On the motive of O'Grady's ten-dimensional hyper-Kähler varieties
aa r X i v : . [ m a t h . AG ] J u l ON THE MOTIVE OF O’GRADY’S TEN-DIMENSIONAL HYPER-K ¨AHLERVARIETIES
SALVATORE FLOCCARI, LIE FU, AND ZIYU ZHANG
Abstract.
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 Voevodskymotive of singular moduli spaces of semistable sheaves on K3 and abelian surfaces as well asthe Chow motive of their crepant resolutions, when they exist. We show that these motivesare in the tensor subcategory generated by the motive of the surface, provided that a crepantresolution exists. This extends a recent result of B¨ulles to the O’Grady-10 situation. In thenon-commutative setting, similar results are proved for the Chow motive of moduli spaces of(semi-)stable objects of the K3 category of a cubic fourfold. As a consequence, we provideabundant examples of hyper-K¨ahler varieties of O’Grady-10 deformation type satisfying thestandard conjectures. In the second part, we study the Andr´e motive of projective hyper-K¨ahlervarieties. We attach to any such variety its defect group, an algebraic group which acts on thecohomology and measures the difference between the full motive and its weight-2 part. Whenthe second Betti number is not 3, we show that the defect group is a natural complement of theMumford–Tate group inside the motivic Galois group, and that it is deformation invariant. Weprove the triviality of this group for all known examples of projective hyper-K¨ahler varieties, sothat in each case the full motive is controlled by its weight-2 part. As applications, we show thatfor any variety motivated by a product of known hyper-K¨ahler varieties, all Hodge and Tateclasses are motivated, the motivated Mumford–Tate conjecture 7.3 holds, and the Andr´e motiveis abelian. This last point completes a recent work of Soldatenkov and provides a different prooffor some of his results.
Contents
1. Introduction 12. Generalities on motives 83. Motives of the stable loci of moduli spaces 134. Motive of O’Grady’s moduli spaces and their resolutions 155. Moduli spaces of objects in 2-Calabi–Yau categories 186. Defect groups of hyper-K¨ahler varieties 207. Applications 27Appendix A. The Kuga–Satake category 30References 321.
Introduction
An important source of constructions of higher-dimensional algebraic varieties is given by takingmoduli spaces of (complexes of) coherent sheaves, subject to various stability conditions, onsome lower-dimensional algebraic varieties. The topological, geometric, algebraic and arithmetic
Mathematics Subject Classification.
Key words and phrases.
Moduli spaces, motives, K3 surfaces, hyper-K¨ahler varieties, Mumford–Tateconjecture. properties of the variety are certainly expected to be reflected in and sometimes even controlthe corresponding properties of the moduli space. Such relations can be made precise in termsof cohomology groups (enriched with Hodge structures and Galois actions for instance) or morefundamentally, at the level of motives . The prototype of such interplay we have in mind isdel Ba˜no’s result [27], which says that the Chow motive of the moduli space M r,d ( C ) of stablevector bundles of coprime rank and degree on a smooth projective curve C is a direct summandof the Chow motive of some power of the curve; in other words, the Chow motive of M r,d ( C ) isin the pseudo-abelian tensor subcategory generated by the Chow motive of C . In [27], a preciseformula for the virtual motive of M r,d ( C ) in terms of the virtual motive of C was obtained, aresult which has been recently lifted to the level of motives in a greater generality by Hoskinsand Pepin-Lehalleur [39].In the realm of compact hyper-K¨ahler varieties [12] [42], this philosophy plays an even more im-portant role: it turns out that taking the moduli spaces of (complexes of) sheaves on Calabi–Yausurfaces or their non-commutative analogues provides the most general and almost exhaustiveway for constructing examples, see [64] [68] [69] [72] [93] [10] [11] [9] and [8] etc. As the firstimportant relationship between the K3 or abelian surface S and a moduli space M := M S ( v ) ofstable (complexes of) sheaves on S with (non-isotropic) Mukai vector v , the second cohomologyof M is identified, as a Hodge lattice, with the orthogonal complement of v in e H ( S, Z ), theMukai lattice of S [70] [72]. Regarding the aforementioned result of del Ba˜no in the curve case,a relation between the motive of S and the motive of M is desired. The first breakthrough inthis direction is the following result of B¨ulles [18] based on the work of Markman [60]. Theorem 1.1 (B¨ulles) . Let S be a projective K3 or abelian surface together with a Brauer class α . Let M be a smooth and projective moduli space of stable objects in D b ( S, α ) with respect tosome Bridgeland stability condition. Then the Chow motive of M is contained in the pseudo-abelian tensor subcategory generated by the Chow motive of S . The analogous result on the level of Grothendieck motives or Andr´e motives was obtained beforeby Arapura [6]. It is also worth pointing out that B¨ulles’ method gives a short and new proofof del Ba˜no’s result using the classical analogue of Markman’s result in the curve case provedby Beauville [13].1.1.
Singular or open moduli spaces and resolutions.
The first objective of the paper isto investigate the situations beyond Theorem 1.1.More precisely, let us fix a projective K3 or abelian surface S together with a Brauer class α , anot necessarily primitive Mukai vector v and a not necessarily generic stability condition σ on D b ( S, α ). We want to understand, in terms of the motive of S , the (mixed) motives [89] of thefollowing varieties (or algebraic spaces ). • The (smooth but in general non-proper) moduli space of σ -stable objects M st := M st S,σ ( v , α ) . • The (proper but in general singular) moduli space of σ -semistable objects M := M S,σ ( v , α ) . • A crepant resolution f M of M , if exists.Here is our expectation for their motives: Conjecture 1.2.
Notation is as above. We work with rational coefficients for cohomology groups and motives. All varieties are defined over the fieldof complex numbers if not otherwise specified. See the recent work [2] for the existence of good moduli spaces.
N THE MOTIVE OF O’GRADY’S TEN-DIMENSIONAL HYPER-K ¨AHLER VARIETIES 3 ( i ) The motives and the motives with compact support (in the sense of Voevodsky) of M st and M are in the triangulated tensor subcategory generated by the motive of S within thecategory of Voevodsky’s geometric motives. ( ii ) The Chow motive of f M (if it exists) is in the pseudo-abelian tensor subcategory generatedby the motive of S within the category of Chow motives. Our first main result below confirms Conjecture 1.2 in the presence of a crepant resolution.Recall that by [45], this happens only in the case of O’Grady’s ten-dimensional example [68](extended by [72]).
Theorem 1.3 (=Corollaries 4.5 and 4.6) . Let S be a projective K3 or abelian surface, let α be aBrauer class, let v ∈ e H ( S ) be a primitive Mukai vector with v = 2 , and let σ be a v -genericstability condition on D b ( S, α ) . Denote by M st (resp. M ) the 10-dimensional moduli space of σ -stable (resp. semistable) objects in D b ( S, α ) with Mukai vector v = 2 v . Let f M be any crepantresolution of M . Then the conclusions of Conjecture 1.2 hold. Note that by the result of Rieß [77], birational hyper-K¨ahler varieties have isomorphic Chowmotives, hence we only need to treat one preferred crepant resolution, namely the one constructedby O’Grady [68].
Remark . The Hodge numbers of hyper-K¨ahler varieties of OG10-type are recently computed by de Cataldo–Rapagnetta–Sacc`a in [24] via the decompositiontheorem and a refinement of Ngˆo’s support theorem. A representation theoretic approach wasdiscovered shortly after by Green–Kim–Laza–Robles [35, Theorem 3.26], where the vanishing ofthe odd cohomology is required to conclude. Note that Theorem 1.3 implies in particular thetriviality of the odd cohomology of hyper-K¨ahler varieties of OG10-type and hence allows [35]to obtain an independent proof of [24, Theorem A]; see [35, Remark 3.30].1.2.
Non-commutative Calabi–Yau “surfaces”.
We see in the above setting that the Calabi–Yau surface plays its role almost entirely through its derived category and the second goal ofthe paper is to extend Theorem 1.1 and the results of § A , i.e. an Ext-finite saturated triangulatedcategory in which the double shift [2] is a Serre functor, equipped with Bridgeland stabilityconditions. Such a category often comes as an admissible subcategory of the derived categoryof a Fano variety, as the “key” component (the so-called Kuznetsov component ) in some semi-orthogonal decomposition. We expect the similar relations as in § A and the (non-commutative) motive of A , hencealso the motive of the Fano variety.To be more precise, let us leave the general technical results to § Y be a smooth cubic fourfold and let Ku( Y ) := hO Y , O Y (1) , O Y (2) i ⊥ = { E ∈ D b ( Y ) | Ext ∗ ( O Y ( i ) , E ) = 0 for i = 0 , , } be its Kuznetsovcomponent, which is a K3 category. One can associate with it a natural Hodge lattice e H (Ku( Y ))using topological K-theory [1]. In [9], a natural stability condition on Ku( Y ) is constructed andby the general theory of Bridgeland [16], we have at our disposal a connected component of themanifold of stability conditions, denoted by Stab † (Ku( Y )).Our second main result generalizes B¨ulles’ Theorem 1.1 to this non-commutative setting: Theorem 1.5 (Special case of Theorem 5.3) . Let Y be a smooth cubic fourfold, let Ku( Y ) be itsKuznetsov component, let v ∈ e H (Ku( Y )) be a primitive Mukai vector, and let σ ∈ Stab † (Ku( Y )) be a v -generic stability condition. Then the Chow motive of the projective hyper-K¨ahler manifold A K3 category is a 2-Calabi–Yau category whose Hochschild homology coincides with that of a K3 surface.
SALVATORE FLOCCARI, LIE FU, AND ZIYU ZHANG M := M Ku( Y ) ,σ ( v ) is in the pseudo-abelian tensor subcategory generated by the Chow motiveof Y .Remark . Note that by the recent work of Li–Pertusi–Zhao [53], the moduli spaces consideredin Theorem 1.5 already include the hyper-K¨ahler fourfold F ( Y ) constructed as Fano variety oflines in Y [14] and the hyper-K¨ahler eightfold Z ( Y ) constructed from twisted cubics in Y (when Y does not contain a plane) [51]. In the first case, the conclusion of Theorem 1.5 can be deducedfrom the earlier work of Laterveer [50]; in the second case, our approach was speculated in [19,Remark 2.7]. Nevertheless, Theorem 1.5 applies to the infinitely many complete families ofprojective hyper-K¨ahler varieties recently constructed by Bayer et al. [8].Just as in § resp. semistable) objects M st := M stKu( Y ) ,σ ( v ) ( resp. M := M Ku( Y ) ,σ ( v )) is ingeneral not proper ( resp. smooth). We expect the following analogy of Conjecture 1.2 in thisnon-commutative setting. Conjecture 1.7 (Special case of Conjecture 5.4) . Notation is as above. ( i ) The motives and the motives with compact support (in the sense of Voevodsky) of M st and M are in the triangulated tensor subcategory generated by the motive of Y within thecategory of Voevodsky’s geometric motives. ( ii ) If there exists a crepant resolution f M → M , then the Chow motive of f M is in the pseudo-abelian tensor subcategory generated by the motive of Y within the category of Chow mo-tives. Analogously to Theorem 1.3, our third result establishes Conjecture 1.7 in the ten-dimensionalsituation studied in [54], where a crepant resolution of M exists (it is again of O’Grady-10deformation type). Recall that e H (Ku( Y )) contains (and it is equal to, if Y is very general) acanonical A -lattice generated by λ and λ (see [9] for the notation). Theorem 1.8.
Notation is as above. Assume that Y is very general. Let the Mukai vector v = 2 v with v = λ + λ and let σ be v -generic. Then the conclusions of Conjecture 1.7 holdtrue for M st , M and any crepant resolution f M of M . As a by-product, we deduce Grothendieck’s standard conjectures [36] [46] for many hyper-K¨ahlervarieties of O’Grady-10 deformation type, cf. [20].
Corollary 1.9.
The standard conjectures hold for all the crepant resolutions f M appeared inTheorem 1.3 and Theorem 1.8. Theorem 1.8 and Corollary 1.9 are proved in the end of § Defect groups of hyper-K¨ahler varieties.
It is easy to see that a general projectivedeformation of (a crepant resolution of) a moduli space of semistable sheaves (or objects) on aCalabi–Yau surface is no longer of this form (even by deforming the surface). If we still want tounderstand the motive of such a hyper-K¨ahler variety X in terms of some tensor constructions ofa “surface-like” (or rather “weight-2”) motive, the right substitution of the surface motive wouldbe the degree-2 motive of X itself. We are therefore interested in the following meta-conjecture. Meta-conjecture 1.10.
Let X be a projective hyper-K¨ahler variety and fix some rigid tensorcategory of motives. If the odd Betti numbers of X vanish, then its motive is in the tensorsubcategory generated by its degree-2 motive. In general, the motive of X lies in the tensorsubcategory generated by the Kuga–Satake construction of its degree-2 motive. In any case, themotive 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 [85], related works are [47] and [81].
