Derived equivalences of hyperk\"ahler varieties

We show that the Looijenga--Lunts--Verbitsky Lie algebra acting on the cohomology of a hyperk\"ahler variety is a derived invariant, and obtain from this a number of consequences for the action on cohomology of derived equivalences between hyperk\"ahler varieties. This includes a proof that derived equivalent hyperk\"ahler varieties have isomorphic $\mathbf{Q}$-Hodge structures, the construction of a rational `Mukai lattice' functorial for derived equivalences, and the computation (up to index 2) of the image of the group of auto-equivalences on the cohomology of certain Hilbert squares of K3 surfaces.

1. Introduction 1.1. Background. We briefly recall the background to our results. We refer to [24] for more details. For a smooth projective complex variety X we denote by DX the bounded derived category of coherent sheaves on X. By a theorem of Orlov [38] any (exact, C-linear) equivalence Φ : DX 1 ∼ → DX 2 comes from a Fourier-Mukai kernel P ∈ D(X 1 × X 2 ), and convolution with the Mukai vector v(P) ∈ H(X 1 × X 2 , Q) defines an isomorphism between the total cohomology of X 1 and X 2 . This isomorphism is not graded, and respects the Hodge structures only up to Tate twists. Nonetheless, Orlov has conjectured [39] that if X 1 and X 2 are derived equivalent, then for every i there exist (non-canonical) isomorphisms H i (X 1 , Q) ∼ = H i (X 2 , Q) of Q-Hodge structures.
Its image is known for varieties with ample or anti-ample canonical class (in which case Aut(DX) is small and well-understood [9]), for abelian varieties [19], and for K3 surfaces. To place our results in context, we recall the description of the image for K3 surfaces. Let X be a K3 surface. Consider the Mukai lattice H(X, Z) := H 0 (X, Z) ⊕ H 2 (X, Z(1)) ⊕ H 4 (X, Z(2)).
It was observed by Mukai [36] that if Φ : DX 1 ∼ → DX 2 is a derived equivalence between K3 surfaces, then Φ H restricts to an isomorphism Φ H : H(X 1 , Z) → H(X 2 , Z) respecting the pairing and Hodge structures. Denote by Aut( H(X, Z)) the group of isometries of H(X, Z) respecting the Hodge structure, and by Aut + ( H(X, Z)) the subgroup (of index 2) consisting of those isometries that respect the orientation on a four-dimensional positive definite subspace of H(X, R). Theorem 1.1 ([36,37,22,40,26]). Let X be a K3 surface. Then the image of ρ X is Aut + ( H(X, Z)).
In this paper, we prove Orlov's conjecture on Q-Hodge structures for hyperkähler varieties, construct a rational version of the Mukai lattice for hyperkähler varieties, and compute (up to index 2) the image of ρ X for certain Hilbert squares of K3 surfaces. The main tool in these results is the Looijenga-Lunts-Verbitsky Lie algebra.
1.2. The LLV Lie algebra and derived equivalences. Let X be a smooth projective complex variety. By the Hard Lefschetz theorem, every ample class λ ∈ NS(X) determines a Lie algebra g λ ⊂ End(H(X, Q)) isomorphic to sl 2 . More generally, this holds for every cohomology class λ ∈ H 2 (X, Q) (algebraic or not) satisfying the conclusion of the Hard Lefschetz theorem. Looijenga and Lunts [34], and Verbitsky [46] have studied the Lie algebra g(X) ⊂ End(H(X, Q)) generated by the collection of the Lie algebras g λ . We will refer to this as the LLV Lie algebra. See § 2.1 for more details.
We say that X is holomorphic symplectic if it admits a nowhere degenerate holomorphic symplectic form σ ∈ H 0 (X, Ω 2 X ). Theorem A ( § 2.4). Let X 1 and X 2 be holomorphic symplectic varieties. Then for every equivalence Φ : DX 1 ∼ → DX 2 there exists a canonical isomorphism of rational Lie algebras Φ g : g(X 1 ) with the property that the map Φ H : H(X 1 , Q) ∼ → H(X 2 , Q) is equivariant with respect to Φ g .
Note that g(X) is defined in terms of the grading and the cup product on H(X, Q), neither of which are preserved under derived equivalences.
To prove Theorem A we introduce a complex Lie algebra g ′ (X) whose definition is similar to the rational Lie algebra g(X), but where the action of H 2 (X, Q) on H(X, Q) is replaced with a natural action of the Hochschild cohomology group HH 2 (X) on Hochschild homology HH • (X). Since Hochschild cohomology and its action on Hochschild homology is known to be invariant under derived equivalences, it follows that g ′ (X) is a derived invariant. We show that if X is holomorphic symplectic, then the isomorphism HH • (X) → H(X, C) (coming from the Hochschild-Kostant-Rosenberg isomorphism) maps g ′ (X) to g(X)⊗ Q C. This is closely related to Verbitsky's 'mirror symmetry' for hyperkähler varieties [46,47]. From this we deduce that the rational Lie algebra g(X) is a derived invariant.

1.3.
A rational Mukai lattice for hyperkähler varieties. Let X be a hyperkähler variety. Consider the Q-vector space H(X, Q) := Qα ⊕ H 2 (X, Q) ⊕ Qβ equipped with the bilinear form b which is the orthogonal sum of the Beauville-Bogomolov form on H 2 (X, Q) and a hyperbolic plane Qα ⊕ Qβ with α and β isotropic and b(α, β) = −1. By analogy with the case of a K3 surface, we will call H(X, Q) the (rational) Mukai lattice of X. Looijenga-Lunts [34] and Verbitsky [46] have shown that the Lie algebra g(X) can be canonically identified with so( H(X, Q)), see § 3.1 for a precise statement. Moreover, Verbitsky [46] has shown that the sub-algebra SH(X, Q) of H(X, Q) generated by H 2 (X, Q) forms an irreducible sub-g(X)-module. Using this, we show that Theorem A implies: Theorem B ( § 4.2). Let X 1 and X 2 be hyperkähler varieties and Φ : DX 1 ∼ → DX 2 an equivalence. Then the induced isomorphism Φ H restricts to an isomorphism Φ SH : SH(X 1 , Q) ∼ → SH(X 2 , Q).
Taking X 1 = X 2 = X in Theorem B we obtain a homomorphism ρ SH X : Aut(DX) −→ GL(SH(X, Q)). The complex structure on a hyperkähler variety X induces a Hodge structure of weight 0 on H(X, Q) given by Denote by Aut H(X, Q) the group of Hodge isometries of H(X, Q).
Theorem C ( § 4.2). Let X be a hyperkähler variety of dimension 2d and second Betti number b 2 . Assume that b 2 is odd or d is odd. Then ρ SH X factors over a map ρ H X : Aut(D(X)) → Aut( H(X, Q)).
Note that all known hyperkähler varieties satisfy the parity conditions in the theorem: there are two infinite series of deformation classes with odd b 2 (generalized Kummers and Hilbert schemes of points), and three exceptional deformation classes with odd d (K3, OG6, OG10) .

