Derived equivalences of hyperk\"ahler varieties

TL;DR

This paper proves that certain algebraic structures acting on the cohomology of hyperk"ahler varieties are preserved under derived equivalences, leading to new invariants and insights into their cohomological behavior.

Contribution

It establishes the invariance of the Looijenga--Lunts--Verbitsky Lie algebra under derived equivalences and derives several consequences for the cohomology of hyperk"ahler varieties.

Findings

Looijenga--Lunts--Verbitsky Lie algebra is a derived invariant.

Derived equivalent hyperk"ahler varieties have isomorphic $ extbf{Q}$-Hodge structures.

Constructed a rational 'Mukai lattice' functorial for derived equivalences.

Abstract

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.

Equations

$${\Phi }^{\mathrm{H}}:\mathrm{H}(X_1,\mathbf{Q})\stackrel{\sim }{\to }\mathrm{H}(X_2,\mathbf{Q})$$

$${\rho }_X:\mathrm{Aut}(\mathcal{D}X)\to \mathrm{GL}(\mathrm{H}(X,\mathbf{Q})),\Phi \mapsto {\Phi }^{\mathrm{H}}.$$

$$\tilde{\mathrm{H}}(X,\mathbf{Z}):={\mathrm{H}}^0(X,\mathbf{Z})\oplus {\mathrm{H}}^2(X,\mathbf{Z}(1))\oplus {\mathrm{H}}^4(X,\mathbf{Z}(2)).$$

$${\Phi }^{\mathfrak{g}}:\mathfrak{g}(X_1)\stackrel{\sim }{\to }\mathfrak{g}(X_2)$$

$$\tilde{\mathrm{H}}(X,\mathbf{Q}):=\mathbf{Q}\alpha \oplus {\mathrm{H}}^2(X,\mathbf{Q})\oplus \mathbf{Q}\beta$$

$${\rho }_X^{\mathrm{SH}}:\mathrm{Aut}(\mathcal{D}X)⟶\mathrm{GL}(\mathrm{SH}(X,\mathbf{Q})).$$

$$\tilde{\mathrm{H}}(X,\mathbf{Q})=\mathbf{Q}\alpha \oplus {\mathrm{H}}^2(X,\mathbf{Q}(1))\oplus \mathbf{Q}\beta .$$

$$\mathrm{H}(X,\mathbf{Q})⟶\mathrm{H}(X^{\prime },\mathbf{Q})\stackrel{{\Phi }^{\mathrm{H}}}{⟶}\mathrm{H}(X^{\prime \prime },\mathbf{Q})⟶\mathrm{H}(X,\mathbf{Q}).$$

$$O^{+}(\Lambda )\subset \mathrm{DMon}(X)\subset O(\Lambda )$$

$$[h,e]=2e,[h,f]=-2f,[e,f]=h$$

$$(\mathrm{ad}e{)}^2:\mathrm{End}(M{)}_{-2}\stackrel{\sim }{\to }\mathrm{End}(M{)}_2$$

$$(\mathrm{ad}e{)}^2:{\mathfrak{g}}_{-2}\hookrightarrow {\mathfrak{g}}_2.$$

$${\mathrm{HH}}^n(X):={\mathrm{Ext}}_{X\times X}^n({\Delta }_{\ast }{\mathcal{O}}_X,{\Delta }_{\ast }{\mathcal{O}}_X)$$

$${\mathrm{HH}}_n(X):={\mathrm{Ext}}_{X\times X}^{d-n}({\Delta }_{\ast }{\mathcal{O}}_X,{\Delta }_{\ast }{\omega }_X).$$

$$I^n:{\mathrm{HH}}^n(X)\stackrel{\sim }{⟶}\underset{i+j=n}{⨁}{\mathrm{H}}^i(X,{\wedge }^j{T}_X)$$

$$I_n:{\mathrm{HH}}_n(X)\stackrel{\sim }{⟶}\underset{j-i=n}{⨁}{\mathrm{H}}^i(X,{\Omega }_X^j).$$

$${\mathrm{HH}}_{\bullet }(X)\stackrel{\sim }{⟶}\mathrm{H}(X,\mathbf{C}).$$

$${\Phi }^{{\mathrm{HH}}^{\bullet }}:{\mathrm{HH}}^{\bullet }(X_1)\stackrel{\sim }{⟶}{\mathrm{HH}}^{\bullet }(X_2),{\Phi }^{{\mathrm{HH}}_{\bullet }}:{\mathrm{HH}}_{\bullet }(X_1)\stackrel{\sim }{⟶}{\mathrm{HH}}_{\bullet }(X_2),$$

