On the Mumford-Tate conjecture for hyperk\"{a}hler varieties

We study the Mumford--Tate conjecture for hyperk\"{a}hler varieties. We show that the full conjecture holds for all varieties deformation equivalent to either an Hilbert scheme of points on a K3 surface or to O'Grady's ten dimensional example, and all of their self-products. For an arbitrary hyperk\"{a}hler variety whose second Betti number is not 3, we prove the Mumford--Tate conjecture in every codimension under the assumption that the K\"{u}nneth components in even degree of its Andr\'{e} motive are abelian. Our results extend a theorem of Andr\'{e}.

1. Introduction 1.0. -Let k ⊂ C be a finitely generated field, with algebraic closurek ⊂ C, and let ℓ be a prime number. Given a smooth and projective variety X over k, Artin's comparison theorem gives a canonical identification of Q ℓ -vector spaces between singular cohomology groups of X(C) and ℓ-adic cohomology groups of Xk.
Both sides come with additional structure, namely, a Hodge structure on the left hand side and a Galois representation on the right hand side. These data are encoded in the corresponding tannakian fundamental groups. The Mumford-Tate conjecture predicts that Artin's comparison isomorphism identifies the two groups. We refer to this statement for i = 2j as the Mumford-Tate conjecture in codimension j for X. Theorem 1.1.
-Let X be a hyperkähler variety of either K3 [m] or OG10-type.
Then, the Mumford-Tate conjecture holds in any codimension for X and for all selfproducts X j .
Our second result establishes the Mumford-Tate conjecture in any codimension for a hyperkähler variety with b 2 > 3 whose even André motive is abelian.
Theorem 1.2. -Let X be a hyperkähler variety such that b 2 (X) > 3. Assume that, for all i ≥ 0, the even component of the André motive of X is an abelian motive.
Then, the Mumford-Tate conjecture holds in any codimension for X. In particular, the Hodge and Tate conjecture for X are equivalent.
The work [KSV19] suggests that any hyperkähler variety has abelian André motive, however, for the time being, this statement remains a conjecture (but see §1.3). By definition, the second Betti number of a hyperkähler variety is always at least 3; all known examples satisfy b 2 > 3 and it is believed that no hyperkähler variety with b 2 = 3 exists [Bea11, Question 4].
1.2. Overview of the contents. -We recall in §2 the statement of the Mumford-Tate conjecture and its motivic version; throughout, we will use the category of motives constructed by André in [And96b]. The following theorem is essentially proven in [And96a], and it has been generalized in [Moo17b].
Then, the motivic Mumford-Tate conjecture in codimension 1 holds for X.
The main tool used in the proof of this result is the Kuga-Satake construction in families, building on ideas due to Deligne [Del71]. The assumption that b 2 (X) > 3 ensures the existence of non-trivial deformations of X, as otherwise the moduli space of hyperkähler varieties deformation equivalent to X would be zero-dimensional.
With X as above, we consider the even part H + B (X) of the singular cohomology with rational coefficients of X(C), so H + B (X) = i H 2i B (X). A crucial ingredient for us is the action of a Q-Lie algebra g tot (X) on H + B (X). This construction is due to Verbitsky [Ver96] and Looijenga-Lunts [LL97]; we recall it in §3. The even singular cohomology of X is the Hodge realization of a motive H + (X), whose motivic Galois group is denoted by G + mot (X). We study the interplay between the actions of this group and of the Lie algebra g tot (X) on H + B (X). This cohomology algebra carries a Hodge structure, whose Mumford-Tate group is denoted by MT + (X); we show in §4 that MT + (X) is a direct factor of the motivic Galois group G + mot (X). Here, we need to assume that b 2 (X) > 3 since we use André's Theorem 1.3.
In §5 we prove that if MT + (X) has finite index in the motivic Galois group G + mot (X), then the Mumford-Tate conjecture holds in arbitrary codimension for X, see Proposition 5.1. The proof of Theorem 1.2 is given in §5.2; this is in fact a direct consequence of the proposition and a general result on abelian motives due to André.