1.4.
Hodge structures of derived equivalent hyperkähler varieties. Another application of Theorem A is the following.
Theorem D ( § 5). Let X 1 and X 2 be derived equivalent hyperkähler varieties. Then for every i the Q-Hodge structures H i (X 1 , Q) and H i (X 2 , Q) are isomorphic.
This confirms Orlov's conjecture for hyperkähler varieties. The proof is inspired by [43].
1.5. Auto-equivalences of the Hilbert square of a K3 surface. In the second half of the paper we consider the problem of determining the image of ρ X for certain hyperkähler varieties. An important difference with the first half of the paper is that integral structures (lattices, arithmetic subgroups, . . . ) will play an important role here.
As a first approximation to determining the image of ρ X , we consider a variation of this problem which is deformation invariant. Let X be a smooth projective complex variety. If X ′ and X ′′ are smooth deformations of X (parametrized by paths in the base), and if Φ : DX ′ ∼ → DX ′′ is an equivalence, then we obtain an isomorphism as the composition We define the derived monodromy group of X to be the subgroup DMon(X) of GL(H(X, Q)) generated by all these isomorphisms. This group contains both the usual monodromy group of X and the image of ρ X : Aut(DX) → GL(H(X, Q)).
If S is a K3 surface, then the result of [26] implies DMon(S) = O + ( H(S, Z)), and that the image of ρ S consists of those elements of DMon(S) that respect the Hodge structure on H(S, Z). Similarly, for an abelian variety A, the results of [19] imply DMon(A) = Spin(H 1 (A, Z)⊕H 1 (A ∨ , Z)), and that the image of ρ A consists of those elements of DMon(A) that respect the Hodge structure on Now let X be a hyperkähler variety of type K3 [2] . We have H(X, Q) = SH(X, Q) and hence by Theorem C the action of Aut(DX) on H(X, Q) factors over a subgroup O( H(X, Q)) of GL(H(X, Q)).
Theorem E ( § 9.4). Let X be a hyperkähler variety deformation equivalent to the Hilbert square of a K3 surface. There is an integral lattice Λ ⊂ H(X, Q) such that See § 9.4 for a precise description of Λ. As an abstract lattice, Λ is isomorphic Crucial in the proof of Theorem E is the derived McKay correspondence due to Bridgeland, King, Reid [11] and Haiman [21]. It provides an ample supply of elements of DMon(X): every deformation of X to the Hilbert square S [2] of a K3 surface S induces an inclusion DMon(S) → DMon(X). As part of the proof, we explicitly compute this inclusion.
We denote by Aut(Λ) the group of isometries of Λ ⊂ H(X, Q) that respect the Hodge structure on H(X, Q). It follows from Theorem E that im(ρ X ) is contained in Aut(Λ) for every X which is deformation equivalent to the Hilbert square of a K3 surface. For some X we can show that the upper bound in the above corollary is close to being sharp. Denote by Aut + (Λ) ⊂ Aut(Λ) the subgroup consisting of those Hodge isometries that respect the orientation of a positive 4-plane in Λ R .

Remark 1.2.
To determine im ρ X up to index 2 for a general hyperkähler of type K3 [2] new constructions of derived equivalences will be needed. Remark 1.3. Theorem E and Theorem F leave an ambiguity of index 2, related to orientations on a maximal positive subspace of H(X, R). In the case of K3 surfaces, it was conjectured by Szendrői [44] that derived equivalences must respect such orientation, and this was proven by Huybrechts, Macrì, and Stellari [26]. Their method is based on deformation to generic (formal or analytic) K3 surfaces of Picard rank 0, and on a complete understanding of the space of stability conditions on those [25]. It is far from clear if such a strategy can be used to remove the index 2 ambiguity for hyperkähler varieties of type K3 [2] . Remark 1.4. That a lattice of signature (4, b 2 − 2) should play a role in describing the image of ρ X for hyperkähler varieties X was expected from the physics literature [16], but it is not clear where the lattice should come from, nor what its precise description should be for general hyperkähler varieties. In the above results, the lattice Λ arises in a rather implicit way, and one may hope for a more concrete interpretation of its elements. Remark 1.5. It is tempting to try to conjecture a description of the group Aut(DX) in terms of an action on a space of stability conditions on X, generalising Bridgeland's work on K3 surfaces [10]. However, there is a representation-theoretic obstruction against doing this naively. The central charge of a hypothetical stability condition on X takes values in H(X, C), yet Theorems E and F suggest the central charge should take values in H(X, C). If X is of type K3 [2] , then H(X, C) and H(X, C) are non-isomorphic irreducible DMon(X)-modules, so that this would require a modification of the notion of stability condition.
1.6. Acknowledgements. I am grateful to Nick Addington, Thorsten Beckmann, Eyal Markman, and Zoë Schroot for many valuable comments on earlier drafts of this paper.

The LLV Lie algebra of a smooth projective variety
In this section we recall the construction of Looijenga and Lunts [34] and Verbitsky [46] of a Lie algebra acting naturally on the cohomology of algebraic varieties. For holomorphic sympletic varieties we show that this Lie algebra is a derived invariant.
2.1. The LLV Lie algebra. Let F be a field of characteristic zero and let M be a Z-graded F -vector space, of finite F -dimension. Denote by h the endomorphism of M that is multiplication by n on M n .
Let e be an endomorphism of M of degree 2. We say that e has the hard Lefschetz property if for every n ≥ 0 the map e n : M −n → M n is an isomorphism. This is equivalent to the existence of an f ∈ End(M ) such that the relations is an isomorphism. It sends f to −2e, so that f is indeed uniquely determined. If e and h lie in g, then g ⊂ End(M ) is graded and the above map restricts to an injective map (ad e) 2 : g −2 ֒→ g 2 . Since h is diagonisable, it is contained in a Cartan sub-algebra of g. The symmetry of the resulting root system implies that dim g −n = dim g n for all n. In particular, the map (ad e) 2 defines an isomorphism between g −2 and g 2 , and we conclude that f lies in g.
Let a be an abelian Lie algebra and e : a → gl(M ), a → e a a Lie homomorphism. We say that e has the hard Lefschetz property if e(a) ⊂ gl(M ) 2 and if there exists some a ∈ a so that e a has the hard Lefschetz property. Note that this is a Zariski open condition on a ∈ a.
If e : a → gl(M ) has the hard Lefschetz property, then we denote by g(a, M ) the Lie algebra generated by the sl 2 -triples (e a , h, f a ) for a ∈ a such that e a has the hard Lefschetz property. We say that (a, M ) is a Lefschetz module if g(a, M ) is semisimple.
Now let X be a smooth projective complex variety of dimension d. Denote by M := H(X, Q)[d] the shifted total cohomology of X (with middle cohomology in degree 0). For a class λ ∈ H 2 (X, Q) consider the endomorphism e λ ∈ End(M ) given by cup product with λ. If λ is ample, then e λ has the hard Lefschetz property, so that the map e : H 2 (X, Q) → gl(M ) has the hard Lefschetz property. We denote the corresponding Lie algebra by g(X) := g(H 2 (X, Q), M ). In other words, g(X) is a semisimple Lie algebra over Q.