$${\mathrm{HH}}_{\bullet }(X_1)$$

$${\mathrm{HH}}^{\bullet }(X)\stackrel{\sim }{⟶}{\mathrm{H}}^{\bullet }(X,{\wedge }^{\bullet }{T}_X)\stackrel{\sim }{⟶}{\mathrm{H}}^{\bullet }(X,{\Omega }_X^{\bullet })\stackrel{\sim }{⟶}{\mathrm{H}}^{\bullet }(X,\mathbf{C}).$$

$$h_p=p-d,h_q=q-d,\text{ on }{\mathrm{H}}^{p,q}.$$

$$e_{\mu }^{\prime }:\mathrm{H}(X,\mathbf{C})\to \mathrm{H}(X,\mathbf{C})$$

$${\Phi }^{{\mathfrak{g}}^{\prime }}:{\mathfrak{g}}^{\prime }(X_1)\stackrel{\sim }{\to }{\mathfrak{g}}^{\prime }(X_2).$$

$${\mathrm{H}}^2(X,\mathbf{C})={\mathrm{H}}^2({\mathcal{O}}_X)\oplus {\mathrm{H}}^1({\Omega }_X^1)\oplus {\mathrm{H}}^0({\Omega }_X^2)$$

$${\mathrm{HH}}^2(X)={\mathrm{H}}^2({\mathcal{O}}_X)\oplus {\mathrm{H}}^1(T_X)\oplus {\mathrm{H}}^0({\wedge }^2{T}_X).$$

$$\sigma =\mathrm{d}{u}_1\wedge \mathrm{d}{v}_1+\cdots +\mathrm{d}{u}_d\wedge \mathrm{d}{v}_d$$

$$\check{\sigma }=\frac{\partial }{\partial u_1}\wedge \frac{\partial }{\partial v_1}+\dots +\frac{\partial }{\partial u_d}\wedge \frac{\partial }{\partial v_d}.$$

$$(\alpha +\sigma ,h,\check{\alpha }+\check{\sigma })\text{and}(\alpha -\sigma ,h,\check{\alpha }-\check{\sigma })$$

$$b=b_X:{\mathrm{H}}^2(X,\mathbf{Q})\times {\mathrm{H}}^2(X,\mathbf{Q})\to \mathbf{Q}$$

$${\int }_X{\lambda }^{2d}=\frac{(2d)!}{2^{d}d!}{c}_{X}b(\lambda ,\lambda {)}^d$$

Full text

This includes a proof that derived equivalent hyperkähler 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 ${\mathcal{D}}X$ the bounded derived category of coherent sheaves on $X$. By a theorem of Orlov [38] any (exact, ${\mathbf{C}}$-linear) equivalence $\Phi\colon{\mathcal{D}}X_{1}\overset{\sim}{\to}{\mathcal{D}}X_{2}$ comes from a Fourier–Mukai kernel ${\mathcal{P}}\in{\mathcal{D}}(X_{1}\times X_{2})$, and convolution with the Mukai vector $v({\mathcal{P}})\in{\rm H}(X_{1}\times X_{2},{\mathbf{Q}})$ defines an isomorphism

[TABLE]

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 ${\rm H}^{i}(X_{1},{\mathbf{Q}})\cong{\rm H}^{i}(X_{2},{\mathbf{Q}})$ of ${\mathbf{Q}}$-Hodge structures.

For every $X$ we have a representation

[TABLE]

Its image is known for varieties with ample or anti-ample canonical class (in which case $\operatorname{Aut}({\mathcal{D}}X)$ 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

[TABLE]

This is a Hodge structure of weight [math], and it comes equipped with a perfect bilinear form $b$ of signature $(4,20)$. For convenience, we denote by $\alpha$ and $\beta$ the natural generators of ${\rm H}^{0}(X,{\mathbf{Z}})$ and ${\rm H}^{4}(X,{\mathbf{Z}}(2))$ respectively, so that we have $\widetilde{\rm H}(X,{\mathbf{Z}})={\mathbf{Z}}\alpha\oplus{\rm H}^{2}(X,{\mathbf{Z}}(1))\oplus{\mathbf{Z}}\beta$. The pairing $b$ is the orthogonal sum of the intersection pairing on ${\rm H}^{2}(X,{\mathbf{Z}}(1))$ and the pairing on ${\mathbf{Z}}\alpha\oplus{\mathbf{Z}}\beta$ given by $b(\alpha,\alpha)=b(\beta,\beta)=0$ and $b(\alpha,\beta)=-1$.