In §6 we complete the proof of our main result Theorem 1.1. By Proposition 4.1, MT + (X) is a direct product factor of G + mot (X); moreover, the complement satisfies various constraints and in particular it commutes with the action of g tot (X), see Lemma 4.3. For the K3 [m] -type, we have a very effective understanding of this action thanks to work of Markman [Mar08], and we deduce from his results that the Mumford-Tate group has finite index in the motivic Galois group. For the OG10type, this finiteness follows instead from the complete description of the g tot (X)representation on the cohomology given by Green-Kim-Laza-Robles [GKLR19]. In both cases, we apply Proposition 5.1 to conclude. In each case, the proof requires a deformation to an explicit example in the given deformation type. We remark that the proof of Theorem 1.1 presented here is different and simpler: it uses neither deformation to a specific example, nor abelianity of the motives involved. We hope that a refinement of this method might lead to a proof of the Mumford-Tate conjecture for arbitrary hyperkähler varieties with b 2 > 3.
1.4. Notation and conventions. -Throughout the whole text, k ⊂ C will be a finitely generated field with algebraic closurek ⊂ C, and ℓ will be a fixed prime number. A hyperkähler variety over k is a smooth projective variety over k such that X(C) is a hyperkähler manifold, as defined in §3.0. Given a complex variety X, we denote by H i (X) its rational singular cohomology groups. The word "motive" always indicates an object of André's category of motives (see §2.4).
Acknowledgements. -I am most grateful to Ben Moonen and Arne Smeets for their careful reading and the many comments, which substantially improved this text.
I am also thankful to the anonymous referee for his/her comments.
2. The Mumford-Tate conjecture 2.0. -We refer to [Moo17a] and the references therein. With notations and assumptions as in §1.4, we let X be a smooth projective variety over the field k. We can extract information about X by looking at various cohomology groups. 2.2. ℓ-adic cohomology. -We write H i ℓ (X) for the i-thétale cohomology group of Xk with Q ℓ -coefficients, which comes with a continuous representation σ ℓ : Gal(k/k) → GL H i ℓ (X) ; we denote by G H i ℓ (X) the Zariski closure of the image of σ ℓ . It is an algebraic group over Q ℓ . If k ′ /k is a field extension, and if X k ′ denotes the base change of X to k ′ , it may happen that G H i ℓ (X k ′ ) becomes smaller than G H i ℓ (X) ; however, the connected component of the identity G H i ℓ (X) 0 is stable under finite field extensions, and there exists a finite field extension k ′ /k such 2.3. -Artin's comparison theorem states that, for all X and i as above, there is a canonical isomorphism of Q ℓ -vector spaces The Mumford-Tate conjecture in codimension j for X is this statement for i = 2j.  We summarize a few known facts about these groups.
-There are natural inclusions -The algebraic group MT(M B ) is connected and reductive. On the other hand, is not known to be reductive, while G mot (Mk) is reductive, but not known to be connected in general.
2.8. -There are contravariant functors H i from the category of smooth projective varieties over k to Mot k , such that, for any smooth projective variety X over k, we have . Therefore, Conjecture 2.2 implies Conjecture 2.1. We refer to Conjecture 2.2 for the motive H 2j (X) as the motivic Mumford-Tate conjecture for X in codimension j.
2.9. Abelian motives. -A motive M ∈ Mot k is abelian if it belongs to the tannakian subcategory generated by the motives of all abelian varieties over k. We will need the following theorem due to André [And96b], which improves Deligne's result on absolute Hodge classes on abelian varieties from [Del82].

Hyperkähler varieties
3.0. -In this section, we work over the complex numbers. A hyperkähler manifold X is a connected, simply connected, compact Kähler manifold admitting a nowhere degenerate holomorphic 2-form which spans H 0,2 (X). At times, we use the expression "hyperkähler variety" instead of writing "projective hyperkähler manifold". The dimension of such a manifold is always even; hyperkähler surfaces are K3 surfaces. The second cohomology group of a hyperkähler manifold X carries a canonical symmetric bilinear form, the Beauville-Bogomolov form, which is non-degenerate and deformation invariant, and yields a morphism of Hodge structures H 2 (X)(1)⊗H 2 (X)(1) → Q.
We refer to [Bea83] and [Huy99] for a proper introduction to the subject.