2.2.
Hochschild homology and cohomology. Let X be a smooth projective variety of dimension d with canonical bundle ω X := Ω d X . Its Hochschild cohomology is defined as HH n (X) := Ext n X×X (∆ * O X , ∆ * O X ) and its Hochschild homology is defined as HH n (X) := Ext d−n X×X (∆ * O X , ∆ * ω X ). Composition of extensions defines maps HH n ⊗HH m → HH n+m and HH n ⊗HH m → HH m−n making HH • (X) into a graded module over the graded ring HH • (X).
The Hochschild-Kostant-Rosenberg isomorphism (twisted by the square root of the Todd class as in [31] and [15]) defines isomorphisms I n : HH n (X) Under these isomorphisms, multiplication in HH • (X) corresponds to the operation induced by the product in ∧ • T X , and the action of HH • (X) on HH • (X) corresponds to the action induced by the contraction action of ∧ • T X on Ω • X , see [13,12]. Together with the degeneration of the Hodge-de Rham spectral sequence, the isomorphism I • defines an isomorphism This map does not respect the grading, rather it maps HH i to the i-th column of the Hodge diamond (normalised so that the 0-th column is the central column ⊕ p H p,p ). Combining with the action of HH • on HH • we obtain an action of the ring HH • (X) on H(X, C). Theorem 2.3. Let Φ : DX 1 ∼ → DX 2 be a derived equivalence between smooth projective complex varieties. Then we have natural graded isomorphisms compatible with the ring structure on HH • and the module structure on HH • , and such that the square Proof. See [12] and [35].
2.3. The Hochschild Lie algebra of a holomorphic symplectic variety. Now assume that X is holomorphic symplectic of dimension 2d. That is, we assume that there exists a symplectic form σ ∈ H 0 (X, Ω 2 X ). Note that this implies that a Zariski dense collection of σ ∈ H 0 (X, Ω 2 X ) will be nowhere degenerate. Through the isomorphism I : HH • (X) → H(X, C), the vector space H(X, C) becomes a module under the ring HH • (X).
This proves the first assertion. For the second it suffices to observe that the module HH • (X, C) is generated by σ d ∈ HH 2d (X) = H 0 (X, Ω 2d X ).
Consider the endomorphisms h p , h q ∈ End(H(X, C)) given by These define the Hodge bi-grading on H(X, C), normalised to be symmetric along the central part H d,d . Note that h = h p + h q . The action of HH n (X) on H(X, C) has degree n for the grading defined by h ′ = h q − h p . Lemma 2.4 and Hard Lefschetz imply: For a Zariski dense-collection of µ ∈ HH 2 (X) the action by µ e ′ µ : H(X, C) → H(X, C) has the hard Lefschetz property with respect to the grading defined by h ′ .
In particular, for every such µ we have a complex subalgebra g µ ⊂ End(H(X, C)) isomorphic to sl 2 , and the collection of such algebras generates a Lie algebra which we denote by g ′ (X) ⊂ End(H(X, C)). From Lemma 2.4 we also obtain: Corollary 2.6. The complex Lie algebras g ′ (X)and g(X)⊗ Q C are isomorphic.
In the next section, we will show something stronger: that g ′ (X) and g(X) ⊗ Q C coincide as sub-Lie algebras of End(H(X, C)). Theorem A then follows by combining this with the following proposition.
Proposition 2.7. Assume that X 1 and X 2 are holomorphic symplectic varieties. Then for every equivalence Φ : It has the property that the map Φ H : Proof. This follows immediately from Theorem 2.3.

2.4.
Comparison of the two Lie algebras and proof of Theorem A. The remainder of this section is devoted to the proof of the following.
. We will use the same symbol λ to denote an element λ ∈ H 2 (X, C) and the endomorphism of End(H(X, C)) given by cup product with λ. Note that we have λ ∈ g(X) ⊗ Q C by construction. Similarly, we will use the same symbol for µ ∈ HH 2 (X) and the resulting µ ∈ End(H(X, C)), given by contraction with µ. We have µ ∈ g ′ (X).
For a symplectic form σ ∈ H 0 (Ω 2 X ) we denote byσ ∈ H 0 (∧ 2 T X ) the image of the form σ ∈ H 0 (Ω 2 X ) under the isomorphism Ω 2 X → ∧ 2 T X defined by σ. In suitable local coordinates, we have Proof. Clearly σ has degree 2 andσ has degree −2 for the grading given by We need to show that [σ,σ] = h p . This follows immediately from a local computation: in the above local coordinates, one verifies that on the standard basis of Ω p the commutator [σ,σ] acts as p − d.
Note that the existence of one nowhere degenerate σ implies that a Zariski dense collection of σ ∈ H 0 (Ω 2 X ) is nowhere degenerate. Lemma 2.10. For a Zariski-dense collection α ∈ H 2 (X, O X ) there is aα ∈ End(H(X, C)) so that (α, h q ,α) is an sl 2 -triple.
Proof. This follows from Lemma 2.9 and Hodge symmetry.
Proof. It suffices to show that this holds for a Zariski dense collection of τ , hence we may assume without loss of generality that τ =σ with σ andσ as in Lemma 2.9. Let α andα be as in Lemma 2.10. Because σ and h p commute with both α and h q , we have that every element of the sl 2 -triple (σ, h p ,σ) commutes with every element of the sl 2 -triple (α, h q ,α). From this, it follows that are sl 2 -triples. Since the elements α ± σ lie in H 2 (X, C), and apparently have the hard Lefschetz property, we conclude that the endomorphismsα ±σ lie in g(X) ⊗ Q C, hence also τ =σ lies in g(X) ⊗ Q C.
Fix a τ ∈ H 0 (X, ∧ 2 T X ) that is nowhere degenerate as an alternating form on Ω 1 X . This defines isomorphisms c τ : Ω 1 X → T X and c τ : Proof. This is again a local computation. If η is a local section of Ω 1 X , then a computation on a local basis shows that [τ, η] = c τ (η) as maps Ω p X → Ω p−1 X . Corollary 2.14. Every element η ′ of H 1 (X, T X ) lies in g(X) ⊗ Q C.
Proof. Every such η ′ is of the form c τ (η) for a unique η ∈ H 1 (Ω 1 X ), and hence the corollary follows from Lemma 2.13, Lemma 2.11, and the fact that η lies in g(X) ⊗ Q C.
We can now finish the comparison of the two Lie algebras.
Proof of Proposition 2.8. By Corollary 2.6 it suffices to show that g ′ (X) is contained in g(X) ⊗ Q C. By Proposition 2.1 it suffices to show that h ′ is contained in g(X) ⊗ Q C, and that for almost every a ∈ HH 2 (X) we have that the action of a on H(X, C) is contained in g(X) ⊗ Q C. This follows from Lemma 2.11, Corollary 2.12, and Corollary 2.14, and the fact that the action of any Together with Proposition 2.7, this proves Theorem A.