It was observed by Mukai [36] that if $\Phi\colon{\mathcal{D}}X_{1}\overset{\sim}{\to}{\mathcal{D}}X_{2}$ is a derived equivalence between K3 surfaces, then $\Phi^{\rm H}$ restricts to an isomorphism $\Phi^{\widetilde{\rm H}}\colon\widetilde{\rm H}(X_{1},{\mathbf{Z}})\to\widetilde{\rm H}(X_{2},{\mathbf{Z}})$ respecting the pairing and Hodge structures. Denote by $\operatorname{Aut}(\widetilde{\rm H}(X,{\mathbf{Z}}))$ the group of isometries of $\widetilde{\rm H}(X,{\mathbf{Z}})$ respecting the Hodge structure, and by $\operatorname{Aut}^{+}(\widetilde{\rm H}(X,{\mathbf{Z}}))$ the subgroup (of index $2$) consisting of those isometries that respect the orientation on a four-dimensional positive definite subspace of $\widetilde{\rm H}(X,{\mathbf{R}})$.

Theorem 1.1** ([36, 37, 22, 40, 26]).**

Let $X$ be a K3 surface. Then the image of $\rho_{X}$ is $\operatorname{Aut}^{+}(\widetilde{\rm H}(X,{\mathbf{Z}}))$.∎

In this paper, we prove Orlov’s conjecture on ${\mathbf{Q}}$-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 $\rho_{X}$ 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 $\lambda\in{\rm NS}(X)$ determines a Lie algebra ${\mathfrak{g}}_{\lambda}\subset\operatorname{End}({\rm H}(X,{\mathbf{Q}}))$ isomorphic to $\operatorname{\mathfrak{sl}}_{2}$. More generally, this holds for every cohomology class $\lambda\in{\rm H}^{2}(X,{\mathbf{Q}})$ (algebraic or not) satisfying the conclusion of the Hard Lefschetz theorem. Looijenga and Lunts [34], and Verbitsky [46] have studied the Lie algebra ${\mathfrak{g}}(X)\subset\operatorname{End}({\rm H}(X,{\mathbf{Q}}))$ generated by the collection of the Lie algebras ${\mathfrak{g}}_{\lambda}$. 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 $\sigma\in{\rm H}^{0}(X,\Omega^{2}_{X})$.

Theorem A** (§ 2.4).**

Let $X_{1}$ and $X_{2}$ be holomorphic symplectic varieties. Then for every equivalence $\Phi\colon{\mathcal{D}}X_{1}\overset{\sim}{\to}{\mathcal{D}}X_{2}$ there exists a canonical isomorphism of rational Lie algebras

[TABLE]

with the property that the map $\Phi^{\rm H}\colon{\rm H}(X_{1},{\mathbf{Q}})\overset{\sim}{\to}{\rm H}(X_{2},{\mathbf{Q}})$ is equivariant with respect to $\Phi^{\mathfrak{g}}$.

Note that ${\mathfrak{g}}(X)$ is defined in terms of the grading and the cup product on ${\rm H}(X,{\mathbf{Q}})$, neither of which are preserved under derived equivalences.

To prove Theorem A we introduce a complex Lie algebra ${\mathfrak{g}}^{\prime}(X)$ whose definition is similar to the rational Lie algebra ${\mathfrak{g}}(X)$, but where the action of ${\rm H}^{2}(X,{\mathbf{Q}})$ on ${\rm H}(X,{\mathbf{Q}})$ is replaced with a natural action of the Hochschild cohomology group ${\rm HH}^{2}(X)$ on Hochschild homology ${\rm HH}_{\bullet}(X)$. Since Hochschild cohomology and its action on Hochschild homology is known to be invariant under derived equivalences, it follows that ${\mathfrak{g}}^{\prime}(X)$ is a derived invariant. We show that if $X$ is holomorphic symplectic, then the isomorphism ${\rm HH}_{\bullet}(X)\to{\rm H}(X,{\mathbf{C}})$ (coming from the Hochschild–Kostant–Rosenberg isomorphism) maps ${\mathfrak{g}}^{\prime}(X)$ to ${\mathfrak{g}}(X)\otimes_{\mathbf{Q}}{\mathbf{C}}$. This is closely related to Verbitsky’s ‘mirror symmetry’ for hyperkähler varieties [46, 47]. From this we deduce that the rational Lie algebra ${\mathfrak{g}}(X)$ is a derived invariant.