Let X be a complex hyperkähler variety of dimension 2n. The rational cohomology H * (X) of X is a graded algebra via cup product. Verbitsky and Looijenga-Lunts studied in [Ver96] and [LL97] a Lie algebra action on H * (X), which we describe below.
3.1. -Let θ ∈ End H * (X) be the degree 0 endomorphism whose action on H j (X) is multiplication by j − 2n, for all j. Given x ∈ H 2 (X), we denote by L x the endomorphism of H * (X) which maps a cohomology class α to the product x ∧ α. We say that a class x ∈ H 2 (X) has the Lefschetz property if, for all positive integers j, and Λ x form an sl 2 -triple, i.e., we have Once it exists, the endomorphism Λ x is uniquely determined, see for instance Proposition 1.4.6 in [GGK + 68, Exposé X].
3.2. -We define g tot (X) as the smallest Lie subalgebra of gl H * (X) containing L x , for all x ∈ H 2 (X), and Λ x , for all x ∈ H 2 (X) with the Lefschetz property. The first Chern class of an ample divisor on X has the Lefschetz property by the Hard Lefschetz theorem. It is shown in [LL97, §(1.9)] that g tot (X) is a semisimple Q-Lie algebra, which is evenly graded by the adjoint action of θ, so that g tot (X) = i g 2i (X). The action of g tot (X) on the cohomology of X preserves the even and odd cohomology, and the Lie subalgebra g 0 (X) consists of the endomorphisms contained in g tot (X) which preserve the grading of H * (X). The construction does not depend on the complex structure of X; therefore, g tot (X) is deformation invariant.  (b) We have g tot (X) = g −2 (X) ⊕ g 0 (X) ⊕ g 2 (X).
Moreover, g 0 (X) ∼ = so(H) ⊕ Q · θ, and θ is central in g 0 (X). The abelian subalgebra g 2 (X) is the linear span of the endomorphisms L x , and g −2 (X) is the span of the Λ x , for x ∈ H 2 (X) with the Lefschetz property.
(c) The Lie subalgebra g 0 (X) acts via derivations on the graded algebra H * (X). The induced action of so(H) ⊂ g 0 (X) on H 2 (X) = H is the standard representation.
The above theorem is proven in [Ver96], and in [LL97, Proposition 4.5]. A proof can also be found in the appendix of [KSV19]. These proofs are carried out with real coefficients, but immediately imply the result with rational coefficients: since g tot (X) is defined over Q, the equality g tot (X)⊗R = so(H)⊗R of Lie subalgebras of gl(H)⊗R shows that the same equality already holds with rational coefficients.
3.4. -We know from Theorem 3.1 that the semisimple part of g 0 (X) is isomorphic to so(H). We denote by the restriction of the representation g 0 (X) → j gl H j (X) to the Lie subalgebra so(H). We also let ρ + : so(H) → i gl H 2i (X) denote the representation induced by ρ on the even cohomology of X. where π 2 is the obvious projection i GL H 2i (X) → GL H 2 (X) .
We refer to [Ver95, §8] for a proof. Note that under the representationρ + , the group SO(H) acts via graded algebra automorphisms on the even cohomology of X, by part (c) of Theorem 3.1.

-
We need to recall one more result. Let W C ∈ End H * (X, C) be the endomorphism which acts on each H p,q (X) as multiplication by i(p − q). It is known that W C is the C-linear extension of an endomorphism W ∈ End H * (X, R) , which is called the Weil operator. This is proven in [Ver96]; see also the appendix to the paper [KSV19].
Corollary 3.4. -The map π 2 is an isomorphism In particular, the weight 0 Hodge structure H + (X) belongs to the tensor subcategory of polarizable Q-Hodge structures generated by H 2 (X)(1).
We first prove a lemma.  -In this section, X is a complex hyperkähler variety; we further assume that b 2 (X) > 3. We consider the weight 0 motive and we denote by G + mot (X) ⊂ i GL H 2i (X)(i) its motivic Galois group. We letπ 2 be the projection G + mot (X) ։ G mot H 2 (X)(1) induced by the inclusion of H 2 (X)(1) into H + (X), and we define P (X) := ker(π 2 ) ⊂ G + mot (X).