Rational cohomology of hyperkähler varieties
3.1. The BBF form and the LLV Lie algebra. Let X be a complex hyperkähler variety of dimension 2d. We denote by its Beauville-Bogomolov-Fujiki, and by c X its Fujiki constant. These are related by We extend b to a bilinear form on by declaring α and β to be orthogonal to H 2 (X, Q), and setting b(α, β) = −1, b(α, α) = 0 and b(β, β) = 0. We equip H(X, Q) with a grading satisfying deg α = −2, deg β = 2, and for which H 2 (X, Q) sits in degree 0. This induces a grading on the Lie algebra so( H(X, Q)).
that maps e λ to e λ for every λ ∈ H 2 (X, Q).
The representation of so( H(X, Q)) on H(X, Q) integrates to a representation of Spin( H(X, Q)) on H(X, Q). Let λ ∈ H 2 (X, Q). Then e λ is nilpotent, and hence B λ := exp e λ is an element of Spin( H(X, Q)). It acts on H(X, Q) as follows: for all r, s ∈ Q and µ ∈ H 2 (X, Q). The action on the total cohomology of X is given by: In particular, if L is a line bundle on X and Φ : DX → DX is the equivalence that maps F to F ⊗ L, then Φ H = B c1(L) .
3.2. The Verbitsky component of cohomology. Let X be a complex hyperkähler variety of dimension 2d. We define the even cohomology of X as the graded Q-algebra H ev (X, Q) := n H 2n (X, Q), and the Verbitsky component of the cohomology of X as the sub-Q-algebra SH(X, [45,8]). The kernel of the Q-algebra homomorphism is generated by the elements λ d+1 with λ ∈ H 2 (X, Q) satisfying b(λ, λ) = 0. Verbitsky also describes the space SH(X, Q) explicitly. Below we normalise this description, and use it to compute the Mukai pairing on SH(X, Q). Proposition 3.5. There is a unique map Proof of Proposition 3.5. Uniqueness is clear. For existence, consider the map given by λ 1 · · · λ n → e λ1 · · · e λn (α d /d!). This map is well-defined since the e λi commute, and by construction it is a morphism of Lefschetz modules satisfying Ψ(1) = α d /d!. It suffices to show that Ψ vanishes on the ideal generated by the λ d+1 for λ ∈ H 2 (X, Q) satisfying b(λ, λ) = 0. Equivalently, it suffices to show that for every x ∈ Sym d H(X, Q) and for every λ (x) = 0. Without loss of generality, we may assume that x is a monomial of the form By degree reasons, we have e k λ (β j λ 1 · · · λ m ) = 0 for k > m. Moreover, it follows from b(λ, λ) = 0 that e k λ (α i ) = 0 for k > i. Combining these, one concludes that e d+1 λ (x) = 0, which is what we had to prove.
Dividing by (2) gives the claimed identity.
Consider the contraction (or Laplacian) operator given by This is a morphism of Lefschetz modules, or equivalently of so( H(X, Q))-modules.
Lemma 3.7. The sequence of Lefschetz modules Proof. Since ∆Ψ(1) = 0, we have ∆ • Ψ = 0. The map ∆ is well-known to be a surjective map of so( H(X, Q))-modules with irreducible kernel. Since Ψ is non-zero and SH(X, Q) is irreducible, it follows that the sequence is exact.
The Mukai pairing [14] on H ev (X, Q) restricts to a pairing b SH on SH(X, Q). It pairs elements of degree m with elements of degree 2d−m, according to the formula The pairing on H(X, Q) induces a pairing on Sym d H(X, Q) defined by ( H(X, Q))-invariant. The map Ψ is almost an isometry, in the following sense.
Proof. Both the Mukai form on SH(X, Q)[2d] and the pairing on Sym d H(X, Q) are so( H(X, Q))-invariant. Since SH(X, Q) is an irreducible so( H(X, Q))-module, it suffices to verify the identity for some x, y ∈ SH(X, which agrees with the identity claimed in the proposition. Remark 3.9. If X is of type K3 [d] then c X = (2d)!/(2 d d!) and Ψ is an isometry.

Action of derived equivalences on the Verbitsky component
In this section we prove Theorems B and C from the introduction.

