k k for some odd k > 1, thanks to Pink [72], we deduce that the Mumford-Tate conjecture holds for any variety motivated by a product of varieties in (i) and (ii) above and hyper-Kähler varieties of the known deformation types. See Moonen [61] for more potential examples.
Appendix A. The Kuga-Satake category Let V be a polarizable rational Hodge structure of K3-type, i.e. V is pure of weight 2 with h 2,0 = h 0,2 = 1 and h p,q = 0 whenever p or q is negative. The Kuga-Satake construction [27] produces an abelian variety KS(V ) closely related to V , which is defined up to isogeny. This isogeny class is not unique, but the main point of this Appendix is to characterize the tannakian subcategory of Hodge structures generated by this abelian variety, which we call the Kuga-Satake category attached to V , KS(V ) := H 1 (KS(V )) ⊂ HS pol Q . In the appendix, all the cohomology groups are with rational coefficients and the notation − means the generated tannakian subcategory inside HS pol Q , if not otherwise specified. We first briefly review the classical construction.
A.1. The Kuga-Satake construction. Choose a polarization q of V , and consider the even Clifford algebra Cl + (V, q). Deligne showed in [27] that there is a unique way to induce a weight-1 effective Hodge structure on Cl + (V, q), which is polarizable and therefore equals in which φ ′ is the restriction of the double cover Spin(V, q) → SO(V, q), and the vertical map on the right is an isomorphism due to the fact that w KS (−1) ∈ Spin(V, q).
A.2. The Kuga-Satake category. Given a tannakian subcategory C ⊂ HS pol Q we denote by C ev the full subcategory of C consisting of objects of even weight. The grading via weights on C is given by a central cocharacter w : G m,Q → MT(C). We let ι := w(−1); it acts as −1 on any Hodge structure of odd weight in C and as the identity on C ev . This means that, whenever C contains a Hodge structure of odd weight, the natural morphism of algebraic groups MT(C) → MT(C ev ) is an isogeny with kernel the order-2 cyclic group generated by ι.
Definition A.2. Let V be a polarizable Hodge structure of K3-type. A Kuga-Satake variety for V is an abelian variety A such that H 1 (A) ev = V . Proof. The only-if part is explained before. Conversely, assume that V ∈ H 1 (A) and that the induced surjection MT(H 1 (A)) → MT(V ) is an isogeny of degree 2. Since V has even weight, this morphism factors over MT( H 1 (A) ev ) → MT(V ), and it follows that the the latter is an isomorphism. Hence, H 1 (A) ev = V . By Lemma A.3 and the discussion in §A.1, the abelian variety KS(V ) is a Kuga-Satake variety for V in the sense of our Definition A.2. It is clear that Kuga-Satake varieties are not unique, but the main observation of the appendix is that the corresponding Kuga-Satake category is so.
Theorem A.4. Let V be a polarizable Hodge structure of K3-type. Then there exists a unique tannakian subcategory KS(V ) of HS pol Q such that V = KS(V ) ev KS(V ).
If A is any Kuga-Satake variety for V , we have H 1 (A) = KS(V ).
Let us first prove the following straightforward lemma. Consider tannakian subcategories C ⊂ D of HS pol Q . Assume that both contains some Hodge structure of odd weight. The inclusion of C in D induces surjective homomorphisms of pro-algebraic groups q : MT(D) → MT(C) and q ev : MT(D ev ) → MT(C ev ). Let π denote the double cover MT(D) → MT(D ev ).
Lemma A.5. In the above situation, π −1 (ker(q ev )) has two connected components, and the component containing the identity equals ker(q). The snake lemma implies the isomorphism ker(q) ∼ = ker(q ev ). Moreover, since ι / ∈ ker(q) by assumption, this completely determines ker(q) as the neutral component of the preimage of ker(q ev ).
Proof of Theorem A.4. Assume given two tannakian subcategories D 1 , D 2 ⊂ HS pol Q , both containing some Hodge structure of odd weight and such that V = D ev i D i for i = 1, 2. Let E be the tannakian subcategory generated by D 1 and D 2 . We have surjective morphisms of proalgebraic groups q i : MT(E) → MT(D i ), i = 1, 2. We claim that these are both isomorphisms. it is apparent that ker(q ev 1 ) = ker(q ev 2 ). Lemma A.5 now implies that ker(q 1 ) = ker(q 2 ) in MT(E). But this precisely means that the subcategories D 1 and D 2 of E coincide, and we conclude that we have D 1 = E = D 2 .
Thanks to André's Theorem 2.4, we can lift Proposition A.4 to the category of André motives. The above discussion leads us naturally to the following question about relations among different Kuga-Satake abelian varieties.
Question: Let A and B be abelian varieties such that H 1 (A) = H 1 (B) in HS pol Q . Does this imply the existence of integers k, l, such that A is dominated by B k and B is dominated by A l ?