1.3. A rational Mukai lattice for hyperkähler varieties

Let $X$ be a hyperkähler variety. Consider the ${\mathbf{Q}}$-vector space

[TABLE]

equipped with the bilinear form $b$ which is the orthogonal sum of the Beauville–Bogomolov form on ${\rm H}^{2}(X,{\mathbf{Q}})$ and a hyperbolic plane ${\mathbf{Q}}\alpha\oplus{\mathbf{Q}}\beta$ with $\alpha$ and $\beta$ isotropic and $b(\alpha,\beta)=-1$. By analogy with the case of a K3 surface, we will call $\widetilde{\rm H}(X,{\mathbf{Q}})$ the (rational) Mukai lattice of $X$. Looijenga–Lunts [34] and Verbitsky [46] have shown that the Lie algebra ${\mathfrak{g}}(X)$ can be canonically identified with $\operatorname{\mathfrak{so}}(\widetilde{\rm H}(X,{\mathbf{Q}}))$, see § 3.1 for a precise statement. Moreover, Verbitsky [46] has shown that the sub-algebra ${\mathrm{SH}}(X,{\mathbf{Q}})$ of ${\rm H}(X,{\mathbf{Q}})$ generated by ${\rm H}^{2}(X,{\mathbf{Q}})$ forms an irreducible sub-${\mathfrak{g}}(X)$-module. Using this, we show that Theorem A implies:

Theorem B** (§ 4.2).**

Let $X_{1}$ and $X_{2}$ be hyperkähler varieties and $\Phi\colon{\mathcal{D}}X_{1}\overset{\sim}{\to}{\mathcal{D}}X_{2}$ an equivalence. Then the induced isomorphism $\Phi^{\rm H}$ restricts to an isomorphism $\Phi^{{\mathrm{SH}}}\colon{\mathrm{SH}}(X_{1},{\mathbf{Q}})\overset{\sim}{\to}{\mathrm{SH}}(X_{2},{\mathbf{Q}}).$

Taking $X_{1}=X_{2}=X$ in Theorem B we obtain a homomorphism

[TABLE]

The complex structure on a hyperkähler variety $X$ induces a Hodge structure of weight [math] on $\widetilde{\rm H}(X,{\mathbf{Q}})$ given by

[TABLE]

Denote by $\operatorname{Aut}\widetilde{\rm H}(X,{\mathbf{Q}})$ the group of Hodge isometries of $\widetilde{\rm H}(X,{\mathbf{Q}})$.

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 $\rho_{X}^{\mathrm{SH}}$ factors over a map $\rho_{X}^{\widetilde{\rm H}}\colon\operatorname{Aut}({\mathcal{D}}(X))\to\operatorname{Aut}(\widetilde{\rm H}(X,{\mathbf{Q}}))$.

See § 3.2 and § 4.2 for an explicit description of the implicit map $\operatorname{Aut}(\widetilde{\rm H}(X,{\mathbf{Q}}))\to\operatorname{GL}({\mathrm{SH}}(X,{\mathbf{Q}}))$.

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 ${\mathbf{Q}}$-Hodge structures ${\rm H}^{i}(X_{1},{\mathbf{Q}})$ and ${\rm H}^{i}(X_{2},{\mathbf{Q}})$ 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 $\rho_{X}$ 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 $\rho_{X}$, we consider a variation of this problem which is deformation invariant. Let $X$ be a smooth projective complex variety. If $X^{\prime}$ and $X^{\prime\prime}$ are smooth deformations of $X$ (parametrized by paths in the base), and if $\Phi\colon{\mathcal{D}}X^{\prime}\overset{\sim}{\to}{\mathcal{D}}X^{\prime\prime}$ is an equivalence, then we obtain an isomorphism as the composition

[TABLE]

We define the derived monodromy group of $X$ to be the subgroup $\operatorname{DMon}(X)$ of $\operatorname{GL}({\rm H}(X,{\mathbf{Q}}))$ generated by all these isomorphisms. This group contains both the usual monodromy group of $X$ and the image of $\rho_{X}\colon\operatorname{Aut}({\mathcal{D}}X)\to\operatorname{GL}({\rm H}(X,{\mathbf{Q}}))$.