4.1.
A representation-theoretical construction. Let K be a field of characteristic different from 2. Let V = (V, b) be a non-zero quadratic space over K. Let d be a positive integer and consider the space The Lie algebra so(V ) acts faithfully on It therefore suffices to show that the image of ψ equals the image of ψ • ϕ.
The adjoint group of so(V ) is SO(V ), and we have a short exact sequence Proposition 4.2. Let V 1 and V 2 be non-zero quadratic spaces. Assume that there is a linear map f : . Then there exists a µ ∈ K × and a similitude ϕ : Proof. LetK be a separable closure of K. After base change toK the quadratic spaces V 1 and V 2 become isometric, hence V 2 determines a class γ in the Galois cohomology group H 1 (Gal(K/K), O(VK). The existence of f shows that γ is mapped to the trivial element under the natural map By Proposition 4.1 (and Hilbert 90), this shows that γ is in the image of the map . But his means (V 1 , λb 1 ) and (V 2 , b 2 ) are isomorphic for any representative λ of a class in K × /(K × ) 2 mapping to γ.
In particular, there exists a similitude ϕ 0 : Proof. This follows immediately from Proposition 4.1.
Proof. Note that SH(X, Q) can be characterized as the minimal sub-g(X)-module of H(X, Q) whose Hodge structure attains the maximal possible level (width). It then follows from Theorem A and from Lemma 3.4 that Φ H restricts to an isomorphism respecting the Lie algebras g(X 1 ) and g(X 2 ). By [14], the map Φ H respects the Mukai pairings, and the theorem follows. Proof. This is clear from the proof of Proposition 3.5: the map Ψ is a morphism of Hodge structures, and so is the quotient map Sym • H(X, Q(1)) → SH(X, Q).
Proposition 4.7. Let X 1 and X 2 be derived equivalent hyperkähler varieties. Then there exists a Hodge similitude ϕ : H(X 1 , Q) ∼ → H(X 2 , Q) and a scalar λ ∈ K × so that the square commutes.
Proof. By Theorem 4.4 and Proposition 4.2 there exists a similitude ϕ and a scalar λ that make the square commute. It remains to check that ϕ respects the Hodge structures. The Hodge structure on H(X i , Q) is given by a morphism h i : C × → O(H(X i , R)), and the preceding lemma implies that the Hodge structure on SH(X i , Q) is given by composing h i with the injective map O(H(X i , R)) → GL(SH(X i , R)). Since ϕ maps the Hodge structure on SH(X 1 , Q) to the Hodge structure on SH(X 2 , Q), we conclude that ϕ maps h 1 to h 2 .
Theorem 4.8 (d odd). Assume that d is odd, and that X 1 and X 2 are deformationequivalent hyperkähler varieties of dimension 2d. Let Φ : DX 1 ∼ → DX 2 be an equivalence. Then there is a unique Hodge isometry Φ H making the square Proof. Since X 1 and X 2 are deformation equivalent, we can choose an isometry ϕ : H(X 1 , Q) ∼ → H(X 2 , Q). Moreover, X 1 and X 2 have the same Fujiki constant, so Sym d ϕ restricts to an isometry between the images of Ψ. Then by Theorem 4.4 and Proposition 4.3, there is a unique isometry ψ ∈ O( H(X 2 , Q)) such that Φ H := ψϕ makes the square commute. Uniqueness forces its formation to be functorial.
That Φ H respects the Hodge structures follows from the same argument as in the proof of Proposition 4.7.
If d is even, then the natural map O( H(X, Q)) → Aut(SH(X, Q), b X , g(X)) is neither injective, nor surjective, and the proof above fails. However, if we moreover assume that b 2 (X) is odd, then one can use the isomorphism SO ×{±1} ∼ = O to salvage the situation a bit.
Define an orientation on X to be the choice of a generator of det H 2 (X, R), up to R × >0 . Equivalently, an orientation is the choice of generator of det H(X, R) up to R × >0 . Define the sign ǫ(ϕ) of a Hodge isometry ϕ : H(X 1 , Q) ∼ → H(X 2 , Q) as ǫ(ϕ) = 1 if ϕ respects the orientations and ǫ(ϕ) = −1 otherwise. A derived equivalence between oriented hyperkähler varieties is a derived equivalence between the underlying unoriented hyperkähler varieties. Theorem 4.9 (d even). Assume that d is even, and that Φ : DX 1 ∼ → DX 2 is a derived equivalence between oriented hyperkähler varieties of dimension 2d. Assume that X 1 and X 2 have odd b 2 , and that the quadratic spaces H 2 (X 1 , Q) and H 2 (X 2 , Q) are isometric. Then there exists a unique Hodge isometry Φ H making the square commute. Morever, the formation of Φ H is functorial for composition of derived equivalences between hyperkähler varieties equipped with orientations.
Proof. This follows from Theorem 4.4 and Proposition 4.3 with essentially the same argument as the proof of Theorem 4.8.
Remark 4.10. If X 1 and X 2 are hyperkähler varieties belonging to one of the known families, and if Φ : DX 1 ∼ → DX 2 is an equivalence, then the hypotheses of either Theorem 4.8 or Theorem 4.9 are satisfied. Indeed, X 1 and X 2 will have the same dimension 2d and because they have isomorphic LLV Lie algebra, they have the same second Betti number b 2 . Going through the list of known families, one sees that this implies that X 1 and X 2 are deformation equivalent. In particular, they have isometric H 2 . Finally, all known hyperkähler varieties of dimension 2d with d even have odd b 2 .
Taking X 1 = X 2 in Theorem 4.8 and Theorem 4.9 yields Theorem C from the introduction:

Hodge structures
In this section we prove Theorem D from the introduction. For a rational quadratic space V we will make use of the algebraic groups SO(V ), Spin(V ), and GSpin(V ) over Spec Q. These groups sit in exact sequences We will write SO(V ), Spin(V ), and GSpin(V ) for the groups of Q-points of these algebraic groups.
Lemma 5.1. Let X be a hyperkähler variety. There exists a unique action of GSpin( H(X, Q)) on H(X, Q) such that (i) the action of Spin( H(X, Q)) ⊂ GSpin( H(X, Q)) integrates the action of g(X) = so( H(X, Q)) (ii) a section λ ∈ G m ⊂ GSpin( H(X, Q)) acts as λ i−2d on H i (X, Q).
Recall from Definition 4.5 that we have equipped H(X, Q) and H ev (X, Q) with Hodge structures of weight 0. Similarly, we equip the odd cohomology of X with a Hodge structure of weight 1 as follows Lemma 5.2. Let g ∈ GSpin ( H(X, Q)). If the action of g on H(X, Q) respects the Hodge structure, then so does its action on H ev (X, Q) and on H odd (X, Q).
Proof. This follows immediately from the fact that the Hodge structure is determined by the action of h ′ ∈ g(X) ⊗ Q C (see § 2.3), and from the faithfulness of the g(X)-module H(X, Q). Theorem 5.3. Let X 1 and X 2 be hyperkähler varieties, and let Φ : DX 1 ∼ → DX 2 be an equivalence. Then for every i the Q-Hodge structures H i (X 1 , Q) and H i (X 2 , Q) are isomorphic.
Proof. Consider the Lie algebra isomorphism Φ g : g(X 1 ) ∼ → g(X 2 ) from Theorem A. By Proposition 4.7, there exists a Hodge similitude φ : Here the vertical maps are the isomorphisms from Theorem 3.1. The K3-type Hodge structure H(X 2 , Q) decomposes as N ⊕ T , with N and T its algebraic and transcendental parts, respectively. The Hodge similitude φ maps the distinguished elements α 1 and β 1 of H(X 1 , Q) to N . By Witt cancellation, there exists a ψ N ∈ SO(N ) and λ, µ ∈ Q × such that ψ N φ(α 1 ) = λα 2 and ψ N φ(β 1 ) = µβ 2 . Extending by the identity, we find a Hodge isometry ψ ∈ SO( H(X 2 , Q)) such that ψφ : H(X 1 , Q) ∼ → H(X 2 , Q) is a graded Hodge similitude. In particular, the induced map ψφ : g(X 1 ) ∼ → g(X 2 ) is graded, and ψφ maps the grading element h 1 ∈ g(X 1 ) to the grading element h 2 ∈ g(X 2 ).
By the exact sequence (5) and by Hilbert 90, the element ψ lifts to an element ψ ∈ GSpin( H(X 2 , Q)). By Lemma 5.1 and Lemma 5.2, it induces automorphisms of the Hodge structures H ev (X 2 , Q) and H odd (X 2 , Q). Now by construction, the composition ψ • Φ H defines isomorphisms which respect both the grading and the Hodge structure, so that they induce isomorphisms of Hodge structures H i (X 1 , Q) ∼ → H i (X 2 , Q), for all i.