N THE MOTIVE OF O’GRADY’S TEN-DIMENSIONAL HYPER-K ¨AHLER VARIETIES 5
Unfortunately, staying within the category of Chow motives (or Voevodsky motives), we areconfronted with several essential difficulties: • As an immediate obstruction, to speak of the degree 2 motive, we have to admit thealgebraicity of the K¨unneth projector, which is part of the standard conjectures. • Even in the case where the standard conjectures are known (for example [20]), theconstruction of the degree 2 part of the Chow motive h ( X ), denoted by h ( X ), is stillconjectural in general: assuming the cohomological K¨unneth projector is algebraic, thereis no canonical way to lift it to an algebraic cycle which is a projector modulo rationalequivalence (see Murre [67]); even when such a candidate construction is available (seefor example [79], [87] [34] in some special cases), it seems too difficult to relate h ( X )and h ( X ) for a general X in the moduli space of hyper-K¨ahler varieties. Nevertheless,let us point out that B¨ulles’ Theorem 1.1 and our extensions Theorems 1.3, 1.5 and 1.8indeed 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 1.10. To circum-vent the aforementioned difficulties we leave the category of Chow motives and work within thecategory of
Andr´e motives [4]. Essentially, this amounts to replacing rational equivalence byhomological equivalence and formally adding the cycles predicted by the standard conjectures;the result is a semisimple abelian Q -linear tannakian category, see § M are encoded in itsmotivic Galois group G mot ( M ). Note that since the Hodge theoretic version of meta-conjecture1.10 holds, its validity at the level of Andr´e motives is implied by Conjecture 2.3 which saysthat all Hodge classes are motivated.Our main contribution in this direction is about a Q -algebraic group, which we call the defectgroup , associated with a projective hyper-K¨ahler variety. Let X be a projective hyper-K¨ahlervariety and let H ( X ) be its Andr´e motive. We have the K¨unneth decomposition H ( X ) = L i H i ( X ). The even motive of X is by definition H + ( X ) = L i H i ( X ). The even defect group of X , denoted by P + ( X ), is defined as the kernel of the surjective morphism of motivic Galoisgroups induced by the natural inclusion H ( X ) ⊂ H + ( X ), namely, P + ( X ) := Ker (cid:0) G mot ( H + ( X )) ։ G mot ( H ( X )) (cid:1) . By definition, P + ( X ) is trivial if and only if H + ( X ) belongs to the tannakian subcategory ofAndr´e motives generated by H ( X ).If all the odd Betti numbers of X vanish, then by convention the defect group of X , denotedby P ( X ), is simply P + ( X ). Otherwise, the role of H ( X ) is naturally taken by a Kuga–Satakeabelian variety A attached to this weight-2 motive, see Definition A.2; the reader may safely takefor A the abelian variety given by the classical Kuga–Satake construction [28]. The Kuga–Satakecategory KS ( X ) := hH ( A ) i is independent of the choice of A , see Theorem A.4; furthermore,provided that b ( X ) = 3, we prove in Lemma 6.10 that the motive H ( A ) belongs to thetannakian subcategory of Andr´e motives generated by H ( X ). We define the defect group of X as the kernel of the corresponding surjective morphism P ( X ) := Ker (cid:0) G mot ( H ( X )) ։ G mot ( H ( A )) (cid:1) of motivic Galois groups. The uniqueness of the Kuga–Satake category ensures that P ( X ) doesnot depend on the choice of A ; by definition, the defect group P ( X ) is trivial if and only if H ( X )belongs to the tannakian category KS ( X ).Recall that the motivic Galois group of H ( X ) contains naturally the Mumford–Tate group
MT( H ∗ ( X )). We show that the defect group is a canonical complement. Theorem 1.11 (=Theorem 6.9, Splitting) . Notation is as before. Assume that b ( X ) = 3 .Then, inside G mot ( H ( X )) , the subgroups P ( X ) and MT( H ∗ ( X )) commute, intersect trivially SALVATORE FLOCCARI, LIE FU, AND ZIYU ZHANG with each other and generate the whole group. In short, we have an equality: G mot ( H ( X )) = MT( H ∗ ( X )) × P ( X ) . Similarly, the even defect group is a direct complement of the even Mumford–Tate group in themotivic Galois group of the even Andr´e motive of X , G mot ( H + ( X )) = MT( H + ( X )) × P + ( X ) . It follows that MT( H + ( X )) is canonically isomorphic to G mot ( H ( X )), and hence to MT( H ( X ))by Andr´e’s results [3] [4]. But this is the first step towards the proof of Theorem 1.11 (see Propo-sition 6.4). Note that the natural morphism G mot ( H ( X )) ։ G mot ( H + ( X )) preserves the directproduct decomposition given in the theorem, so that P + ( X ) is a quotient of P ( X ).Theorem 1.11 can be seen as a structure result for the motivic Galois group. The proof is givenin § § P ( X ). Corollary 1.12 (=Corollary 6.11) . For any projective hyper-K¨ahler variety X with b ( X ) = 3 ,the following conditions are equivalent: ( i + ) The even defect group P + ( X ) is trivial. ( ii + ) The even Andr´e motive H + ( X ) is in the tannakian subcategory generated by H ( X ) . ( iii + ) H + ( X ) is abelian. ( iv + ) Conjecture 2.3 holds for H + ( X ) : MT( H + ( X )) = G mot ( H + ( X )) . Similarly, if some odd Betti number of X is not zero, we have the following equivalent conditions: ( i ) The defect group P ( X ) is trivial. ( ii ) The Andr´e motive H ( X ) is in the tannakian subcategory generated by H (KS( X )) , where KS( X ) is any Kuga–Satake abelian variety associated to 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 1.12, Conjecture 2.3 for hyper-K¨ahler varieties and the meta-conjecture1.10 for their Andr´e motives are all equivalent to the following conjecture.
Conjecture 1.13.
The defect group of any projective hyper-K¨ahler variety is trivial.Remark . We are not able to prove Conjecture 1.13 in general so far,but only for all the known examples of hyper-K¨ahler varieties (Corollary 1.16 below). However,( i ) we will show in Corollary 7.2 that Conjecture 1.13 is implied by the following conjecture:an Andr´e 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 coho-mology ring is compatible with the ring structure as well as the Looijenga–Lunts–VerbitskyLie algebra action [55] [85], and most importantly, it is a deformation invariant. Theorem 1.15 (=Theorem 6.12, Deformation invariance of defect groups) . Let S be a smoothquasi-projective variety and X → S be a smooth proper morphism with fibers being projectivehyper-K¨ahler manifolds with b = 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. Applications to “known” hyper-K¨ahler varieties.
In the sequel, a hyper-K¨ahler vari-ety is called known , if it is deformation equivalent to Hilbert schemes of K3 surfaces (K3 [ n ] -type)[12], generalized Kummer varieties associated to abelian surfaces (Kum n -type) [12], O’Grady’s 6-dimensional examples (OG6-type) [69], or O’Grady’s 10-dimensional examples (OG10-type) [68].First, we can prove Conjecture 1.13 for all known hyper-K¨ahler varieties. Corollary 1.16.
The defect group is trivial for all known hyper-K¨ahler varieties.
N THE MOTIVE OF O’GRADY’S TEN-DIMENSIONAL HYPER-K ¨AHLER VARIETIES 7
Combining this with Corollary 1.12, we have the following consequences, providing evidences tothe meta-conjecture 1.10 in the world of Andr´e motives.
Corollary 1.17.
Let X be a known projective hyper-K¨ahler variety. Then ( i ) its Andr´e motive is abelian; ( ii ) for any m ∈ N , all Hodge classes of H ∗ ( X m , Q ) are motivated (hence absolutely Hodge); ( iii ) if X is of K3 [ n ] , OG6, or OG10-type, then H ( X ) ∈ hH ( X ) i ; ( iv ) if X is of Kum n -type, then H ( X ) ∈ hH (KS( X )) i and H + ( X ) ∈ hH ( X ) i . The item ( i ) on the abelianity of Andr´e motive is proved for K3 [ n ] -type by Schlickewei [78], forKum n -type and OG6-type in the recent work of Soldatenkov [80].Second, we can prove the Mumford–Tate conjecture for all known hyper-K¨ahler varieties definedover a finitely generated field extension of Q ; see § [ n ] -type, it has been proven in [32]. In fact, what we obtain in the followingTheorem 1.18 is a stronger result in two aspects: • we identify the Mumford–Tate group and the Zariski closure of the image of the Galoisrepresentation via a third group, namely the motivic Galois group. This is the so-called motivated Mumford–Tate conjecture • we can treat products. In general, it is far from obvious to deduce the Mumford–Tateconjecture for a product of varieties from the conjecture for the factors. Thanks to thework of Commelin [21] this can be done when the Andr´e motives of the varieties involvedare abelian. Theorem 1.18 (Special case of Theorem 7.9) . Let k be a finitely generated subfield of C .For any smooth projective k -variety that is motivated by a product of known hyper-K¨ahlervarieties, the motivated Mumford–Tate conjecture 7.3 holds. In particular, the Tate conjectureand the Hodge conjecture are equivalent for such varieties.Remark . The combination of Corollary 1.12 and Theorem 1.15 (plusthe fact that two deformation equivalent hyper-K¨ahler varieties can be connected by algebraicfamilies) implies that the abelianity of the Andr´e motive of hyper-K¨ahler varieties is a deforma-tion invariant property (Corollary 7.2( i )). When finalizing the paper, we discovered the recentupdate of Soldatenkov’s preprint [80], where he also obtained this result, as well as Corollary1.17( i ), except for the O’Grady-10 case. We attribute the overlap to him. The proofs and pointsof view are somewhat different: [80] makes a detailed study of the Kuga–Satake constructionin families, while our argument does not involve the Kuga–Satake construction when the oddcohomology is trivial, but relies on Andr´e’s theorem [3] on the abelianity of H . As a bonus ofemphasizing the usage of defect groups in our study, on the one hand, there seems to be somepromising approaches mentioned in Remark 1.14 to show the abelianity of the Andr´e motive ofhyper-K¨ahler varieties in general; on the other hand, even if Conjecture 1.13 (hence the abelian-ity) turned out to fail for some deformation family of hyper-K¨ahler varieties, our method canstill control their Andr´e motives by its degree-2 part together with information on the Andr´emotive of one given member in that family, see Corollary 7.2 ( ii ) , ( iii ). Convention:
From § § C if not otherwise stated. We work exclusively with rational coefficients for cohomology groupsand Chow groups, as well as for the corresponding categories of motives. For simplicity, thenotation CHM ( resp. DM, AM) stands for the category of rational Chow motives ( resp. rationalgeometric motives in the sense of Voevodsky, rational Andr´e motives) over a base field k , whichare usually denoted by CHM( k ) Q ( resp. DM gm ( k ) Q , AM( k )) in the literature. A smooth projective variety X is said to be motivated by another smooth projective variety Y if its Andr´emotive H ( X ) belongs to hH ( Y ) i , the tannakian subcategory of Andr´e motives generated by H ( Y ); or equivalently, H ( X ) is a direct summand of the Andr´e motive of a power of Y . Note that any non-zero divisor of Y gives riseto a splitting injection Q ( − → H ( Y ), hence hH ( Y ) i is automatically stable by Tate twists. SALVATORE FLOCCARI, LIE FU, AND ZIYU ZHANG
Acknowledgement:
We want to thank Chunyi Li for helpful discussions and Ben Moonenfor his careful reading of the draft. We also thank the referee for all the helpful comments forimprovement. 2.
Generalities on motives
In this section, we recall various categories of motives that we will be using, gather some of theirbasic properties, and explain some relations between them. Most of the content is standard andwell-documented, except Proposition 2.2 and results of § Chow motives.
Let SmProj k be the category of smooth projective varieties over an arbi-trary base field k . Let CHM be the category of Chow motives with rational coefficients, equippedwith the functor h : SmProj opk → CHM . We follow the notation and conventions of [5]. CHM is a pseudo-abelian rigid symmetric tensorcategory, whose objects consist of triples (
X, p, n ), where X is a smooth projective variety ofdimension d X over the base field k , p ∈ CH d X ( X × k X ) with p ◦ p = p , and n ∈ Z . Morphisms f : M = ( X, p, n ) → N = ( Y, q, m ) are elements γ ∈ CH d X + m − n ( X × k Y ) such that γ ◦ p = q ◦ γ = γ . The tensor product of two motives is defined in the obvious way by the fiberproduct 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 , X denotes the class of the diagonal inside X × k X , and the unitmotive is denoted by := h (Spec( k )). In particular, we have CH l ( X ) = Hom( ( − l ) , h ( X )).The Tate motive of weight − i is the motive ( i ) := (Spec( k ) , ∆ Spec( k ) , i ). A motive is said tobe 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 generatedby M is by definition the smallest full subcategory of CHM containing M that is stable underisomorphisms, direct sums, direct summands, tensor products and duality. We denote thissubcategory by h M i 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 asplitting injection ( − → h ( X ); therefore when X has a non-zero divisor, h h ( X ) i CHM containsthe 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 perfectbase field k . Let DM be Voevodsky’s triangulated category of geometric motives over k withrational coefficients [89]. 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 withcompact 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 rigidtensor triangulated category, where the duality functor is determined by the so-called motivicPoincar´e duality , which says that for any connected smooth k -variety X of dimension d , M ( X ) ∨ ≃ M c ( X )( − d )[ − d ] . (1)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 ,CH i ( X ) = Hom( ( i )[2 i ] , M c ( X )) . (2)An important property we will use is the localization distinguished triangle [89]: let Z be aclosed subscheme of X ∈ Sch k , then there is a distinguished triangle in DM: M c ( Z ) → M c ( X ) → M c ( X \ Z ) → M c ( Z )[1] . (3) N THE MOTIVE OF O’GRADY’S TEN-DIMENSIONAL HYPER-K ¨AHLER VARIETIES 9
Given a mixed motive M ∈ DM, the tensor triangulated subcategory of DM generated by M ,denoted by h M i DM is the smallest full subcategory of DM containing M that is stable underisomorphisms, direct sums, tensor products, duality and cones (hence also shifts and directsummands). By definition, h M i DM is a pseudo-abelian rigid tensor triangulated category. Againfor a smooth projective variety X admitting a non-zero effective divisor, h M ( X ) i DM containsall Tate motives, and hence it is also stable by Tate twists.By [89], there is a fully faithful tensor functorCHM op −→ DM , 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 ( − i ) in CHM is sent to ( i )[2 i ]. In this paperwe identify CHM op with its essential image in DM. Question . If a Chow motive can be obtained from another Chowmotive by performing tensor operations and cones in DM, can it be obtained already withinCHM by performing tensor operations therein?The following observation gives a positive answer to this question.
Proposition 2.2.