We will first establish some preliminary results.
Proof. -We have a commutative diagram Here, ι + and ι 2 denote the natural inclusions; π 2 and ι 2 are isomorphisms due to Corollary 3.4 and Theorem 1.3 respectively. We can now take s = ι + • (ι 2 •π 2 ) −1 . Proof. -Note that P (X) acts on H + (X) via algebra automorphisms since the cupproduct is induced by an algebraic cycle, namely, the small diagonal δ ⊂ X 3 ; moreover, by definition, its action preserves the grading and is trivial on H 2 (X). Hence, if p ∈ P (X), then p commutes with θ and L x , for x ∈ H 2 (X). Further, if x has the Lefschetz property, then p commutes with Λ x as well: indeed, L x , θ and pΛ x p −1 form an sl 2 -triple, and this forces pΛ x p −1 = Λ x , see §3.1. As the various operators L x and Λ x , for x ∈ H 2 (X), generate the Lie subalgebra g tot (X) ⊂ gl H + (X) , we conclude that P (X) commutes with the whole of g tot (X).
Proof of Proposition 4.1. -By Lemma 4.2, P (X) · MT + (X) = G + mot (X), and the two subgroups have trivial intersection. By Lemma 3.5 and the above Lemma 4.3, P (X) and MT(X) + commute. It follows that G + mot (X) is the direct product of these two subgroups.

A sufficient condition
5.0. -With notations and assumptions as in §1.4, let X be a hyperkähler variety over k, and assume that b 2 (X) > 3. Consider the weight 0 motive and write G + mot (X) for its motivic Galois group. Let H + B (X) and H + ℓ (X) denote respectively the Hodge and ℓ-adic realization of H + (X). We write MT + (X) for MT H + B (X) and G + ℓ (X) for G H + ℓ (X) . We identify H + B (X) ⊗ Q ℓ with H + ℓ (X) via Artin's comparison isomorphism. Then both MT + (X) ⊗ Q ℓ and G + ℓ (X) are identified with subgroups of GL H + ℓ (X) .

-Conjecturally, under
Artin's isomorphism we have MT + (X) ⊗ Q ℓ ∼ = G + ℓ (X) 0 . We refer to this statement as the Mumford-Tate conjecture for H + (X); it implies the Mumford-Tate conjecture in codimension j for X and for all integers j, and, if X has trivial odd cohomology, it also implies the Mumford-Tate conjecture in any codimension for any self-power X k . Recall from §2.7 that G + ℓ (X) 0 is a subgroup of G + mot (Xk) ⊗ Q ℓ ∼ = G + mot (X C ) ⊗ Q ℓ , and that, by Proposition 4.1, we have an equality G + mot (X C ) = P (X) × MT + (X) of subgroups of GL H + B (X) .
Proof. -Consider the commutative diagram The horizontal arrows on the bottom are isomorphisms due to Theorem 1.3, and the vertical map on the left is an isomorphism thanks to Corollary 3.4. By Proposition 4.1 we have G + mot (Xk) = P (X) × MT + (X); if P (X) is finite, it follows that we have G + mot (Xk) 0 = MT + (X). Hence, replacing in the above diagram G + mot (Xk) with its connected component of the identity, also the leftmost arrow on the top row becomes an isomorphism. Thus all arrows in the diagram become isomorphisms, and we obtain Moreover, if P (X) is trivial then G + mot (Xk) is connected and equal to MT + (X), and therefore the motivic Mumford-Tate conjecture holds in this case. If m = 1, then X is the original K3 surface; we will assume m ≥ 2. In this case dim X = 2m, the odd cohomology of X vanishes, and the second Betti number equals 23, [Göt90]. We say that X is of OG10-type if it is deformation equivalent to O'Grady's ten dimensional hyperkähler variety constructed in [O'G99]. In this case the odd Betti numbers of X vanish as well, and we have b 2 (X) = 24, see [dCRS19].
6.1. -Let Aut H + (X) be the group of automorphisms of the graded Q-algebra H + (X) = i H 2i (X). Let K(X) ⊂ Aut H + (X) be the kernel of the natural restriction map Aut H + (X) → GL H 2 (X) . The group P (X) acts via algebra automorphisms, and, by construction, its action is trivial in degree 2. Hence, we have P (X) ⊂ K(X).