6.
Topological K-theory 6.1. Topological K-theory and the Mukai vector. In this paragraph we briefly recall some basic properties of topological K-theory of projective algebraic varieties. See [2,3,1] for more details.
For every smooth and projective X over C we have a Z/2Z-graded abelian group This is functorial for pull-back and proper pushforward, and carries a product structure. The group K 0 top (X) is the Grothendieck group of topological vector bundles on the differentiable manifold X an . Pull-back agrees with pull-back of vector bundles, and the product structure agrees with the tensor product of vector bundles. There is a natural Z/2Z-graded map v top X : K top (X) → H(X, Q), which in even degree is given by v top There is a 'forgetful map' K 0 (X) → K top (X) from the Grothendieck group of algebraic vector bundles (or equivalently of the triangulated category DX). This is compatible with pull back, multiplication, and proper pushfoward. The mukai vector v X : 6.2. Equivariant topological K-theory. The above formalism largely generalizes to an equivariant setting. Again we briefly recall the most important properties, see [42,4,5,29] for more details. If X is a smooth projective complex variety equipped with an action of a finite group G we denote by K 0 G (X) the Grothendieck group of G-equivariant algebraic vector bundles on X, or equivalently the Grothendieck group of the bounded derived category D G X of G-equivariant coherent O X -modules. This is functorial for pullback along G-equivariant maps and push-forward along G-equivariant proper maps.
Similarly, we have the G-equivariant topological K-theory where K 0 top,G (X) is the Grothendieck group of topological G-equivariant vector bundles.
There is a natural map K 0 G (X) → K 0 top,G (X) compatible with pull-back and tensor product. If f : X → Y is proper and G-equivariant, then we have a pushforward map f * : K top,G (X) → K top,G (Y ). There is a Riemann-Roch theorem [4,29], stating that the square commutes. Now assume that we have a finite group G acting on X, and a finite group H acting on Y . If P is an object in D G×H (X × Y ), then convolution with P induces a functor Φ P : D G X → D H Y , see [41] for more details. Similarly, convolution with the class of P in K 0 top,G×H (X × Y ) induces a map Φ K P : K top,G (X) → K top,H (Y ). These satisfy the usual Fourier-Mukai calculus, and moreover they are compatible, in the sense that the square

Cohomology of the Hilbert square of a K3 surface
Let S be a K3 surface and X = S [2] its Hilbert square. In the coming few paragraphs we recall the structure of the cohomology of X in terms of the cohomology of S. See [6,18,23] for more details.
with the property that c 1 (L) is mapped to c 1 (L 2 ), and δ is mapped to c 1 (E). We will use this isomorphism to identify H 2 (S, Z) ⊕ Zδ with H 2 (X, Z). The Beauville-Bogomolov form on H 2 (X, Z) satisfies Cup product defines an isomorphism Sym 2 H 2 (X, Q) ∼ → H 4 (X, Q). By Poincaré duality, there is a unique q X ∈ H 4 (X, Q) representing the Beauville-Bogomolov form, in the sense that for all λ 1 , λ 2 ∈ H 2 (X, Z). Multiplication by q X defines an isomorphism H 2 (X, Q) → H 6 (X, Q) and for all λ 1 , λ 2 , λ 3 in H 2 (X, Q) we have in H 6 (X, Q). Finally, for all λ ∈ H 2 (X, Q) the Fujiki relation 7.3. Todd class of the Hilbert square.
Proposition 7.1. Td X = 1 + 5 2 q X + 3[pt]. Proof. See also [23, §23.4]. Since the Todd class is invariant under the monodromy group of X, we necessarily have for some s, t ∈ Q. By Hirzebruch-Riemann-Roch, for every line bundle L on S with c 1 (L) = λ we have By the relations (9) and (7) Comparing the two expressions yields the result.