Notation is as before. Let M be a Chow motive. Then we have an equalityof subcategories of DM : h M i CHM = h M i DM ∩ CHM . Proof.
The argument is due to Wildeshaus [90, Proposition 1.2], which we reproduce here forthe convenience of the readers. This statement is also independently discovered by Hoskins–Pepin-Lehalleur recently in [40]. In [15] Bondarko introduced the notion of weight structureson triangulated categories and constructs a bounded non-degenerate weight structure w on DMwhose heart DM w =0 consists of the Chow motives CHM op . As h M i DM is generated by thesubcategory h M i CHM and the latter, being a subcategory of CHM, is negative in the sense of[15, Definition 4.3.1], we can apply [15, Theorem 4.3.2 II] to conclude that there exists a uniquebounded weight structure v on h M i DM whose heart h M i v =0DM is h M i CHM . By shifting, we see thatfor any n ∈ Z , h M i v = n DM ⊂ DM w = n . (4)We claim that for any n , we have h M i v > n DM ⊂ DM w > n ∩h M i DM and h M i v n DM ⊂ DM w n ∩h M i DM . (5)Indeed, it suffices to show the first inclusion in the case n = 0. Given any object N of h M i v > , bythe boundedness of v , N can be obtained by a finite sequence of successive extensions of objectsof h M i DM with non-negative v -weight, which have non-negative w -weight by (4). Therefore N ∈ DM w > , and the claim is proved.Now we show that the inclusions in (5) are actually equalities. Given N ∈ DM w > ∩h M i DM , wehave Hom( N, N ′ ) = 0 for all N ′ ∈ h M i v − since, by the second inclusion in (5), N ′ ∈ DM w − .Therefore N ∈ h M i v > . The first equality is proved; the argument for the second one is similar.As a consequence, h M i CHM = h M i v =0DM = DM w =0 ∩h M i DM = CHM ∩h M i DM . (cid:3) Obviously, the proof shows that the same result holds if we start with a subcategory of CHMinstead of just an object. The case of the subcategory of abelian motives is exactly [90, Propo-sition 1.2], from which we borrowed the argument above.Note that due to the abstract machinery of weight structures, the above proof does not givea constructive way to eliminate the usage of cones if a Chow motive is explicitly expressed interms of a second one by tensor operations and cones.
Andr´e 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 equiv-alence (here we use the rational singular homology group of the associated complex analyticspace) in the construction of Chow motives ( § Grothendieckmotives , denoted GRM, which comes with a canonical full functor CHM → GRM. The categoryof Grothendieck motives is conjectured to be semisimple and abelian; Jannsen [44] showed thatit is the case if and only if numerical equivalence agrees with homological equivalence, which isone of Grothendieck’s standard conjectures.The standard conjectures being difficult, in [4] an unconditional theory was proposed by Andr´e,refining Deligne’s category of absolute Hodge motives [30]. He replaced in the construction ofGrothendieck 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 usingalgebraic cycles and the Hodge ∗ -operator. The resulting category of Andr´e motives is denotedby AM, and it is a semisimple abelian category. The canonical faithful functor GRM → AM isan isomorphism if the standard conjectures hold true for all smooth projective varieties.The virtue of AM is that it works well with the tannakian formalism. There are natural functors:SmProj op H −→ AM r −→ HS pol Q F −→ Vect Q , where H is the functor that associates to a variety its Andr´e motive, HS pol Q is the category ofpolarizable rational Hodge structures, r is the Hodge realization functor, and F is the forgetfulfunctor. The composition of r ◦ H is equal to the functor H attaching to a smooth projectivevariety its rational cohomology group. It is easy to see that the functors r and F are conservative.2.3.1. Mumford–Tate group and motivic Galois group.
It is well-known that HS pol Q is a neutraltannakian semisimple abelian category with fiber functor F . Given a polarizable rational Hodgestructure V , let h V i HS be the full tannakian subcategory of HS pol Q generated by V . The restrictionof F to this subcategory is again a fiber functor. The Mumford–Tate group of V , denoted byMT( V ), is by definition the automorphism group of the tensor functor F | h V i HS and h V i HS isequivalent to the category of representations of MT( V ). Note that as V is assumed to bepolarizable, MT( V ) is reductive. Mumford–Tate groups are known to be always connected. TheMT( 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 Andr´emotive M ∈ AM, the tannakian subcategory h M i AM is again neutral tannakian, with fiberfunctor F ◦ r | h M i AM ; the tensor automorphism group of this functor is denoted by G mot ( M ) andcalled the motivic Galois group of M . The tannakian category h M i AM is then equivalent to thecategory of representations of this reductive group, and the G mot ( M )-invariants in any tensorconstruction of r ( M ) are precisely the motivated classes.2.3.2. Motivated vs. Hodge.
Let k ⊂ C be in addition algebraically closed. For any M ∈ AM, asall motivated cycles are Hodge classes, the tensor invariants of the motivic Galois group are alltensor invariants of the Mumford–Tate group. Both groups being reductive, we have a canonicalinclusion MT( r ( M )) ⊂ G mot ( M ), by [30, Proposition 3.1]. The Hodge conjecture implies thatthe converse should hold as well. Conjecture 2.3 (Hodge classes are motivated) . Let k be an algebraically closed subfield of C .For any M ∈ AM , we have an equality of subgroups of GL (cid:0) r ( M ) (cid:1) : MT (cid:0) r ( M ) (cid:1) = G mot ( M ) . Since Mumford–Tate groups are connected, Conjecture 2.3 predicts in particular that G mot ( M )should also be connected; already this statement is a difficult open problem.The most significant evidence to this conjecture is Andr´e’s result in [4] saying that on abelianvarieties, all Hodge classes are motivated, strengthening the previous result of Deligne [30] onabsolute Hodge classes. Let us state the result in the following form: N THE MOTIVE OF O’GRADY’S TEN-DIMENSIONAL HYPER-K ¨AHLER VARIETIES 11
Theorem 2.4 ([4, Theorem 0.6.2]) . Conjecture 2.3 holds for any abelian Andr´e motives. Moreprecisely, over an algebraically closed field k ⊂ C , for any M ∈ AM ab , the rigid tensor subcate-gory of AM generated by the motives of abelian varieties, we have MT (cid:0) r ( M ) (cid:1) = G mot ( M ) . Relative Andr´e motives and monodromy: proper setting.
Another remarkableaspect of Andr´e motives is their behaviour under deformations. The results presented beloware essentially due to Andr´e (based on Deligne [29]) in the projective setting and formalized byMoonen [62, § k be an uncountableand algebraically closed subfield of C . The starting point is the following observation. Lemma 2.5.
The contra-variant functor H : SmProj k → AM extends naturally to the category SmProp k of smooth proper varieties.Proof. Let X be a smooth and proper (non-necessarily projective) algebraic variety defined over k . Consider its Nori motive H Nori ( X ) = L i H i Nori ( X ). For each i ∈ N , H i Nori ( X ) carries a weightfiltration W • , inducing the weight filtration on its Hodge realization [41, Theorem 10.2.5]. Inparticular, r (cid:0) Gr Wl H i Nori ( X ) (cid:1) = Gr Wl H i ( X ) . However, the Hodge structure H i ( X ) is pure , Gr Wl H i ( X ) is zero for all l = 0. By the conser-vativity of r , the Nori motive Gr Wl 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 pureNori motives is equivalent to the category of Andr´e motives. (cid:3) The following result generalizes Andr´e’s deformation principle for motivated cycles [4, Th´eor`eme0.5] to the proper setting (but always with projective fibers). It has been obtained recently bySoldatenkov [80, Proposition 5.1]. We include here an alternative proof with the point beingthat Andr´e’s original proof actually works, when combined with Lemma 2.5.
Theorem 2.6 (Andr´e–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 ( S, R i f ∗ Q ( i )) , and assume thatthere exists s ∈ S such that the restriction ξ s ∈ H i ( X s , Q ( i )) of ξ to the fibre over s ismotivated. Then, for all s ∈ S , the class ξ s ∈ H i ( X s , Q ( i )) is motivated.Proof. As in [4], we can assume that S is a smooth affine curve. Choose a smooth compacti-fication 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 [29, 4.1.1] ensures that the image of the morphism of Hodgestructures j ∗ s : H i ( X , Q ( i )) → H i ( X s , Q ( i )) coincides with the subspace of monodromy in-variants. Andr´e’s proof uses the morphism j ∗ s induced on Andr´e motives, and conclude thatthe subspace of monodromy invariants at s ∈ S is a submotive which does not depend on thechosen point. Now, in our case X is not necessarily projective, but still has a well-defined Andr´emotive H ( X ) = L i H i ( X ) by Lemma 2.5, and j ∗ s is a morphism of Andr´e motives. Then wecan conclude via the same argument as in Andr´e [4]. (cid:3) The following definition extends slightly the usual notion of families of Andr´e motives.
Definition 2.7 ( cf. [62, Definition 4.3.3]) . Let S be a smooth connected quasi-projective variety.An Andr´e motive ( resp. generalized Andr´e motive ) over S is a triple ( X /S, e, n ) with • f : X → S a smooth projective ( resp. proper) morphism with connected projective fibers, • e a global section of R d ( f × f ) ∗ Q X × S X ( d ), where d is the relative dimension of f , • n an integer,such that for some s ∈ S (or equivalently by Theorem 2.6, for any s ∈ S ), the value e ( s ) ∈ H d ( X s × X s , Q ( d )) is a motivated projector. This can be easily seen in the following way: by Chow’s lemma, one can find a blow-up e X → X with e X smooth and projective. Then by the projection formula, H i ( X ) is a direct summand of the pure Hodge structure H i ( X ), hence is also pure. These objects, with morphisms defined in the usual way, form a tannakian semisimple abeliancategory denoted by AM( S ) ( resp. g AM( S )). Obviously, a generalized Andr´e motive over a pointis nothing else but an Andr´e motive introduced before. There is a natural realization functorfrom the category of generalized Andr´e motives over S to the tannakian category of algebraic variations of Q -Hodge structures in the sense of Deligne [29, Definition 4.2.4]:AM( S ) ⊂ g AM( S ) r −→ VHS a Q ( S ) ⊂ VHS pol Q ( S ) . By construction, for any smooth proper morphism f : X → S with projective fibers and anyinteger i , we have a generalized Andr´e motive H i ( X /S ) whose realization is R i f ∗ Q ∈ VHS a Q ( S ).Given a (generalized) Andr´e motive M/S ∈ g AM( S ), we aim to study the variation of motivicGalois groups G mot ( M s ) and Mumford–Tate groups MT( r ( M ) s ) when s varies in S . Consider themonodromy representation π ( S, s ) → GL( r ( M ) s ) associated to the local system underlying therealization 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 finite ´etale cover of S ; Deligne [29,Theorem 4.2.6] proved that G mono ( M/S ) s is a semisimple Q -algebraic group. The variation ofthese groups with s determines a local system of algebraic groups G mono ( M/S ). Theorem 2.8 ( cf. [62, § . Let S be as above and let M/S be a generalized Andr´e motive over S . There exists two local systems of reductive algebraic groups MT( r ( M ) /S ) and G mot ( M/S ) over S with the following properties:(i) we have inclusions of local systems of algebraic groups: G mono ( M/S ) ⊂ MT( r ( M ) /S ) ⊂ G mot ( M/S ) ⊂ GL( r ( M ) /S ); (ii) for a very general (i.e., outside of a countable union of closed subvarieties of S ) point s ∈ S , we have MT( r ( M ) s ) = MT( r ( M ) /S ) s and G mot ( M s ) = G mot ( M/S ) s ;(iii) for all s ∈ S , we have MT( r ( M ) s ) ⊂ MT( r ( M ) /S ) s and G mot ( M s ) ⊂ G mot ( M/S ) s ;(iv) for all s ∈ S , we have G mono ( M/S ) s · MT( r ( M ) s ) = MT( r ( M ) /S ) s and G mono ( M/S ) s · G mot ( M s ) = G mot ( M/S ) s . In particular, each of the inclusion in ( iii ) is an equality if and only if G mono ( M/S ) s iscontained respectively in MT( r ( M ) s ) and G mot ( M s ) .The local system MT( r ( M ) /S ) is called the generic Mumford–Tate group of r ( M ) /S , and G mot ( M/S ) is called the generic motivic Galois group of M/S .Proof.
There exists a non-empty Zariski open subset U ⊂ S such that the restriction of M/S to U is an Andr´e motive over U . The desired conclusions hold for the restricted family over U byTheorems 4.1.2, 4.1.3, 4.3.6, and 4.3.9 in Moonen’s survey [62]; hence, we get two local systemsof algebraic groups over U with the properties above. The fundamental group of S is a quotientof that of U . Since ( i ) holds over U , we can extend the generic Mumford–Tate and motivicGalois 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 ) areimmediate 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 genericMumford–Tate group is similar. Up to a base change of the family M/S by a finite ´etale coverof S , we may assume that the algebraic monodromy group is connected. Let s ∈ S be any pointsuch that G mono ( M/S ) s is contained in G mot ( M s ); this is the case for a very general point, by( i ) and ( ii ). The monodromy group acts on G mot ( M s ) by conjugation, and this defines a localsystem of algebraic groups G mot ( M s /S ) with fiber isomorphic to the motivic Galois group atthe point s . Consider any tensor construction T /S = (
M/S ) ⊗ m ⊗ ( M/S ) ∨ , ⊗ n , and let ξ s be A variation of Q -Hodge structures over S is called algebraic if the restriction to some non-empty Zariski opensubset U of S is a direct summand of a variation of the form R i f ∗ Q ( j ) for some smooth projective morphism f : X → U and some integer j . N THE MOTIVE OF O’GRADY’S TEN-DIMENSIONAL HYPER-K ¨AHLER VARIETIES 13 the cohomology class of a motivated cycle in r ( T ) s . The class ξ s is monodromy invariant, andtherefore it extends to a global section ξ of the local system underlying r ( T ) /S . By Theorem 2.6the restriction ξ s is motivated for any s ∈ S . By the reductivity of the groups involved, we deducethat for any s ∈ S we have G mot ( M s ) ⊂ G mot ( M s /S ) s , and we conclude by ( ii ) that the lattermust be equal to G mot ( M/S ) s . This proves ( iii ) and that if G mono ( M/S ) s ⊂ G mot ( M s ) thenG mot ( M s ) = G mot ( M/S ) s .( iv ). By ( i ) and ( iii ), we clearly have G mono ( M/S ) s · G mot ( M s ) ⊂ G mot ( M/S ) s . Since bothsides 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 ) s · G mot ( M s ), then itis the class of a motivated cycle which is monodromy invariant. By Theorem 2.6, it extends to aglobal section ξ of r ( T ) /S such that ξ s ′ is motivated at any s ′ ∈ S . It follows that ξ s is invariantfor G mot ( M/S ) s . The proof of the assertion regarding the Mumford–Tate group is similar. (cid:3) Relations.