If $S$ is a K3 surface, then the result of [26] implies $\operatorname{DMon}(S)=\operatorname{O}^{+}(\widetilde{\rm H}(S,{\mathbf{Z}}))$, and that the image of $\rho_{S}$ consists of those elements of $\operatorname{DMon}(S)$ that respect the Hodge structure on $\widetilde{\rm H}(S,{\mathbf{Z}})$. Similarly, for an abelian variety $A$, the results of [19] imply $\operatorname{DMon}(A)=\operatorname{Spin}({\rm H}^{1}(A,{\mathbf{Z}})\oplus{\rm H}^{1}(A^{\vee},{\mathbf{Z}}))$, and that the image of $\rho_{A}$ consists of those elements of $\operatorname{DMon}(A)$ that respect the Hodge structure on ${\rm H}^{1}(A,{\mathbf{Z}})\oplus{\rm H}^{1}(A^{\vee},{\mathbf{Z}})$.

Now let $X$ be a hyperkähler variety of type ${\mathrm{K3}}^{[2]}$. We have ${\rm H}(X,{\mathbf{Q}})={\mathrm{SH}}(X,{\mathbf{Q}})$ and hence by Theorem C the action of $\operatorname{Aut}({\mathcal{D}}X)$ on ${\rm H}(X,{\mathbf{Q}})$ factors over a subgroup $\operatorname{O}(\widetilde{\rm H}(X,{\mathbf{Q}}))$ of $\operatorname{GL}({\rm H}(X,{\mathbf{Q}}))$.

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 $\Lambda\subset\widetilde{\rm H}(X,{\mathbf{Q}})$ such that

[TABLE]

inside $\operatorname{O}(\widetilde{\rm H}(X,{\mathbf{Q}}))$.

See § 9.4 for a precise description of $\Lambda$. As an abstract lattice, $\Lambda$ is isomorphic to ${\rm H}^{2}(X,{\mathbf{Z}})\oplus U$, but its image in $\widetilde{\rm H}(X,{\mathbf{Q}})$ is not ${\mathbf{Z}}\alpha\oplus{\rm H}^{2}(X,{\mathbf{Z}})\oplus{\mathbf{Z}}\beta$.

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 $\operatorname{DMon}(X)$: every deformation of $X$ to the Hilbert square $S^{[2]}$ of a K3 surface $S$ induces an inclusion $\operatorname{DMon}(S)\to\operatorname{DMon}(X)$. As part of the proof, we explicitly compute this inclusion.

We denote by $\operatorname{Aut}(\Lambda)$ the group of isometries of $\Lambda\subset\widetilde{\rm H}(X,{\mathbf{Q}})$ that respect the Hodge structure on $\widetilde{\rm H}(X,{\mathbf{Q}})$. It follows from Theorem E that $\operatorname{im}(\rho_{X})$ is contained in $\operatorname{Aut}(\Lambda)$ 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 $\operatorname{Aut}^{+}(\Lambda)\subset\operatorname{Aut}(\Lambda)$ the subgroup consisting of those Hodge isometries that respect the orientation of a positive $4$-plane in $\Lambda_{\mathbf{R}}$.

Theorem F** (§ 10.2).**

Let $S$ be a complex K3 surface and $X=S^{[2]}$. Assume that ${\rm NS}(X)$ contains a hyperbolic plane. Then we have $\operatorname{Aut}^{+}(\Lambda)\subset\operatorname{im}(\rho_{X})\subset\operatorname{Aut}(\Lambda)$.

Remark 1.2**.**

To determine $\operatorname{im}\rho_{X}$ up to index $2$ for a general hyperkähler of type $\mathrm{K3}^{[2]}$ 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 $\widetilde{\rm H}(X,{\mathbf{R}})$. 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 [math], 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 $\mathrm{K3}^{[2]}$.