Derived McKay correspondence
8.1. The derived McKay correspondence. As in § 7.1, we consider a K3 surface S, its Hilbert square X = S [2] , the maps p : Z → S × S and q : Z → X, and the group G = {1, σ} acting on S × S and Z.
The derived McKay correspondence [11] is the triangulated functor BKR : given as the composition where the first functor maps F to q * F equipped with the trivial G-linearization. By [11, 1.1] and [21, 5.1] the functor BKR is an equivalence of categories. Its inverse is given by: The inverse equivalence of BKR is given by Proof. This follows from combining [33, 4.1] with the projection formula for q and the fact that q * E ∼ = O Z (−R). Now let S 1 and S 2 be K3 surfaces with Hilbert squares X 1 and X 2 . As was observed by Ploog [40], any equivalence Φ : is an isomorphism.
Proof. (See also [11, § 10]). This is a purely formal consequence of the calculus of equivariant Fourier-Mukai transforms sketched in § 6.2. The functor BKR and its inverse are given by kernels P ∈ D G (X × S × S) and Q ∈ D G (S × S × X). The map BKR top is given by convolution with the class of P in K 0 top,G (X × S × S). The identities in K 0 (X × X) and K 0 G×G (S × S × S × S) witnessing that P and Q are mutually inverse equivalences induce analogous identities in K 0 top . These show that convolution with the class of Q defines a two-sided inverse to BKR top .
Consider the map ψ K : K 0 top (X) → K 0 top (S × S) G obtained as the composition of BKR top and the forgetful map from K 0 top,G (S × S) to K 0 top (S × S). Also, consider the map ] denotes the class of the topological vector bundle F ⊠F equipped with ± the natural G-linearisation.
By construction, these maps are 'functorial' in DS, in the following sense: → DS 2 is a derived equivalence between K3 surfaces, and Φ [2] : DX 1 ∼ → DX 2 the induced equivalence between their Hilbert squares, then the squares Proposition 8.4. The sequence Proof. In the proof, we will implicitly identify K top,G (S × S) and K top (X).
Note that the map θ K is additive. Indeed, let F 1 and F 2 be (topological) vector bundles on S. Then the cross term , which vanishes because the matrices ( 0 1 1 0 ) and 0 −1 are conjugated over Z. The composition ψ K θ K vanishes, and since the Schur multiplier of G is trivial, the map ψ K is surjective. Computing the Q-dimensions one sees that it suffices to show that θ K is injective.
Pulling back to the diagonal and taking invariants defines a map . This composition computes to This coincides with the second Adams operation, which is injective on K 0 top (S)⊗ Z Q, since it has eigenvalues 1, 2, and 4. We conclude that θ K is injective, and the proposition follows.
Remark 8.5. One can show that the sequence A computation in the cohomology of the Hilbert square. This paragraph contains the technical heart of our computation of the derived monodromy of the Hilbert square of a K3 surface.
Proposition 8.6. The square Proof. Since K 0 top (S) ⊗ Z Q is additively generated by line bundles, it suffices to show (11) v top for a topological line bundle L with λ = c 1 (L). Deforming S if necessary, we may assume that L is algebraic. Using Proposition 8.1 and the fact that the natural map is an isomorphism of O X -modules we find We conclude that θ K maps L to [ We compute its image under v X . Using the formula for the Todd class from Proposition 7.1 we find v X (θ K (L)) = (1 + 5 4 q X + · · · ) exp(λ)(1 − e −δ ). Since 1 − e −δ has no term in degree 0, the degree 8 part of the square root of the Todd class is irrelevant, so we have v X (θ K (L)) = (1 + 5 4 q X ) exp(λ)(1 − e −δ ). By the Fujiki relation (9) from § 7.2, we have λ 3 δ = 0, so the above can be rewritten as v X (θ K (L)) = (1 + 5 4 q X ) · δ + λδ + λ 2 2 δ · 1 − e −δ δ .
Since we have q X δλ = b(δ, λ) = 0, we can rewrite this further as Comparing this with the right-hand-side of (11) we see that it suffices to show Q). This boils down to the identities in H 6 (X, Q) and H 8 (X, Q) respectively. These follow easily from the relations (7), (8), and (9) in § 7.2.
9. Derived monodromy group of the Hilbert square of a K3 surface 9.1. Derived monodromy groups. Let X be a smooth projective complex variety. We call a deformation of X the data of a smooth projective variety X ′ , a proper smooth family X → B , a path γ : [0, 1] → X and isomorphisms X ∼ → X γ(0) and X ′ ∼ → X γ(1) . We will informally say that X ′ as a deformation of X, the other data being implicitly understood. Parallel transport along γ defines an isomorphism H(X, Q) ∼ → H(X ′ , Q). If X ′ and X ′′ are deformations of X, and if φ : X ′ → X ′′ is an isomorphism of projective varieties, then we obtain a composite isomorphism We call such isomorphism a monodromy operator for X, and denote by Mon(X) the subgroup of GL(H(X, Q)) generated by all monodromy operators.
If X ′ and X ′′ are deformations of X, and if Φ : DX ′ ∼ → DX ′′ is an equivalence, then we obtain an isomorphism We call such isomorphism a derived monodromy operator for X, and denote by DMon(X) the subgroup of GL(H(X, Q)) generated by all derived monodromy operators.
By construction, the derived monodromy group is deformation invariant. It contains the usual monodromy group, and the image of ρ X , and we have a commutative square of groups Aut(X) Aut(DX) Mon(X) DMon(X). ρX Remark 9.1. The above definition is somewhat ad hoc, and should be considered a poor man's derived monodromy group. This is sufficient for our purposes. A more mature definition should involve all non-commutative deformations of X. Proof. Indeed, if Φ : DS 1 → DS 2 is an equivalence, then preservers the Mukai form, as well as a natural orientation on four-dimensional positive subspaces (see [26, § 4.5]). Also any deformation preserves the Mukai form and the natural orientation, so any derived monodromy operator will land in O + ( H(S, Z)).
The converse inclusion can be easily obtained from the Torelli theorem, together with the results of [40,22] on derived auto-equivalences of K3 surfaces. Alternatively one can use that the group O + ( H(S, Z)) is generated by relfections in −2 vectors δ. By the Torelli theorem, any such −2-vector will become algebraic on a suitable deformation S ′ of S, and by [32] there exists a spherical object E on S ′ with Mukai vector v(E) = δ. The spherical twist in E then shows that reflection in δ is indeed a derived monodromy operator. 9.2. Action of DMon(S) on H(X, Q). By the derived McKay correspondence, any derived equivalence Φ S : DS 1 ∼ → DS 2 between K3 surfaces induces a derived equivalence Φ X : DX 1 ∼ → DX 2 between the corresponding Hilbert squares. By Propositions 8.3 and 8.4, the induced map Φ H X only depends on Φ H S . Since any deformation of a K3 surface S induces a deformation of X = S [2] , we conclude that we have a natural homomorphism DMon(S) −→ DMon(X), and hence an action of DMon(S) on H(X, Q). In this paragraph, we will explicitly compute this action. As a first approximation, we determine the DMon(S)-module structure of H(X, Q), up to isomorphism. Proof. This follows from Propositions 8.3 and 8.4.