We summarize in the diagram below the natural functors relating the variouscategories of motives we discussed above. For the sake of completeness, we inserted in thediagram also Nori’s category of mixed motives MM
Nori , whose pure part is the abelian categoryof Andr´e motives by Arapura’s result in [7], see also [41] for a recent account.SmProj opk (cid:31) (cid:127) / / (cid:127) _ (cid:15) (cid:15) h & & ▲▲▲▲▲▲▲▲▲▲▲ SmProp opk H & & ◆◆◆◆◆◆◆◆◆◆◆◆◆ + + ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ H ∗ , , ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ CHM / / (cid:127) _ (cid:15) (cid:15) AM r / / (cid:127) _ (cid:15) (cid:15) HS pol Q F / / (cid:127) _ (cid:15) (cid:15) Vect Q MM Nori r / / MHS Q F / / Vect Q DM op C / / D b (MM Nori ) ⊕ H i O O r / / D b (MHS Q ) F / / ⊕ H i O O D b (Vect Q ) ⊕ H i O O Sch opk M qqqqqqqqqqqq ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ ❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝ (6)Here the comparison functor C is due to Harrer [38, Theorem 7.4.17].3. Motives of the stable loci of moduli spaces
In this section, we generalize an argument of B¨ulles [18] to give a relationship between the motiveof the (in general quasi-projective) moduli space of stable sheaves on a K3 or abelian surfaceand the motive of the surface.Let S be a projective K3 surface or abelian surface. Denote by f NS( S ) = H ( S, Z ) ⊕ NS( S ) ⊕ H ( S, Z ) the algebraic Mukai lattice, equipped with the following Mukai pairing: for any v =( r, l, s ) and v ′ = ( r, l, s ′ ) in f NS( S ), h v , v ′ i := ( l, l ′ ) − rs ′ − r ′ s ∈ Z . Given a Brauer class α , a Mukai vector v ∈ f NS( S ) with v > σ of the α -twisted derived category D b ( S, α ), let M st be the moduli space of σ -stableobjects in D b ( S, α ) with Mukai vector v . By [64], M st is a smooth quasi-projective holomorphicsymplectic variety of dimension 2 m := v + 2. To understand the (mixed) motive of M st , let usfirst recall the following result of Markman, extended by Marian–Zhao. Theorem 3.1 ([60] [59] [58]) . Let E and F be two (twisted) universal families over M st × S .Then ∆ M st = c m ( − E xt ! π ( π ∗ ( E ) , π ∗ ( F ))) ∈ CH m ( M st × M st ) , where m is the dimension of M st and E xt ! π ( π ∗ ( E ) , π ∗ ( F )) denotes the class of the complex Rπ , ∗ ( π ∗ ( E ) ∨ ⊗ L π ∗ ( F )) in the Grothendieck group of M st × M st , where π ij ’s are the naturalprojections from M st × S × M st .Pointer to references. For the case of Gieseker-stable sheaves, [60, Theorem 1] states the resultfor the cohomology class, but the proof gives the equality in Chow groups. Indeed, in [59,Theorem 8], the statement is for Chow groups. Moreover, the assumption on the existence of auniversal family can be dropped ([59, Proposition 24]): it suffices to replace in the formula thesheaves E and F by certain universal classes in the Grothendieck group K ( S ×M st ) constructedin [59, Definition 26]. More recently, it is shown in [58] that the technique of Markman can beadapted to obtain the result in the full generality as stated. (cid:3) As a consequence, we can obtain the following analogue of [18, (3), p.6]
Proposition 3.2 (Decomposition of the diagonal) . There exist finitely many integers k i andcycles γ i ∈ CH e i ( M st × S k i ) , δ i ∈ CH d i ( S k i × M st ) , such that ∆ M st = X δ i ◦ γ i ∈ CH m ( M st × M st ) , here dim M st = 2 m = e i + d i − k i for all i .Proof. We follow the proof of [18, Theorem 1]. First of all, we observe that by Lieberman’sformula (see [5, § ∗ ( M st × M st ) (with respect to the ring structure given by the composition of correspon-dences) I = h β ◦ α | α ∈ CH ∗ ( M st × S k ) , β ∈ CH ∗ ( S k × M st ) , k ∈ N i ⊆ CH ∗ ( M st × M st )is closed under the intersection product, hence is a Q -subalgebra of CH ∗ ( M st × M st ). A com-putation similar to [18, (2), p.6] using the Grothendieck–Riemann–Roch theorem shows thatch( − [ E xt ! π ( π ∗ ( E ) , π ∗ ( F ))]) = − ( π ) ∗ ( π ∗ α · π ∗ β )where α = ch( E ∨ ) · π ∗ p td( S ) and β = ch( F ) · π ∗ p td( S ) . It follows that ch n ( − [ E xt ! π ( π ∗ ( E ) , π ∗ ( F ))]) ∈ I for any n ∈ N . An induction argument thenshows that c n ( − [ E xt ! π ( π ∗ ( E ) , π ∗ ( F ))]) ∈ I for each n ∈ N . In particular, combined withTheorem 3.1, ∆ M st is in I , which is equivalent to the conclusion. (cid:3) In terms of mixed motives, one can reformulate Proposition 3.2 as follows.
Corollary 3.3 (Factorization of the comparison map) . In the category DM of mixed motives,the canonical comparison morphism M ( M st ) → M c ( M st ) can be factorized as the followingcomposition: M ( M st ) → M i M ( S k i )( e i − k i )[2 e i − k i ] → M c ( M st ) , 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 ) isequal to the space Hom DM ( M ( M st ) , M ( S k i )( j − k i )[2 j − k i ])as well as to the space Hom DM ( M ( S k i )(2 m − j )[4 m − j ] , M c ( M st )) . (cid:3) N THE MOTIVE OF O’GRADY’S TEN-DIMENSIONAL HYPER-K ¨AHLER VARIETIES 15
Remark . In Proposition 3.2, if one denotes γ = ⊕ γ i and δ = ⊕ δ i , thenwe get the following morphisms of mixed Hodge structures. H ∗ c ( M st ) γ −→ M i H ∗ ( S k i )(2 k i − e i ) δ −→ H ∗ ( M st ) , where the composition is precisely the comparison morphism from the compact support coho-mology to the usual cohomology. Remark . In the case that S is an abelian surface,the moduli space M st is isotrivially fibered over S × b 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 Theorem3.1 seems to be unknown for K st .4. Motive of O’Grady’s moduli spaces and their resolutions
In this section, we study the motive of O’Grady’s 10-dimensional hyper-K¨ahler varieties [68].Those are symplectic resolutions of certain singular moduli spaces of sheaves on K3 or abeliansurfaces. We first recall the construction.4.1.
Symplectic resolution of the singular moduli space.
Let S be a projective K3 surfaceor abelian surface, let α be a Brauer class, and let v = 2 v be a Mukai vector, such that v ∈ f NS( S ) is primitive with v = 2. Let σ be a v -generic stability condition on the α -twistedderived category D b ( S, α ) (for example, a v -generic polarization). We write M st = M S,σ ( v , α ) st for the smooth and quasi-projective moduli space of σ -stable objects in D b ( S, α ) with Mukaivector v , and M = M S,σ ( v , α ) ss for the (singular) moduli space of σ -semistable objects with the same Mukai vector. In [68],O’Grady constructed a symplectic resolution f M of M (see also [45]), which is a projective (irre-ducible if S is a K3 surface) holomorphic symplectic manifold of dimension 10, not deformationequivalent to the fifth Hilbert schemes of the surface S . We know that these hyper-K¨ahlervarieties are all deformation equivalent [72].Let us briefly recall the geometry of M . We follow the notations in [68], see also [52] and [61, § M admits a filtration M ⊃ Σ ⊃ Ωwhere Σ = Sing( M ) = M \ M st ∼ = Sym ( M S,σ ( v , α ))is the singular locus of M , which consists of strictly σ -semistable objects; andΩ = Sing(Σ) ∼ = M S,σ ( v , α )is the singular locus of Σ, hence the diagonal in Sym ( M S,σ ( v , α )). Notice that M S,σ ( v , α )is a smooth projective holomorphic symplectic fourfold deformation equivalent to the Hilbertsquares of S .In [68], O’Grady produced a symplectic resolution f M of M in three steps. As the explicitgeometry 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 onlysingularity of M is an A -singularity along the strict transform Σ of Σ. In fact, Σ is smooth,satisfying Σ ∼ = Hilb ( M S,σ ( v , α )) , with the morphism Σ → Σ being the corresponding Hilbert-Chow morphism, whose exceptionaldivisor is precisely the intersection of Ω and Σ in M . Step 2.
We blow up M along Σ to obtain a (non-crepant) resolution c M of M . The exceptionaldivisor b Σ is thus a P -bundle over Σ. We denote by b Ω the strict transform of Ω. Then c M is asmooth projective compactification of M st , with boundary ∂ c M = c M \ M st = b Ω ∪ b Σbeing the union of two smooth hypersurfaces which intersect transversally.
Step 3.
Lastly, an extremal contraction of c M contracts b Ω as a P -bundle to e Ω, which is a3-dimensional quadric bundle (more precisely, the relative Lagrangian Grassmannian fibrationassociated to the tangent bundle) over Ω. The space obtained is denoted by f M , which is shownto be a symplectic resolution of M . Remark . By the main result of Lehn–Sorger [52], O’Grady’s symplectic resolution can alsobe obtained by a single blow-up of M along its (reduced) singular locus Σ. The exceptionaldivisor e Σ is nothing else but the image of b Σ under the contraction in the third step describedabove, which is singular along e Ω, the preimage of Ω. If we blow up f M along e Ω, we will obtainagain c M , with the exceptional divisor being b Ω and the strict transform of e Σ being b Σ. In short,the order of blow-ups can be “reversed”; see the following commutative diagram from [61, § e Ω f M = c M = Bl Σ M u u ❦❦❦❦❦❦❦❦❦❦❦❦❦ ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ f M = Bl Σ M ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ M = Bl Ω M u u ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ M . The motive of O’Grady’s resolution.
We will compute the Chow motives of the bound-ary components of c M , then describe the Chow motives of the resolutions c M and f M . We startwith the following observation. Lemma 4.2.
Let X be a smooth projective variety. The Chow motive h (Hilb ( X )) belongs to h h ( X ) i CHM , the pseudo-abelian tensor subcategory of
CHM generated by h ( X ) .Proof. We assume dim X = n . Let ∆ X ⊆ X × X be the diagonal, then by [57, § h (Bl ∆ X ( X × X )) = h ( X ) ⊕ (cid:0) ⊕ n − i =1 h ( X )( − i ) (cid:1) . Since Hilb ( X ) = Bl ∆ X ( X × X ) / Z , its motive is the Z -invariant part h (Hilb ( X )) = h (Bl ∆ X ( X × X )) Z which is a direct summand of h (Bl ∆ X ( X × X )), hence is contained in the desired subcategory. (cid:3) Lemma 4.3.
The Chow motives h ( b Σ) , h ( b Ω) and h ( b Σ ∩ b Ω) are all contained in the subcategory h h ( S ) i CHM .Proof.
By O’Grady’s construction, b Σ is a P -bundle over Σ ∼ = Hilb ( M S,σ ( v , α )). It followsfrom [57, §
7] that h ( b Σ) = h (Σ) ⊕ h (Σ)( − . By [18, Theorem 0.1], h ( M S,σ ( v , α )) is in the tensor subcategory of Chow motives generatedby h ( S ). It follows by Lemma 4.2 that h (Σ) is also in this subcategory, therefore so is h ( b Σ).Again by O’Grady’s construction, b Ω is a P -bundle over e Ω. It follows that h ( b Ω) = h ( e Ω) ⊕ h ( e Ω)( − ⊕ h ( e Ω)( − . N THE MOTIVE OF O’GRADY’S TEN-DIMENSIONAL HYPER-K ¨AHLER VARIETIES 17
Moreover, since e Ω is a 3-dimensional quadric bundle over Ω, by [86, Remark 4.6], we have that h ( e Ω) = h (Ω) ⊕ h (Ω)( − ⊕ h (Ω)( − ⊕ h (Ω)( − . Since Ω ∼ = M S,σ ( v , α ), it follows by [18, Theorem 0.1] that h (Ω) is in the thick tensor subcate-gory of Chow motives generated by h ( S ), hence the same is true for h ( e Ω) and h ( b Ω).Similarly, the intersection b Σ ∩ b Ω is a smooth conic bundle over e Ω, again by [86, Remark 4.6], itsmotive is in the tensor subcategory generated by that of e Ω. One concludes as for b Ω. (cid:3) Here comes the key step of the proof.
Proposition 4.4.