Remark 1.4**.**

That a lattice of signature $(4,b_{2}-2)$ should play a role in describing the image of $\rho_{X}$ 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 $\Lambda$ 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 $\operatorname{Aut}({\mathcal{D}}X)$ 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 ${\rm H}(X,{\mathbf{C}})$, yet Theorems E and F suggest the central charge should take values in $\widetilde{\rm H}(X,{\mathbf{C}})$. If $X$ is of type $\mathrm{K3}^{[2]}$, then ${\rm H}(X,{\mathbf{C}})$ and $\widetilde{\rm H}(X,{\mathbf{C}})$ are non-isomorphic irreducible $\operatorname{DMon}(X)$-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.

2. 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 ${\mathbf{Z}}$-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 $n\geq 0$ the map $e^{n}\colon M_{-n}\to M_{n}$ is an isomorphism. This is equivalent to the existence of an $f\in\operatorname{End}(M)$ such that the relations

[TABLE]

hold in $\operatorname{End}(M)$. Thus, $(e,h,f)$ forms an $\operatorname{\mathfrak{sl}}_{2}$-tripe and defines a Lie homomorphism $\operatorname{\mathfrak{sl}}_{2}\to\operatorname{End}(M)$.

Proposition 2.1**.**

Assume that $e$ has the hard Lefschetz property. Then the element $f$ satisfying (1) is unique, and if $e$ and $h$ lie in a semi-simple sub-Lie algebra ${\mathfrak{g}}\subset\operatorname{End}(M)$, then so does $f$.

Proof of Proposition 2.1.

The action of $\operatorname{ad}e$ on $\operatorname{End}(M)$ has the hard Lefschetz property for the grading defined by $\operatorname{ad}h$. In particular

[TABLE]

is an isomorphism. It sends $f$ to $-2e$, so that $f$ is indeed uniquely determined.

If $e$ and $h$ lie in ${\mathfrak{g}}$, then ${\mathfrak{g}}\subset\operatorname{End}(M)$ is graded and the above map restricts to an injective map

[TABLE]

Since $h$ is diagonisable, it is contained in a Cartan sub-algebra of ${\mathfrak{g}}$. The symmetry of the resulting root system implies that $\dim{\mathfrak{g}}_{-n}=\dim{\mathfrak{g}}_{n}$ for all $n$. In particular, the map $(\operatorname{ad}e)^{2}$ defines an isomorphism between ${\mathfrak{g}}_{-2}$ and ${\mathfrak{g}}_{2}$, and we conclude that $f$ lies in ${\mathfrak{g}}$. ∎

Let ${\mathfrak{a}}$ be an abelian Lie algebra and $e\colon{\mathfrak{a}}\to\operatorname{\mathfrak{gl}}(M),\,a\mapsto e_{a}$ a Lie homomorphism. We say that $e$ has the hard Lefschetz property if $e({\mathfrak{a}})\subset\operatorname{\mathfrak{gl}}(M)_{2}$ and if there exists some $a\in{\mathfrak{a}}$ so that $e_{a}$ has the hard Lefschetz property. Note that this is a Zariski open condition on $a\in{\mathfrak{a}}$.

If $e\colon{\mathfrak{a}}\to\operatorname{\mathfrak{gl}}(M)$ has the hard Lefschetz property, then we denote by ${\mathfrak{g}}({\mathfrak{a}},M)$ the Lie algebra generated by the $\operatorname{\mathfrak{sl}}_{2}$-triples $(e_{a},h,f_{a})$ for $a\in{\mathfrak{a}}$ such that $e_{a}$ has the hard Lefschetz property. We say that $({\mathfrak{a}},M)$ is a Lefschetz module if ${\mathfrak{g}}({\mathfrak{a}},M)$ is semisimple.

Now let $X$ be a smooth projective complex variety of dimension $d$. Denote by $M:={\rm H}(X,{\mathbf{Q}})[d]$ the shifted total cohomology of $X$ (with middle cohomology in degree [math]). For a class $\lambda\in{\rm H}^{2}(X,{\mathbf{Q}})$ consider the endomorphism $e_{\lambda}\in\operatorname{End}(M)$ given by cup product with $\lambda$. If $\lambda$ is ample, then $e_{\lambda}$ has the hard Lefschetz property, so that the map $e\colon{\rm H}^{2}(X,{\mathbf{Q}})\to\operatorname{\mathfrak{gl}}(M)$ has the hard Lefschetz property. We denote the corresponding Lie algebra by ${\mathfrak{g}}(X):={\mathfrak{g}}({\rm H}^{2}(X,{\mathbf{Q}}),M)$.