Since g(X) is a purely topological invariant, it is preserved under deformations. In particular, Theorem 4.11 implies that we have an inclusion DMon(X) ⊂ O( H(X, Q)). We conclude there exists a unique map of algebraic groups h making the square Theorem 9.4. The map h in the square (12) is given by The proof of this theorem will occupy the remainder of this paragraph.
Consider the unique homomorphism of Lie algebras ι : g(S) → g(X) that respects the grading and maps e λ to e λ for all λ ∈ H 2 (S, Q) ⊂ H 2 (X, Q). Under the isomorphism of Theorem 3.1 this corresponds to the map so( H(S, Q)) → so( H(X, Q)) induced by the inclusion of quadratic spaces H(S, Q) ⊂ H(X, Q).
Recall from Section 8.3 the map θ H : H(S, Q) → H(X, Q).
We are now ready to prove the main result of this paragraph.
Proof of Theorem 9.4. By Proposition 8.6 the map θ H is equivariant for the action of DMon(S). Lemma 9.5 then implies that for all g ∈ SO( H(S, Q)). We have an orthogonal decomposition H(X, Q) = B −δ/2 ( H(S, Q)) ⊕ C with C of rank 1. Since SO( H(S, Q)) is normal in O( H(S, Q)), the action of O( H(S, Q)) (via h) must preserve this decomposition. With respect to this decomposition h must then be given by where the ǫ i (g) : O( H(S, Q)) → {±1} are quadratic characters. This leaves four possibilities for h. One verifies that ǫ 1 = ǫ 2 = det g is the only possibility compatible with Proposition 9.3 and Lemma 9.6, and the theorem follows.
9.3. A transitivity lemma. In this section we prove a lattice-theoretical lemma that will play an important role in the proofs of Theorem E and Theorem F. Let b : L × L → Z be an even non-degenerate lattice. Let U be a hyperbolic plane with basis consisting of isotropic vectors α, β satisfying b(α, β) = −1.
Lemma 9.7. Let L be an even lattice containing a hyperbolic plane. Let G ⊂ O(U ⊕ L) be the subgroup generated by γ and by the B λ for all λ ∈ L. Then for all δ ∈ U ⊕ L with δ 2 = −2 and for all g ∈ O(U ⊕ L) there exists a g ′ ∈ G such that g ′ g fixes δ.
Hence G contains the subgroup E U (L) ⊂ O(U ⊕ L) of unimodular transvections with respect to U . By [20,Prop. 3.3], there exists a g ′ ∈ E U (L) mapping gδ to δ.
9.4. Proof of Theorem E. Let X be a hyperkähler variety of type K3 [2] . Let δ ∈ H 2 (X, Z) be any class satisfying δ 2 = −2 and b(δ, λ) ∈ 2Z for all λ ∈ H 2 (X, Z). For example, if X = S [2] , we may take δ = c 1 (E) as in § 7.2. Consider the integral lattice The sub-group Λ ⊂ H(X, Q) does not depend on the choice of δ. In this section, we will prove Theorem E. More precisely, we will show: We start with the lower bound. Proof. Since the derived monodromy group is invariant under deformation, we may assume without loss of generality that X = S [2] for a K3 surface S and δ = c 1 (E) as in § 7.2. The shift functor [1] on DX acts as −1 on H(X, Q), which coincides with the action of −1 ∈ O( H(X, Q)). In particular, −1 ∈ O + (Λ) lies in DMon(X), so it suffices to show that SO + (Λ) is contained in DMon(X).
Let G ⊂ O( H(X, Q)) be the subgroup generated by h(γ) and the isometries B λ for λ ∈ H 2 (X, Z). Clearly G is contained in DMon(X).
Let g be an element of SO + (Λ), and consider the image gB δ/2 δ of B δ/2 δ. By Lemma 9.7 there exists a g ′ ∈ G ⊂ DMon(X) so that g ′ g fixes B δ/2 δ. But then g ′ g acts on (B δ/2 δ) ⊥ = B δ/2 Zα ⊕ H 2 (S, Z) ⊕ Zβ with determinant 1, and preserving the orientation of a maximal positive subspace. In particular, g ′ g lies in the image of DMon(S) → DMon(X), and we conclude that g lies in DMon(X).
The proof of the upper bound is now almost purely group-theoretical. Proof. More generally, this holds for any even lattice Λ with the property that the quadratic form q(x) = b(x, x)/2 on the Z-module Λ is semi-regular [30,§ IV.3]. For such Λ, the group schemes Spin(Λ) and SO(Λ) are smooth over Spec Z, see e.g. [28]. In particular, for every prime p the subgroups Spin(Λ ⊗ Z p ) and SO(Λ ⊗ Z p ) of Spin(Λ ⊗ Q p ) resp. SO(Λ ⊗ Q p ) are maximal compact subgroups. It follows that the groups Spin(Λ) = Spin(Λ ⊗ Q) Let Γ ⊂ SO(Λ ⊗ Q) be a maximal arithmetic subgroup containing SO + (Λ). Let Γ be its inverse image in Spin(Λ ⊗ Q), so that we have an exact sequence Since the group Γ is arithmetic and contains Spin(Λ), we have Γ = Spin(Λ). Moreover, Γ normalizes SO + (Λ) = ker(Γ → Q × /2), and as the normalizer of an arithmetic subgroup of SO(Λ⊗Q) is again arithmetic, we have that Γ must equal the normalizer of SO + (Λ). But then Γ contains SO(Λ), and we conclude Γ = SO(Λ). Proof. DMon(X) preserves the integral lattice K top (X) in the representation H(X, Q) of O( H(X, Q)), and hence is contained in an arithmetic subgroup of O( H(X, Q)) = SO( H(X, Q))×{±1}. By Proposition 9.9 it contains SO + (Λ)×{±1}, so we conclude from the preceding proposition that DMon(X) must be contained in O(Λ).
Together with Proposition 9.9 this proves Theorem 9.8. 10. The image of Aut(DX) on H(X, Q) 10.1. Upper bound for the image of ρ X . We continue with the notation of the previous section. In particular, we denote by X a hyperkähler variety of type K3 [2] , and by Λ ⊂ H(X, Q) the lattice defined in § 9.4. We equip H(X, Q) with the weight 0 Hodge structure H(X, Q) = Qα ⊕ H 2 (X, Q(1)) ⊕ Qβ.
We denote by Aut(Λ) ⊂ O(Λ) the group of isometries of Λ that preserve this Hodge structure.
Theorem 10.2. Let S be a K3 surface and let X be the Hilbert square of S. Assume that NS(X) contains a hyperbolic plane. Then Aut + (Λ) ⊂ im ρ X ⊂ Aut(Λ).
Proof. In view of Proposition 10.1 we only need to show the lower bound. The argument for this is entirely parallel to the proof of Proposition 9.9. Recall that we have Λ = B δ/2 Zα ⊕ H 2 (S, Z((1)) ⊕ Zδ ⊕ Zβ .
Denote by G ⊂ Aut(Λ) the subgroup generated by ρ X (γ X ) and the isometries B λ = ρ X (− ⊗ L) with L a line bundle of class λ ∈ NS(X). Clearly G is contained in the image of ρ X . Note that G acts on the lattice Λ alg := B δ/2 Zα ⊕ NS(X) ⊕ Zβ and that by our assumption NS(X) contains a hyperbolic plane.
Let g ∈ Aut + (Λ). By Lemma 9.7 applied to L = NS(X), there exists a g ′ ∈ G such that g ′ g fixes B δ/2 δ. But then g ′ g acts on (B δ/2 δ) ⊥ = B δ/2 Zα ⊕ H 2 (S, Z) ⊕ Zβ with determinant 1, and preserving the Hodge structure and the orientation of a maximal positive subspace. In particular, g ′ g lies in the image of Aut(DS), and we conclude that g lies in im ρ X .