The Chow motive h ( c M ) belongs to h h ( S ) i CHM . We give two proofs with the same starting point, namely Proposition 3.2. The difference is thatthe first one is elementary by staying in the category of Chow motives and is geometric so thatin principle it gives rise to an explicit expression of the Chow motive h ( c M ) in terms of h ( S ); thesecond one is quicker by using mixed motives and Proposition 2.2, but it is hopeless to deduceany concrete relation between these two motives via this approach. First proof of Proposition 4.4.
By Proposition 3.2, we have[∆ M st ] = X δ i ◦ γ i ∈ CH ( M st × M st ) , where γ i ∈ CH e i ( M st × S k i ) and δ i ∈ CH d i ( S k i × M st ). Let b γ i ∈ CH e i ( c M × S k i ) and b δ i ∈ CH d i ( S k i × c M ) be any closure of cycles representing γ i and δ i respectively. Then the supportof the class ∆ c M − X b δ i ◦ b γ i ∈ CH ( c M × c M )lies in the boundary ( c M × ∂ c M ) ∪ ( ∂ c M × c M ), hence we can write in CH ( c M × c M )∆ c M = X b δ i ◦ b γ i + Y b Σ + Y b Ω + Z b Σ + Z b Ω (7)for some algebraic cycles Y b Σ ∈ CH ( c M × b Σ), Y b Ω ∈ CH ( c M × b Ω), Z b Σ ∈ CH ( b Σ × c M ) and Z b Ω ∈ CH ( b Ω × c M ).For each i , the cycles b γ i and b δ i can be viewed as morphisms of motives h ( c M ) b γ i −→ h ( S k i )( n i ) b δ i −→ h ( c M )for n i = e i − m = 2 k i − d i . On the other hand, we denote by j b Σ and j b Ω the closed embeddingof b Σ and b Ω in c M respectively. Then we have morphisms of motives h ( c M ) Y b Σ −→ h ( b Σ) ( j b Σ ) ∗ −→ h ( c M ) , h ( c M ) Y b Ω −→ h ( b Ω) ( j b Ω ) ∗ −→ h ( c M ) , h ( c M ) j ∗ b Σ −→ h ( b Σ)( − Z b Σ −→ h ( c M ) , h ( c M ) j ∗ b Ω −→ h ( b Ω)( − Z b Ω −→ h ( c M ) . It follows by (7) that the sum of all the above compositions add up to the identity on h ( c M ).Hence h ( c M ) is a direct summand of (cid:16) ⊕ i h ( S k i )( n i ) (cid:17) ⊕ h ( b Σ) ⊕ h ( b Ω) ⊕ h ( b Σ)( − ⊕ h ( b Ω)( − . Combining this with Lemma 4.3, we finish the proof. (cid:3)
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 with compactsupport of each stratum is in some triangulated tensor subcategory of DM, then so is the motivewith compact support of the ambiant space; conversely, if the motive with compact supportof the ambiant scheme as well as those of all but one strata are in some triangulated tensorsubcategory of DM, then so is the motive with compact support of the remaining stratum.Now from the geometry recalled in § c M has a stratification with four strata M st , b Ω \ b Σ, b Σ \ b Ω, b Ω ∩ b Σ. By Lemma 4.3, Proposition 3.2 and the previous paragraph, the motiveswith compact support of c M as well as those of its strata and their closures are in h M ( S ) i DM .Since c M is smooth and projective, its motive lies in the subcategory of Chow motives, hence in h h ( S ) i CHM by Proposition 2.2. (cid:3)
Corollary 4.5.
The Chow motive h ( f M ) is contained in h h ( S ) i CHM .Proof.
Since c M is a blow-up of f M along a smooth center, it follows by [57, §
9] that h ( f M ) is adirect summand of h ( c M ). Then the conclusion follows from Proposition 4.4 together with thefact that h h ( S ) i CHM is closed under direct summands. (cid:3)
Corollary 4.6.
The mixed motives M ( M st ) , M c ( M st ) and M ( M ) ≃ M c ( M ) all belong to h M ( S ) i DM , the triangulated tensor subcategory of DM generated by M ( S ) .Proof. Recall first that c M has a stratification with strata being M st , b Ω ∩ b Σ, b Σ \ b Ω and b Ω \ b Σ.By Lemma 4.3 and Proposition 4.4, together with a repeated use of the distinguished trianglefor motives with compact support ([89, P.195]) yields that the motives with compact supportof all strata as well as their closures are in h M ( S ) i DM . This proves the claim for M c ( M st ) and M c ( M ). The remaining claim for M ( M st ) follows from the motivic Poincar´e duality (1). (cid:3) Corollary 4.7.
There are infinitely many projective hyper-K¨ahler varieties of O’Grady-10 de-formation type whose Chow motive is abelian.Proof.
By Corollary 4.5, it suffices to see that there are infinitely many projective K3 surfaceswith abelian Chow motives. To this end, we can take for example the Kummer K3 surfaces orK3 surfaces with Picard number at least 19 by [71]. (cid:3)
Remark . When S is an abelian surface, the previouslyconsidered moduli spaces M st , M , c M and f M are all isotrivally fibered over S × b S , via themap E ( c ( E ) , alb( c ( E ))). Let us denote the corresponding fibers by K st , K , b K , e K , calledKummer moduli spaces of sheaves. Except for some special cases like generalized Kummervarieties (see [34]), the analogy of Proposition 4.4, Corollary 4.5 and Corollary 4.6 are unknownfor those fibers in general. The missing ingredient is the analogy of Markman’s Theorem 3.1,see Remark 3.5.5. Moduli spaces of objects in 2-Calabi–Yau categories
As is alluded to in the introduction, many projective hyper-K¨ahler varieties are constructed asmoduli spaces of objects in some 2-Calabi–Yau categories, and it is natural to wonder how themotive of the moduli space is related to the “motive” of this category, whatever it means .The most prominent case of 2-Calabi–Yau category is the K3 category constructed as theKuznetsov component of the derived category of a smooth cubic fourfold. However, given therapid development of the study of stability conditions for many other 2-Calabi–Yau categories,we decided to treat them here in a broader generality. The prudent reader can stick to the cubicfourfold case without missing the point. The motive of a differential graded category can certainly be made precise: it is the theory of non-commutativemotives, see Tabuada [82] for a recent account. However, we will take a more naive approach here, which givesmore precise information by keeping the Tate twists.
N THE MOTIVE OF O’GRADY’S TEN-DIMENSIONAL HYPER-K ¨AHLER VARIETIES 19
Let Y be a smooth projective variety and A an admissible triangulated subcategory of D b ( Y ),the bounded derived category of coherent sheaves on Y . Assume that A is , thatis, the Serre functor of A is the double shift [2]. Example 5.1.
Here are some interesting examples we have in mind:( i ) Y is a K3 surface or abelian surface, and A = D b ( Y ).( ii ) Y is a smooth cubic fourfold and A is the Kuznetsov component defined as the semi-orthogonal complement of the exceptional collection hO Y , O Y (1) , O Y (2) i , see [48].( iii ) Y is a Gushel–Mukai variety [37] [65] [25] of even dimension n = 4 or 6, and A is theKuznetsov component defined as the semi-orthogonal complement of the exceptional col-lection hO Y , U ∨ , O Y (1) , U ∨ (1) , · · · , O Y ( n − , U ∨ ( n − i , 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 Pl¨ucker polarization, see [49].( iv ) Y is a smooth hyperplane section of Gr(3 , A is the semi-orthogonal complement of the exceptional collection hB Y , B Y (1) , · · · , B Y (8) i , where B Y is the restriction of the exceptional collection B of length 12 in the rectangularLefschetz decomposition of Gr(3 ,
10) constructed by Fonarev [33], see [56, § A is non-empty, which is expected for all thecases in Example 5.1 and is established and studied for K3 and abelian surfaces in [17] (see also[92]), for the Kuznetsov component of cubic fourfolds by [9], and for the Kuznetsov componentof Gushel–Mukai fourfolds by [73]. We denote the distinguished connected component of thestability manifold by Stab † ( A ).As in [1], we can define a lattice structure on the topological K-theory of A , denoted by e H ( A ),see [56, § v ∈ e H ( A ), and σ ∈ Stab † ( A ), one can form M st := M A ,σ ( v ) st ( resp. M := M A ,σ ( v )) the moduli space of σ -stable ( resp. σ -semistable) objects in A with Mukaivector v , which is a smooth quasi-projective holomorphic symplectic variety ( resp. proper andpossibly singular symplectic variety).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. ( i ) Let E and F ∈ D b ( M st × Y ) be two universal families. Then ∆ M st = c m ( − E xt ! π ( π ∗ ( E ) , π ∗ ( F ))) ∈ CH m ( M st × M st ) , where m is the dimension of M st , E xt ! π ( π ∗ ( E ) , π ∗ ( F )) denotes the class of the complex Rπ , ∗ ( π ∗ ( E ) ∨ ⊗ L π ∗ ( F )) in the Grothendieck group of M st × M st , where π ij ’s are thenatural projections from M st × Y × M st . ( ii ) There exist finitely many integers k i and cycles γ i ∈ CH( M st × Y k i ) , δ i ∈ CH( Y k i × M st ) ,such that ∆ M st = X i δ i ◦ γ i ∈ CH m ( M st × M st ) . Proof.
The proof of ( i ) is similar to the proof of Markman’s theorem [60] or rather its extensionin [58]. Their proof only uses standard properties for stable objects and the Serre duality forK3 surfaces, which both hold for A .The proof of ( ii ) is exactly the same as in Proposition 3.2 (B¨ulles’ argument), by replacing S by Y everywhere. (cid:3) We first consider the situation where the stability agrees with semi-stability. Then v must beprimitive and σ is v -generic. In this case, M is a smooth and projective hyper-K¨ahler variety, if it is not empty. Once we have the decomposition of the diagonal in Proposition 5.2 ( ii ), thesame proof as in [18] yields the following generalization of Theorem 1.1. Theorem 5.3.
Let Y be a smooth projective variety and let A be an admissible triangulatedsubcategory of D b ( Y ) such that A is 2-Calabi–Yau. Let v be a primitive element in the topologicalK-theory of A and let σ ∈ Stab † ( A ) be a v -generic stability condition. If M := M A ,σ ( v ) isnon-empty, then its Chow motive is in the pseudo-abelian tensor subcategory generated by theChow motive of Y . As a non-commutative analogue of Conjecture 1.2, we formulate the following conjecture.
Conjecture 5.4.
In the same situation as in Theorem 5.3, except that v is not necessarilyprimitive and σ is not necessarily generic. If M st := M A ,σ ( v ) st and M := M A ,σ ( v ) arenon-empty, then their motives and motives with compact support are in the tensor triangulatedsubcategory generated by the motive of Y . If moreover M admits a crepant resolution f M , thenthe Chow motive of f M is in the pseudo-abelian tensor subcategory generated by the Chow motiveof Y . For evidence for Conjecture 5.4, we restrict to the case where Y is a very general cubic fourfoldand A is its Kuznetsov component. Let λ and λ be the cohomological Mukai vectors of theprojections into A of O Y (1) and O Y (2) respectively. Then the topological K-theory of A is an A -lattice with basis { λ , λ } , equipped with a K3-type Hodge structure [1]. Then for a genericstability condition σ (see [9]), there is an O’Grady-type crepant resolution of the singular modulispace M A ,σ (2 λ + 2 λ ), which is of O’Grady-10 deformation type, see [54]. Our result is thatConjecture 5.4 holds true in this case. See Theorem 1.8 in the introduction for the precisestatement. Proof of Theorem 1.8.
The argument is more or less the same as in §
4: the singular locus ofthe moduli space of semistable objects M (2 v ) is Sym ( M ( v )), whose singular locus is thediagonal M ( v ). By the same procedure of blow-ups as in § c M together with a stratification such that the motive with compact support of all stratabelong to the tensor triangulated subcategory generated by the motive of M ( v ), hence also tothe subcategory generated by h ( Y ), by Theorem 5.3. The rest of the proof is the same as § (cid:3) Proof of Corollary 1.9.
In the situation of Theorem 1.3 ( resp.
Theorem 1.8), f M is motivated by the surface S ( resp. the cubic fourfold Y ) in the sense of Arapura [6, Lemma 1.1]: indeed,by applying the full functor CHM → GRM from the category of Chow motives to that ofGrothendieck motives, our main result implies that the Grothendieck motive of f M is in thepseudo-abelian tensor subcategory generated by the Grothendieck motive of S ( resp. Y ). Sincethe Lefschetz standard conjecture is known for S and Y , we can invoke Arapura’s result [6,Lemma 4.2] to obtain the standard conjectures for f M . (cid:3) Defect groups of hyper-K¨ahler varieties
In this section we study the Andr´e motives of projective hyper-K¨ahler varieties with b = 3. Forany such X , we construct the defect group P ( X ), and prove Theorem 6.9 (=Theorem 1.11) andCorollary 6.11 (=Corollary 1.12). In the next section we will apply these results to the knownexamples of hyper-K¨ahler varieties.The starting point and a main tool of our study is the following general theorem due to Andr´e. Theorem 6.1 ([3]) . Let X be a projective hyper-K¨ahler variety such that b ( X ) = 3 . Then theAndr´e motive H ( X ) is abelian. In particular, Conjecture 2.3 holds for H ( X ) . We review the Lie algebra action constructed by Looijenga–Lunts [55] and Verbitsky [85] oncohomology groups of varieties, as well as its remarkable properties when applied to compact
N THE MOTIVE OF O’GRADY’S TEN-DIMENSIONAL HYPER-K ¨AHLER VARIETIES 21 hyper-K¨ahler manifolds. This action is crucial for the proof of Theorem 6.9. To ease the notation,the coefficient field Q in all cohomology groups is suppressed.6.1. The Looijenga–Lunts–Verbitsky (LLV) Lie algebra.