To conclude the proof of Theorem 1.1 it therefore suffices to establish the following.
Proposition 6.1. -Assume X is a hyperkähler variety of either K3 [m] or OG10type. Then K(X) is a finite group.
As we are going to explain, this is a consequence of results due to Markman [Mar08] in the first case and due to Green-Kim-Laza-Robles [GKLR19] in the second case.
for α of degree 2n + 2q. It is shown in [LL97, Proposition 1.6 and its proof], that the Lie algebra g tot (X) preserves infinitesimally the Poincaré pairing, and that φ restricts to a non-degenerate pairing on every g tot (X)-submodule of H + (X).
6.3. -The group K(X) acts on H + (X) via graded algebra automorphisms and it acts trivially in degree 2; it follows that K(X) preserves the pairing φ. Moreover, the argument used to prove Lemma 4.3 shows that this group commutes with g tot (X).
Proof of Proposition 6.1 for the OG10-type. -Assume X is of OG10-type. The representation of g tot (X) on the cohomology has been fully described in [GKLR19, Theorem 1.1-(iv)]. We have where V 1 is the subalgebra generated by H 2 (X) and V 2 is an absolutely irreducible g tot (X)-representation. We deduce that K(X) is a subgroup of by Schur's lemma. Further, we know that the pairing φ restricts to a non-degenerate invariant form on V 2 , and we deduce that K(X) ⊂ {1, −1}. 6.4. -As apparent from the proof, for the OG10-type the decomposition of the cohomology into g tot (X)-isotypical components already imposes the desired finiteness result. The analogous decomposition for the K3 [m] -type becomes more and more complicated as the dimension increases, see [GKLR19]. Nevertheless, it becomes more manageable once the algebra structure is taken into account, thanks to the following result of Markman. From now on, we assume that X is a variety of K3 [m] -type.
For l ≥ 0, we let A 2l ⊂ H + (X) be the subalgebra generated by j≤l H 2j (X). Note that A 2l = H + (X) for l ≥ m. Recall from Corollary 3.2 that we have a representationρ + of SO(H) on H + (X).
Note that this implies C 2 = H 2 (X) and C 2i = 0 for i > m. Each C 2i is in particular a subrepresentation for g 0 (X) and, hence, for SO(H). Moreover, the g tot (X)-module generated by C 2i is orthogonal to A 2i−2 with respect to φ. Proof of Proposition 6.1 for the K3 [m] -type. -We claim first of all that each subspace C 2i is stable under the action of K(X). In fact, since K(X) acts via graded algebra automorphisms, the subalgebras A 2l are K(X)-stable for all l. Since φ is K(X)-invariant, it follows that the orthogonal complement to each A 2l is preserved as well; as K(X) acts compatibly with the grading, it indeed stabilizes C 2i , for all i.
The subspaces C 2i generate the cohomology by Theorem 6.2.(b), and K(X) commutes with the representationρ. Hence, we have Let V ⊂ C 2i be an irreducible-SO(H) representation. By Theorem 6.2.(c), the representation V is absolutely irreducible and it appears in C 2i with multiplicity one; it follows that V is stable under K(X) as well. By Schur's lemma, each element of K(X) acts on g tot (X) · V via multiplication by some rational number. On the other hand K(X) preserves the form φ, whose restriction to the g tot (X)-module generated by V is non-degenerate, and therefore the action of K(X) on g tot (X) · V factors through {1, −1} ∼ = Z/2Z. Using again Theorem 6.2.(c), we conclude that, for all i, End(C 2i ) SO(H) × is a subgroup Z/2Z 2 , and hence we have Theorem 1.1 is proved.
Remark 6.3. -The conclusion of Proposition 6.1 does not hold for the remaining deformation types Kum m and OG6. This can be checked using the description of the g tot (X)-representation of the cohomology given in [GKLR19]: in fact, for these deformation types, there are g tot (X)-representations which appear in the cohomology with higher multiplicities, which cannot be explained only by taking into account the algebra structure on the cohomology.