Let X be a 2 m -dimensionalcompact hyper-K¨ahler variety. A cohomology class x ∈ H ( X ) is said to satisfy the Lefschetzproperty if the maps given by cup-product L jx : H m − j ( X ) → H m + j ( X ) sending α to x j ∪ α ,are isomorphisms for all j >
0. The Lefschetz property for a class x in H ( X ) is equivalent tothe existence of a sl -triple ( L x , θ, Λ x ), where θ ∈ End (cid:0) H ∗ ( X ) (cid:1) is the degree-0 endomorphismwhich acts as multiplication by k − m on H k ( X ) for all k ∈ N . Moreover, in this case Λ x isuniquely determined by L x and θ . Note that the first Chern class of an ample divisor on X hasthe 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 -triples ( L x , θ, Λ x ) as above for all cohomology classes x ∈ H ( X ) satisfyingthe Lefschetz property. It is shown in [55, § (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 complexstructure; therefore, g LLV ( X ) is deformation invariant.Let us denote by H the space H ( X ) equipped with the Beauville–Bogomolov quadratic form[12]. Let e H denote the orthogonal direct sum of H and a hyperbolic plane U = h v, w i equippedwith the form v = w = 0 and vw = −
1. We summarize the main properties of the Liealgebra g LLV ( X ). Theorem 6.2 (Looijenga–Lunts–Verbitsky) . ( i ) There is an isomorphism of Q -Lie algebras g LLV ( X ) ∼ = so ( e H ) , which maps θ ∈ g LLV ( X ) to the element of so ( e H ) which acts as multiplication by − on v ,by on w , and by on H . Hence, we have g LLV ( X ) = g − ( X ) ⊕ g ( X ) ⊕ g ( X ) . Moreover, g ( X ) ∼ = so ( H ) ⊕ Q · θ , is the centralizer of θ in g LLV ( X ) . The abelian subalgebra g ( X ) is the linear span of the endomorphisms L x , for x ∈ H ( X ) , and g − ( X ) is the spanof the Λ x , for all x ∈ H ( X ) with the Lefschetz property. ( ii ) The Lie subalgebra so ( H ) ⊂ g ( X ) acts by derivations on the graded algebra H ∗ ( X ) . Theinduced action of so ( H ) on H ( X ) is the standard representation. ( iii ) Let ρ : so ( H ) → gl ( H ∗ ( X )) be the induced representation of so ( H ) ⊂ g ( X ) . Then theWeil operator W is an element of ρ (cid:0) so ( H ) (cid:1) ⊗ R . The above theorem is proved in [85], and in [55, Proposition 4.5], see also the appendix of [47].These proofs are carried out with real coefficients, but immediately imply the result with rationalcoefficients: since g LLV ( X ) is defined over Q , the equality g LLV ( X ) ⊗ R = so ( e H ) ⊗ R of Liesubalgebras of gl ( e H ) ⊗ R implies that the same equality already holds with rational coefficients. Remark . Let ρ + : so ( H ) → gl ( H + ( X )) be the induced representation on theeven cohomology. It follows from [84, Corollary 8.2] that ρ + integrates to a faithful representation e ρ + : SO( H ) → Y i GL (cid:0) H i ( X ) (cid:1) , such that the induced representation on H ( X ) is the standard representation. If the oddcohomology of X is non-trivial, ρ integrates to a faithful representation e ρ : Spin( H ) → Y i GL (cid:0) H i ( X ) (cid:1) , and the kernel of the action of Spin( H ) on the even cohomology is an order-2 subgroup h ι i ,where ι is the non-trivial element in the kernel of the double cover Spin( H ) → SO( H ) and e ρ ( ι ) Here the Weil operator refers to the derivation of the usual Weil operator, which acts on H p,q ( X ) as multi-plication by i p − q . Hence, W acts on each H p,q ( X ) as multiplication by i ( p − q ). acts on H i ( X ) via multiplication by ( − i . Note also that the action induced by e ρ and e ρ + isvia algebra automorphisms, thanks to Theorem 6.2( ii ).6.2. Splitting of the motivic Galois group.
Let H ( X ) ( resp. H + ( X )) be the full ( resp. even)rational cohomology group of X equipped with Hodge structure. The natural inclusions of H ( X ) into H + ( X ) and H ∗ ( X ) induce surjective morphisms of Mumford–Tate groups π +2 : MT( H + ( X )) → MT( H ( X )); π : MT( H ∗ ( X )) → MT( H ( X )) . Let ι ∈ GL( H ∗ ( X )) act on each H i ( X ) via the multiplication by ( − i for all i . Proposition 6.4.
The notation is as above. ( i ) The morphism π +2 is an isomorphism. In particular, the Hodge structure H + ( X ) belongsto the tensor subcategory of HS pol Q generated by H ( X ) . ( ii ) If X has non-vanishing odd cohomology, the morphism π is an isogeny with kernel h ι i ≃ Z / Z . Moreover, if A is any Kuga–Satake variety for H ( X ) in the sense of DefinitionA.2, we have h H ∗ ( X ) i HS = h H ( A ) i HS . The natural choice for A is the abelian variety obtained through the Kuga–Satake constructionon H ( X ) equipped with the Beauville-Bogomolov form, see § A.1; let us remark that also theconstruction of [47] yields a Kuga–Satake variety for H ( X ) in our sense.The proof of the proposition will be given after some preliminary results. Recall ([28]) that thealgebraic group CSpin( H ) is the quotient of G m × Spin( H ) in which we identify the element − ∈ G m with the non-trivial central element ι of Spin( H ). We introduce a representation σ : CSpin( H ) → Y i GL( H i ( X ))by σ = w · e ρ , where e ρ : Spin( H ) → Q i GL( H i ( X )) is the representation from Remark 6.3 and w : G m → GL( H i ( X )) is the weight cocharacter, i.e. w ( λ ) acts on H i ( X ) as multiplication by λ i , for all i and all λ . This is a priori a representation of G m × Spin( H ), but it indeed factorsthrough CSpin( H ) since by Remark 6.3 we have w ( −
1) = e ρ ( ι ). We also set σ + : CSpin( H ) → Y i GL( H i ( X )) and σ : CSpin( H ) → GL( H ( X ))to be the induced representations on the even cohomology and on H ( X ) respectively. Lemma 6.5. (i) The homomorphism σ + : CSpin( H ) → Q i GL( H i ( X )) is an isogeny of de-gree 2 onto its image. The natural projection pr +2 : Q i GL( H i ( X )) → GL( H ( X )) mapsthe image of σ + isomorphically onto the image of σ .(ii) If X has non-vanishing odd cohomology, the representation σ : CSpin( H ) → Q i GL( H i ( X )) is faithful, and the projection pr : Q i GL( H i ( X )) → GL( H ( X )) induces a degree 2isogeny between the image of σ and the image of σ .Proof. By Remark 6.3 and the explicit description of w , the kernels of σ + and σ both coincidewith the central subgroup of order 2 generated by ( − ,
1) = (1 , ι ). This proves part ( i ). If X has non-vanishing odd cohomology, σ is faithful by Remark 6.3, and the second assertionfollows. (cid:3) Remark . Note that the twisted representation σ ′ = w ′ · e ρ where w ′ ( λ ) acts on H i ( X ) viamultiplication by λ i − m is the representation obtained via integration of g → Q i gl ( H i ( X )).The point of introducing the above representation is that it controls the Mumford–Tate group. Lemma 6.7.
The Mumford–Tate group
MT( H ∗ ( X )) is contained in the image of σ . N THE MOTIVE OF O’GRADY’S TEN-DIMENSIONAL HYPER-K ¨AHLER VARIETIES 23
Proof.
Let G = Im( σ ). Since both MT( H ∗ ( X )) and G are reductive, by [30, Proposition 3.1] itsuffices to check that for any tensor construction T = M i H ∗ ( X ) ⊗ m i ⊗ (cid:0) H ∗ ( X ) ∨ (cid:1) ⊗ n i , any element α of T that is invariant for G is also fixed by MT( H ∗ ( X )). Let α ∈ T be such aninvariant 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 ) forsome integer p . However, since w ( G m ) also acts trivially on α , we must have p = 0; hence α isa Hodge class of type (0, 0) and is thus fixed by the Mumford–Tate group. (cid:3) 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 : Q i GL( H i ( X )) → GL( H ( X )). Lemma 6.5implies in particular that the restriction of pr +2 to Im( σ + ) is injective; hence its restriction tothe subgroup 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 : MT( H ∗ ( X )) → MT( H ( X )) is contained in the kernel of pr : Im( σ ) → Im( σ ). By Lemma 6.5, this is anorder 2 central subgroup of Im( σ ), generated by w ( −
1) = ι . Clearly w ( −
1) is contained inMT( X ), and it follows that π is an isogeny of degree 2 whose kernel is generated by ι .Finally, let A be any Kuga–Satake abelian variety for H ( X ), in the sense of Definition A.2.Then h H ( A ) i HS is the unique tannakian subcategory such that h H ( X ) i HS = h H ( A ) i evHS ( h H ( A ) i HS , by Theorem A.4. Therefore it is enough to show that h H ∗ ( X ) i HS also satisfies thisproperty. Consider the commutative diagramMT( H ∗ ( X )) MT( H ( X ))MT( h H ∗ ( X ) i evHS ) . π ev π π ev2 We have just proven that π 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 ev2 is an isomorphism and hence h H ∗ ( X ) i evHS = h H ( X ) i HS . (cid:3) The following observation will be used in the proof of Theorem 6.9.
Lemma 6.8.
Let G be a group acting on H ∗ ( X ) via graded algebra automorphisms. If G actstrivially on H ( X ) , then the G -action commutes with the action of the LLV Lie algebra ( § Let g ∈ G . By assumption, g commutes with θ and L x , for any x ∈ H ( X ). Moreover, if x has the Lefschetz property, then g commutes with Λ x as well: indeed, L x = gL x g − , θ = gθg − and g Λ x g − form an sl -triple, and since Λ x is uniquely determined by the elements L x and θ , we must have g Λ x g − = Λ x . One can conclude since the various operators L x and Λ x , for x ∈ H ( X ), generate the Lie algebra g LLV ( X ). (cid:3) We now turn to the proof of the main result of this section.
Theorem 6.9 (Splitting) . Let X be a projective hyper-K¨ahler variety with b ( X ) = 3 . Then,inside G mot ( H ( X )) , the subgroups P ( X ) and MT( H ∗ ( X )) commute, intersect trivially with eachother and generate the whole group. In short, we have an equality: G mot ( H ( X )) = MT( H ∗ ( X )) × P ( X ) . Similarly, the even defect group is a direct complement of the even Mumford–Tate group in themotivic Galois group of the even Andr´e motive of X , G mot ( H + ( X )) = MT( H + ( X )) × P + ( X ) . Proof.
We first treat the even motive. We have a commutative diagramG mot ( H + ( X )) G mot ( H ( X ))MT( H + ( X )) MT( H ( X )) π +2 , mot i + i π +2 Here, i + and i denote the natural inclusions; π +2 and i are isomorphisms due to Proposition 6.4and Theorem 6.1 respectively. It follows that we have a section s = i + ◦ ( i ◦ π +2 ) − of π +2 , mot ,whose image is MT( H + ( X )).Recall that P + ( X ) is defined as the kernel of the map π +2 , mot . We deduce that G mot ( H + ( X )) isthe semidirect product of its subgroups P + ( X ) and MT( H + ( X )), which intersect trivially. Inorder 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 theimage of the representation σ + . Since P + ( X ) preserves the grading on H + ( X ), its action clearlycommutes with the weight cocharacter w . Note that every element of G mot ( H + ( X )) acts viaalgebra 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 triviallyon H ( 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 e 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 ( X ), see Appendix A. By Lemma 6.10 below, the motive H ( A ) belongs to hH ( X ) i AM .We consider the commutative diagramG mot ( H ( X )) G mot ( H ( A ))MT( H ∗ ( X )) MT( H ( A )) π A, mot i i A π A 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 theKuga–Satake abelian variety A ; this group is by definition the defect group of X . As above, wededuce the existence of a section of π A, mot with image MT( H ∗ ( X )), and to conclude we need toshow that P ( X ) and MT( H ∗ ( X )) commute. To this end, we consider the commutative diagramwith exact rows1 P ( X ) G mot ( H ( X )) G mot ( H ( A )) 11 Q ( X ) G mot ( H ( X )) G mot ( H ( X )) 1 π A, mot = π , mot The group Q ( X ) commutes with the action of g LLV , by Lemma 6.8, and it therefore commuteswith the Mumford–Tate group, thanks to Lemma 6.7. Since P ( X ) is a subgroup of Q ( X ), italso commutes with MT( H ∗ ( X )), and we have G mot ( H ( X )) = P ( X ) × MT( H ∗ ( X )). Also notethat we have Q ( X ) ∩ MT( H ∗ ( X )) = h ι i , and that Q ( X ) = P ( X ) × h ι i . (cid:3) In the previous proof, we have used the following result. See Appendix A for the notation.
Lemma 6.10.
Assume that the odd cohomology of X does not vanish and b ( X ) = 3 . Let A be any Kuga–Satake variety (Definition A.2) for the Hodge structure H ( X ) . Then H ( A ) ∈hH ( X ) i AM .Proof. Note that since H ( X ) is an abelian motive by Andr´e’s Theorem 6.1, any Kuga-Satakevariety A for H ( X ) satisfies hH ( A ) i ev = hH ( X ) i , see Corollary A.6. Choose any such A , N THE MOTIVE OF O’GRADY’S TEN-DIMENSIONAL HYPER-K ¨AHLER VARIETIES 25 and consider the Andr´e motive H ( X ) ⊕ H ( A ). The inclusions of the summands H ( X ) and H ( A ) determine surjective homomorphisms q : G mot ( H ( X ) ⊕ H ( A )) → G mot ( H ( X )), and q A : G mot ( H ( X ) ⊕ H ( A )) → G mot ( H ( A )). The desired conclusion is equivalent to the inclusionker( q ) ⊂ ker( q A ). In fact, this precisely means that the tannakian category generated by H ( A )is contained in hH ( X ) i , which then implies that q is an isomorphism. We consider the analogousmorphisms for the even parts q ev : G mot (cid:0) hH ( X ) ⊕ H ( A ) i ev (cid:1) → G mot (cid:0) hH ( X ) i ev (cid:1) ,q ev A : G mot (cid:0) hH ( X ) ⊕ H ( A ) i ev (cid:1) → G mot (cid:0) hH ( A ) i ev (cid:1) . The conclusion of Lemma A.5 holds for Andr´e motives as well. Therefore, the preimageof ker( q ev ) (respectively, of ker( q ev A )) under the morphism G mot ( H ( X ) ⊕H ( A )) → G mot ( hH ( X ) ⊕H ( A ) i ev ) equals h ι i × ker( q ) (respectively, h ι i × ker( q A )), and it suffices to show that ker( q ev ) ⊂ ker( q ev A ). To this end, consider the commutative diagram with short exact rows1 ker( q ev A ) G mot ( hH ( X ) ⊕ H ( A ) i ev ) G mot ( hH ( A ) i ev ) 11 K G mot ( hH ( X ) i ev ) G mot ( H ( X )) 1 j q ev A q ev ∼ = π ev2 , mot The rightmost vertical map is an isomorphism by assumption. The snake lemma now yieldsthat ker( j ) = ker( q ev ), which shows that ker( q ev ) ⊂ ker( q ev A ). (cid:3) What does the defect group measure?
With the structure result of the motivic Galoisgroup being proved in Theorem 6.9, we can deduce that the defect group indeed grasps theessential difficulty of meta-conjecture 1.10 for Andr´e motives.
Corollary 6.11.
For any projective hyper-K¨ahler variety X with b ( X ) = 3 , the followingconditions are equivalent: ( i + ) The even defect group P + ( X ) is trivial. ( ii + ) The even Andr´e motive H + ( X ) is in the tannakian subcategory generated by H ( X ) . ( iii + ) H + ( X ) is abelian. ( iv + ) Conjecture 2.3 holds for H + ( X ) : MT( H + ( X )) = G mot ( H + ( X )) . Similarly, if some odd Betti number of X is not zero, we have the following equivalent conditions: ( i ) The defect group P ( X ) is trivial. ( ii ) The Andr´e motive H ( X ) is in the tannakian subcategory generated by H (KS( X )) , where KS( X ) is any Kuga–Satake abelian variety associated to H ( X ) . ( iii ) H ( X ) is abelian. ( iv ) Conjecture 2.3 holds for H ( X ) : MT( H ∗ ( X )) = G mot ( H ( X )) .Proof. We first treat the even motive. It follows immediately from Theorem 6.9 that ( i + ) and( iv + ) are equivalent.( i + ) implies ( ii + ): By the definition of P + ( X ), if it is trivial, then the natural surjectionG mot ( H + ( X )) → G mot ( H ( X )) is an isomorphism. Then ( ii + ) follows from the Tannaka duality.The implication from ( ii + ) to ( iii + ) follows from the fact that H ( X ) is abelian, which is Andr´e’sTheorem 6.1.Finally, ( iii + ) implies ( iv + ) thanks to Andr´e’s Theorem 2.4.In the presence of non-vanishing odd Betti numbers, the proof is similar to the even case: theequivalence of ( i ) and ( iv ) is immediate from Theorem 6.9. ( ii ) obviously implies ( iii ); ( iii )implies ( iv ) by Andr´e’s Theorem 2.4. Finally, let us show that ( i ) implies ( ii ): if P ( X ) is trivialthen G mot ( H ( X )) → G mot ( H ( A )) is an isomorphism, where A is any Kuga–Satake variety for H ( X ) in the sense of Definition A.2. Therefore, H ( X ) is in hH ( A ) i AM by Tannaka duality. (cid:3) Deformation invariance.
We have seen in the above proof that the action of the defectgroup commutes with the LLV-Lie algebra. We prove now that defect groups are deformationinvariant in algebraic families. The relevant notation and results are recalled in § f : X → S be a smooth and proper family over a non-singular quasi-projective variety S suchthat all fibres X s are projective hyper-K¨ahler varieties with b = 3. We have naturally thefollowing generalized Andr´e motives over S (Definition 2.7): H ( X /S ), H i ( X /S ) and H + ( X /S ).Up to replacing S by an ´etale cover, we can assume that the algebraic monodromy groupG mono ( H ( X /S )) is connected. Theorem 6.12 (Deformation invariance of defect groups) . Let S be a smooth quasi-projectivevariety and X → S be a smooth proper morphism with fibers being projective hyper-K¨ahler man-ifolds with b = 3 . Then for any s, s ′ ∈ S , the defect groups P ( X s ) and P ( X s ′ ) are canonicallyisomorphic, and similarly for the even defect groups.Proof. We prove first the invariance of the even defect group. For any point s ∈ S , we haveG mot ( H + ( X s )) = MT( H + ( X s )) × P + ( X s ) by Theorem 6.9. Let s ∈ S be a very generalpoint. By Theorem 2.8 ( i ) and ( ii ), we have G mono ( H + ( X /S )) s ⊂ MT( H + ( X s )). Hence,the monodromy acts trivially on P + ( X s ), which therefore extends to a constant local system P + ( X /S ) such that we have a splittingG mot ( H + ( X /S )) = MT( H + ( X /S )) × P + ( X /S )of local systems of algebraic groups over S . The local system P + ( X /S ) is identified with thekernel of the natural morphism of generic motivic Galois groupsG mot ( H + ( X /S )) ։ G mot ( H ( X /S )) . For any s ∈ S we have the inclusion of G mot ( H + ( X s )) into G mot ( H + ( X /S )) s , which restricts tothe inclusions MT( H + ( X s )) ֒ → MT( H + ( X /S )) s and P + ( X s ) ֒ → P + ( X /S ) s .It is enough to show that, for all s ∈ S , the equality P + ( X s ) = P + ( X /S ) s holds.By Theorem 2.8( iv ), we haveG 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 { } × MT( H + ( X /S )) s ⊂ P + ( X /S ) s × MT( H + ( X /S )) s = G mot ( H + ( X /S )) s , and therefore we haveG mono ( H + ( X /S )) s · G mot ( H + ( X s )) == G mono ( H + ( X /S )) s · (cid:0) P + ( X s ) × MT + ( X s ) (cid:1) = P + ( X s ) × (cid:0) G mono ( H + ( X /S )) s · MT + ( X s ) (cid:1) , which forces P + ( X s ) = P + ( X /S ) s .In presence of non-vanishing Betti numbers in odd degree, the proof is similar. Again, choosinga very general point s ∈ S , we obtain a local system P ( X /S ) with fiber P ( X s ) such thatG mot ( H ( X /S )) = MT( H ∗ ( X /S )) × P ( X /S ) . The Kuga-Satake construction can be performed in families, see [28], to obtain a smooth properfamily
A → S such that A s is a Kuga-Satake variety for H ( X s ) in the sense of Definition A.2,for all s . Thanks to Lemma 6.10 we have a natural morphism of generic motivic Galois groupsG mot ( H ( X /S )) ։ G mot ( H ( A /S ))which does not depend on any choice involved in the construction of A ; the local system P ( X /S )is identified with the kernel of the morphism above.It follows that for any s ∈ S the inclusion G mot ( H ( X s )) ֒ → G mot ( H ( X /S )) s restricts to inclusionsMT( H ∗ ( X s )) ֒ → MT( H ∗ ( X /S )) s and P ( X s ) ֒ → P ( X /S ) s . Now we conclude via the sameargument given for the even case. (cid:3) N THE MOTIVE OF O’GRADY’S TEN-DIMENSIONAL HYPER-K ¨AHLER VARIETIES 27 Applications
Andr´e motives of hyper-K¨ahler varieties.
As we have seen in Theorem 6.12, thedefect group does not change along smooth proper algebraic families. In fact, the defect groupis invariant in the whole deformation class.
Corollary 7.1.
Let X and X ′ be two deformation equivalent projective hyper-K¨ahler varietieswith b = 3 . Then their defect groups are isomorphic: P + ( X ) ∼ = P + ( X ′ ) and P ( X ) ∼ = P ( X ′ ) .Proof. Pick two deformation equivalent projective hyper-K¨ahler varieties X and X ′ with b = 3.It has been shown by Soldatenkov ([80, § f i : Y i → S i , i = 0 , , . . . , k over smooth quasi-projective curves S i and points a i , b i ∈ S i together with isomorphisms X ∼ = Y a , Y ib i ∼ = Y i +1 a i +1 , for i = 1 , , . . . , k − , and Y kb k ∼ = X ′ . 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 6.12. (cid:3) Corollary 7.2.
Fix a deformation class of compact hyper-K¨ahler manifolds with b = 3 .(i) (Soldatenkov [80] ) If one projective hyper-K¨ahler variety in the deformation class hasabelian Andr´e motive, then so does any other projective member in this class.(ii) There exists an Andr´e motive D + depending only on the deformation class, with Hodgerealization being of Tate type, and such that for any projective hyper-K¨ahler variety X inthis deformation class we have hH + ( X ) i AM = hH ( X ) , D + i AM . (iii) Similarly, if some odd Betti number is non-zero in the chosen deformation class, thereexists an Andr´e motive D depending only on the deformation class, with Hodge realizationbeing of Tate type, and such that for any projective X in the chosen deformation class wehave hH ( X ) i AM = hH (KS( X )) , Di AM , where KS( X ) is any Kuga–Satake variety for H ( X ) (Definition A.2).Proof. ( i ) follows from the combination of Corollary 6.11 and Corollary 7.1.( ii ). This follows via a reinterpretation of Theorem 6.9 in terms of a defect motive. Recall thatwe have G mot ( H + ( X )) = MT( H + ( X )) × P + ( X ). The category Rep( P + ( X )) can be seen as thesubcategory of hH + ( X ) i AM on which MT( H + ( X )) acts trivially, i.e. , it consists of the motives in hH + ( X ) i whose realization is of Tate type. By [30, Proposition 3.1], the category Rep( P + ( X ))is generated as a tannakian category by any faithful representation of P + ( X ); choosing one suchrepresentation determines a motive D + ( X ) ∈ hH + ( X ) i , such that inside AM, hH + ( X ) i = hD + ( X ) , H ( X ) i . Let now
X → S be a smooth proper family with fibres projective hyper-K¨ahler varietieswith b = 3 over a smooth quasi-projective base S . We assume that the monodromy group G mono ( X /S ) is connected. We consider the generalized Andr´e motive H + ( X /S ) over S , withrealization H + ( X /S ); by Theorem 6.12, we have a splitting of local systems of algebraic groupsG mot ( H + ( X /S )) = MT( H + ( X /S )) × P + ( X /S ) such that P + ( X s ) = P + ( X /S ) s for all s ∈ S .We choose a tensor construction T + ( X /S ) = H + ( X /S ) ⊗ a ⊗ H + ( X /S ) ∨ , ⊗ b such that the sub-space W + ( X s ) ⊂ T + ( X s ) of MT( H + ( X /S )) s -invariants is a faithful P + ( X s )-representation.Since W + ( X s ) is stable for the action of G mot ( H + ( X /S )) s , we obtain generalized Andr´e mo-tives W + ( X /S ) ⊂ T + ( X /S ) over S , with realizations W + ( X /S ) ⊂ T + ( X /S ). For all s ∈ S wehave hH + ( X s ) i = hW + ( X s ) , H ( X s ) i .Since the monodromy group is connected, by Theorem 2.8(i) the local system W + ( X /S ) isconstant. Now Theorem 2.6 implies that for any two points s , s ∈ S we have an isomorphism ofmotives W + ( X s ) ∼ = W + ( X s ). In fact, let 0 be any point of S and let D + = W + ( X ). Let D + /S be the constant generalized Andr´e motive over S with fibre D + , supported onto the constantlocal system D + /S . Then the identity id : W + ( X ) → ( D + /S ) is monodromy invariant andmotivated, and hence it extends to a global section ξ of the local system Hom( W + ( X /S ) , D + /S )such that ξ s is the realization of an isomorphism of motives W + ( X s ) ∼ = D + s , for any s ∈ S ; hence,we have hH + ( X s ) i = hD + , H ( X s ) i . Thanks to [80, § b = 3 via finitely many families as above anditerate the argument given.(iii). Same argument as above. (cid:3) We can now prove that the defect group of any known projective hyper-K¨ahler variety is trivial.
Proof of Corollary 1.16.
The second Betti numbers of known hyper-K¨ahler varieties are as fol-lows: 22 for K3 surfaces [43]; 23 and 7 for varieties of K3 [ n ] -type and of generalized Kummer typerespectively, see [12]; 24 for varieties of OG10-type and 8 for those of OG6-type, as computed byRapagnetta in [76] and [75]. Hence, the triviality of the defect group is a deformation invariantproperty by Corollary 7.1. It is therefore enough to find in each of the known deformation classesa representative whose defect group is trivial, or equivalently, whose Andr´e motive is abelian.( i ) For K3 surfaces, this is Andr´e [4, Th´eor`eme 0.6.3].( ii ) For the K3 [ n ] -type, the motivic decomposition of de Cataldo–Migliorini [22], together withthe case of K3 surfaces ( i ), implies that the Andr´e motive of a Hilbert scheme of a K3 surfaceis abelian.( iii ) For the generalized Kummer type, using the work of Cataldo–Migliorini [23] on semi-smallresolutions, a motivic decomposition for a generalized Kummer variety associated to an abeliansurface in terms of abelian motives was obtained in [91] and [34, Corollary 6.3].( iv ) For the O’Grady-6 deformation type, it follows from [61], as observed by Soldatenkov [80]:in [61], some hyper-K¨ahler variety of this deformation type was constructed as the quotient ofsome hyper-K¨ahler variety of K3 [3] -type by a birational involution (with well-understood inde-terminacy loci). One can then conclude by ( ii ).( v ) For O’Grady-10 deformation type, we use Corollary 4.7. (cid:3) Motivated Mumford–Tate conjecture.
We first recall a strengthening of the Mumford–Tate conjecture involving motivic Galois groups, see Moonen’s survey [62, § k is a finitely generated subfield of C , and ℓ is a prime number. Attached to a smoothprojective variety X defined over k , we have on the one hand the rational singular (Betti) co-homology H ∗ B ( X ) = L i H i B ( X ) := L i H i ( X an C , Q ), naturally equipped with a Hodge structure,and on the other hand the ℓ -adic ´etale cohomology H ∗ ℓ ( X ) = L i H iℓ ( X ) := L i H i ´et ( X k , Q ℓ ),which is a continuous Q ℓ -representation of Gal( k/k ). These two cohomology theories providerealization functors from AM( k ), the category of Andr´e motives over Spec( k ): r B : AM( k ) → HS pol Q ; r ℓ : AM( k ) → Rep Q ℓ (cid:0) Gal( k/k ) (cid:1) . 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 . Thisalgebraic group is not necessarily connected, but becomes so after a finite field extension of k . Itis not known to be reductive in general. The category of Q ℓ -Galois representations is a neutraltannakian abelian category, and the tannakian subcategory h V i is equivalent to the category offinite 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 riseto an isomorphism of Q ℓ -algebraic groups γ : GL( r B ( M )) ⊗ Q ℓ ∼ = GL( r ℓ ( M )), under whichG mot ( M C ) ⊗ Q ℓ is identified with G mot ,ℓ ( M k ), where the latter is the motivic Galois group ofthe tannakian category h M k i AM( k ) with fiber functor r ℓ composed with the forgetful functor.The following conjecture is a motivic extension of the Mumford–Tate conjecture [66]. N THE MOTIVE OF O’GRADY’S TEN-DIMENSIONAL HYPER-K ¨AHLER VARIETIES 29
Conjecture 7.3 (Motivated Mumford–Tate conjecture) . The canonical isomorphism γ inducesidentifications of Q ℓ -algebraic groups MT( r B ( M )) ⊗ Q ℓ = G mot ( M C ) ⊗ Q ℓ ∼ = G mot ,ℓ ( M k ) = G ( r ℓ ( M )) . Remark . The first equality is the content of Conjecture 2.3 and the last equality is theanalogous statement saying that all Tate classes are motivated. The original statement of theMumford–Tate conjecture only predicts that under γ we haveMT( H ∗ B ( X )) ⊗ Q ℓ = G ( H ∗ ℓ ( X )) , for any smooth and projective variety X over k .Let us define a hyper-K¨ahler variety over k to be a smooth projective variety X over k such that X C is a hyper-K¨ahler variety. The following result confirms Conjecture 7.3 for the degree-2 partof the motive of X , see Moonen [63] for some generalizations. Theorem 7.5 (Andr´e [3]) . Let X be a hyper-K¨ahler variety defined over k with b = 3 . Thenthe motivated Mumford–Tate conjecture holds for the Andr´e motive H ( X ) . Let X be as above. Then G mot ( H ( X k )) ∼ = G mot ( H ( X C )) = MT( H ∗ B ( X )) × P ( X ) by Theorem 6.9. Proposition 7.6. If P + ( X ) is finite (resp. trivial), then the Mumford–Tate conjecture (resp. themotivated Mumford–Tate conjecture) holds for the motive H + ( X ) . If P ( X ) is finite (resp. triv-ial), then the Mumford–Tate conjecture (resp. the motivated Mumford–Tate conjecture) holds forthe 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 diagramMT( H +B ( X )) ⊗ Q ℓ G mot ( H + ( X k )) ⊗ Q ℓ G ( H + ℓ ( X )) MT( H ( X )) ⊗ Q ℓ G mot ( H ( X k )) ⊗ Q ℓ G ( H ℓ ( X )) ∼ = ∼ = ∼ = ∼ = The two horizontal morphisms on the bottom are isomorphisms due to Theorem 7.5, the verticalmap on the left is an isomorphism thanks to Proposition 6.4 and the top left arrow is anisomorphism by Theorem 6.9 since P + ( X ) is finite by assumption. It follows that all arrows inthe diagram are isomorphisms, and so G ( H + ℓ ( X )) ∼ = G mot ( H + ( X k )) ⊗ 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 moti-vated 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, whichimplies that also P + ( X ) is finite. We consider another commutative diagramMT( H ∗ B ( X )) ⊗ Q ℓ G mot ( H ( X k )) ⊗ Q ℓ G ( H ∗ ℓ ( X )) MT( H +B ( X )) ⊗ Q ℓ G mot ( H + ( X k )) ⊗ Q ℓ G ( H + ℓ ( X )) ∼ = ∼ ∼ = ∼ = The horizontal arrows on the bottom are isomorphisms due to the above; the top left horizontalmap is an isomorphism by Theorem 6.9, since P ( X ) is finite by assumption, while the leftmostvertical arrow is an isogeny due to Proposition 6.4. It follows that also the middle vertical arrowis an isogeny. We deduce that G mot ( H ( X k )) ⊗ Q ℓ and G ( H ∗ ℓ ( X )) are connected algebraicgroups of the same dimension over Q ℓ . Hence, the inclusion G ( H ∗ ℓ ( X )) ֒ → G mot ( H ( X k )) ⊗ Q ℓ is an isomorphism. If P ( X ) is actually trivial, then G mot ( H ( X k )) is connected, and we concludethat the motivated Mumford–Tate conjecture holds for the full Andr´e motive H ( X ). (cid:3) Definition 7.7.
Let k ⊂ C be a finitely generated field. Define C k to be the tannakian subcate-gory of AM( k ) generated by the motives of all hyper-K¨ahler varieties whose associated complexmanifold is of one of the four known deformation types. Remark . Note that this category contains already the motive of cubic fourfolds, as they aremotivated by their Fano varieties of lines (see for example [50]). Very likely, C k also containsthe motive of some interesting Fano varieties whose cohomology is of K3-type, for instance,Gushel–Mukai varieties [37] [65], Debarre–Voisin Fano varieties [26] and many more [31]. Theorem 7.9.
The motivated Mumford–Tate conjecture holds for any motive M ∈ C k . Inparticular, for any smooth projective variety motivated by a product of projective hyper-K¨ahlervarieties of known deformation type, the Hodge conjecture and the Tate conjecture are equivalent.Proof. By Commelin [21, Theorem 10.3], the subcategory of abelian Andr´e motives satisfyingthe Mumford–Tate conjecture is a tannakian subcategory. Therefore, it suffices to check theabelianity and the Mumford–Tate conjecture for the generators of C k .By Corollary 1.16 the defect group of any hyper-K¨ahler variety X of known deformation type istrivial. Hence, the motive H ( X ) ∈ C k is abelian by Corollary 6.11, and the motivated Mumford–Tate conjecture holds for its Andr´e motive by Proposition 7.6. (cid:3) Remark . Thanks to [21], we can put even more generators in the category C k to obtain newevidence for the Mumford–Tate conjecture. Since the conjecture is known to hold for( i ) geometrically simple abelian varieties of prime dimension, by Tankeev [83],( ii ) abelian varieties of dimension g with trivial endomorphism ring over k such that 2 g isneither a k -th power for some odd k > (cid:0) kk (cid:1) for some odd k >
1, thanksto Pink [74],we deduce that the Mumford–Tate conjecture holds for any variety motivated by a product ofvarieties in ( i ) and ( ii ) above and hyper-K¨ahler varieties of the known deformation types. SeeMoonen [63] 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 , = h , = 1 and h p,q = 0 whenever p or q is negative. The Kuga–Satake construction[28] 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 thetannakian subcategory of Hodge structures generated by this abelian variety, which we call the Kuga–Satake category attached to V , KS ( V ) := h H (KS( V )) i ⊂ HS pol Q . In the appendix, all the cohomology groups are with rational coefficients and the notation h−i means the generated tannakian subcategory inside HS pol Q , if not otherwise specified. We firstbriefly review the classical construction.A.1. The Kuga–Satake construction.
Choose a polarization q of V , and consider the Cliffordalgebra Cl( V, q ). Deligne showed in [28] that there is a unique way to induce a weight-1 effectiveHodge structure on Cl(
V, q ), which is polarizable and therefore equals H (KS( V )) for someabelian variety KS( V ), well-defined up to isogeny. The key relation between V and KS( V ) isthe fact that the natural action of V on Cl( V, q ) via left multiplication yields an embedding ofHodge structures V (1) ֒ → H (KS( V )) ⊗ H (KS( V )) ∨ . For abelian Andr´e motives, the Mumford–Tate conjecture is equivalent to its motivated version 7.3, thanksto Andr´e’s result Theorem 2.4.
N THE MOTIVE OF O’GRADY’S TEN-DIMENSIONAL HYPER-K ¨AHLER VARIETIES 31
Consider the weight cocharacters w V : G m → GL( V ) and w KS( V ) : G m → GL( H (KS( V ))),defined by w V ( λ ) = λ · id and w KS ( λ ) = λ · id respectively, for all λ ; we have MT( V ) ⊂ w V ( G m ) · SO(
V, q ) and MT( H (KS( V ))) ⊂ w KS ( G m ) · Spin(
V, q ), The inclusion h V i ⊂ h H (KS( V )) inducesa surjective morphism φ : MT( V ) → MT( H (KS( V ))), which we claim is a double cover. Indeed,there is a commutative diagram with exact rows1 Spin( V, q ) ∩ MT( H (KS( V ))) MT( H (KS( V ))) G m
11 SO(
V, q ) ∩ MT( V ) MT( V ) G m φ ′ φ ∼ = in which φ ′ is the restriction of the double cover Spin( V, q ) → SO(
V, q ), and the vertical map onthe right is an isomorphism due to the fact that w KS ( − ∈ Spin(
V, q ). Remark
A.1 . The above construction can be performed given any non-degenerate quadraticform q on V such that the restriction of q ⊗ R to (cid:0) H , ( V ) ⊕ H , ( V ) (cid:1) ∩ ( V ⊗ R ) is positivedefinite and q ( σ ) = 0 for any σ ∈ H , ( V ), see [43, §
4, Remark 2.3].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 weightson C is given by a central cocharacter w : G m, Q → MT( C ). We let ι := w ( − − 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 groupsMT( 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 H ( A ) i ev = h V i . Lemma A.3 (Equivalent definition) . An abelian variety A is a Kuga–Satake variety for V ifand only if V ∈ h H ( A ) i and the induced surjective morphism MT( H ( A )) → MT( V ) is anisogeny of degree .Proof. The only-if part is explained before. Conversely, assume that V ∈ h H ( A ) i and that theinduced surjection MT( H ( A )) → MT( V ) is an isogeny of degree 2. Since V has even weight,this morphism factors over MT( h H ( A ) i ev ) → MT( V ), and it follows that the the latter is anisomorphism. Hence, h H ( A ) i ev = h V i . (cid:3) By Lemma A.3 and the discussion in § A.1, the abelian variety KS( V ) is a Kuga–Satake varietyfor 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 uniquetannakian subcategory KS ( V ) of HS pol Q such that h V i = KS ( V ) ev ( KS ( V ) . If A is any Kuga–Satake variety for V , we have h H ( A ) i = KS ( V ) . Let us first prove the following straightforward lemma. Consider tannakian subcategories C ⊂ D of HS pol Q . Assume that both contain 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, the morphism π : MT( D ) → MT( D ev ) induces an isomor-phism ker( q ) ∼ = ker( q ev ) , and π − (ker( q ev )) = h ι i × ker( q ) . Proof.
Consider the commutative diagram with exact rows1 h ι i MT( D ) MT( D ev ) 11 h ι i MT( C ) MT( C ev ) 1 . ∼ = q π q ev The snake lemma implies the isomorphism ker( q ) ∼ = ker( q ev ). Moreover, since ι / ∈ ker( q ) byassumption and it is central in MT( D ), we have π − (ker q ev ) = ι × ker( q ). (cid:3) Proof of Theorem A.4.
Assume given two tannakian subcategories D , D ⊂ HS pol Q , both con-taining some Hodge structure of odd weight and such that h V i = D ev i ( D i for i = 1 ,
2. Let E be the tannakian subcategory generated by D and D . We have surjective morphisms of pro-algebraic groups q i : MT( E ) → MT( D i ), i = 1 ,
2. We claim that these are both isomorphisms.From the commutative diagram MT( E ev ) MT( D ev1 )MT( D ev2 ) MT( V ) q ev1 q ev2 ∼ = ∼ = it is apparent that ker( q ev1 ) = ker( q ev2 ). Lemma A.5 now implies that ker( q ) = ker( q ) in MT( E ).But this precisely means that the subcategories D and D of E coincide, and we conclude thatwe have D = E = D . (cid:3) Thanks to Andr´e’s Theorem 2.4, we can lift Theorem A.4 to the category of abelian Andr´emotives.
Corollary A.6 (Motivic Kuga–Satake category) . If M ∈ AM is an abelian Andr´e motive whoseHodge realization is of K3 -type, then there exists a unique tannakian subcategory KS ( M ) of AM such that h M i AM = KS ( M ) ev ( KS ( M ) . Moreover, KS ( M ) = hH ( A ) i AM for any Kuga–Satake variety A (Definition A.2) for the Hodgestructure r ( M ) . The above discussion leads us naturally to the following question about relations among differentKuga–Satake abelian varieties.
Question:
Let A and B be abelian varieties such that h H ( A ) i = h H ( B ) i in HS pol Q . Does thisimply the existence of integers k, l , such that A is dominated by B k and B is dominated by A l ? References
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