Proposition 2.2** ([34, 1.6, 1.9]).**

$({\rm H}^{2}(X,{\mathbf{Q}}),M)$* is a Lefschetz module.∎*

In other words, ${\mathfrak{g}}(X)$ is a semisimple Lie algebra over ${\mathbf{Q}}$.

2.2. Hochschild homology and cohomology

Let $X$ be a smooth projective variety of dimension $d$ with canonical bundle $\omega_{X}:=\Omega_{X}^{d}$. Its Hochschild cohomology is defined as

[TABLE]

and its Hochschild homology is defined as

[TABLE]

Composition of extensions defines maps ${\rm HH}^{n}\otimes{\rm HH}^{m}\to{\rm HH}^{n+m}$ and ${\rm HH}^{n}\otimes{\rm HH}_{m}\to{\rm HH}_{m-n}$ making ${\rm HH}_{\bullet}(X)$ into a graded module over the graded ring ${\rm HH}^{\bullet}(X)$.

The Hochschild–Kostant–Rosenberg isomorphism (twisted by the square root of the Todd class as in [31] and [15]) defines isomorphisms

[TABLE]

and

[TABLE]

Under these isomorphisms, multiplication in ${\rm HH}^{\bullet}(X)$ corresponds to the operation induced by the product in $\wedge^{\bullet}T_{X}$, and the action of ${\rm HH}^{\bullet}(X)$ on ${\rm HH}_{\bullet}(X)$ corresponds to the action induced by the contraction action of $\wedge^{\bullet}T_{X}$ on $\Omega_{X}^{\bullet}$, see [13, 12].

Together with the degeneration of the Hodge–de Rham spectral sequence, the isomorphism $I_{\bullet}$ defines an isomorphism

[TABLE]

This map does not respect the grading, rather it maps ${\rm HH}_{i}$ to the $i$-th column of the Hodge diamond (normalised so that the [math]-th column is the central column $\oplus_{p}{\rm H}^{p,p}$). Combining with the action of ${\rm HH}^{\bullet}$ on ${\rm HH}_{\bullet}$ we obtain an action of the ring ${\rm HH}^{\bullet}(X)$ on ${\rm H}(X,{\mathbf{C}})$.

Theorem 2.3**.**

Let $\Phi\colon{\mathcal{D}}X_{1}\overset{\sim}{\to}{\mathcal{D}}X_{2}$ be a derived equivalence between smooth projective complex varieties. Then we have natural graded isomorphisms

[TABLE]

compatible with the ring structure on ${\rm HH}^{\bullet}$ and the module structure on ${\rm HH}_{\bullet}$, and such that the square

[TABLE]

commutes.

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 $\sigma\in{\rm H}^{0}(X,\Omega^{2}_{X})$. Note that this implies that a Zariski dense collection of $\sigma\in{\rm H}^{0}(X,\Omega^{2}_{X})$ will be nowhere degenerate.

Through the isomorphism $I\colon{\rm HH}_{\bullet}(X)\to{\rm H}(X,{\mathbf{C}})$, the vector space ${\rm H}(X,{\mathbf{C}})$ becomes a module under the ring ${\rm HH}^{\bullet}(X)$.

Lemma 2.4**.**

${\rm HH}^{\bullet}(X)\cong{\rm H}^{\bullet}(X,{\mathbf{C}})$* as graded rings, and ${\rm H}(X,{\mathbf{C}})$ is free of rank one as ${\rm HH}^{\bullet}(X)$-module.*

Proof.

A symplectic form $\sigma$ defines an isomorphism $\Omega^{1}_{X}\overset{\sim}{\to}T_{X}$, and hence an isomorphism of algebras $\wedge^{\bullet}\Omega^{1}_{X}\overset{\sim}{\to}\wedge^{\bullet}T_{X}$. Combining this with the Hochschild–Kostant–Rosenberg isomorphism $I$ and the degeneration of the Hodge–de Rham spectral sequence we obtain a chain of isomorphisms of graded rings

[TABLE]

This proves the first assertion. For the second it suffices to observe that the module ${\rm HH}_{\bullet}(X,{\mathbf{C}})$ is generated by $\sigma^{d}\in{\rm HH}_{2d}(X)={\rm H}^{0}(X,\Omega^{2d}_{X})$. ∎

Consider the endomorphisms $h_{p},h_{q}\in\operatorname{End}({\rm H}(X,{\mathbf{C}}))$ given by

[TABLE]

These define the Hodge bi-grading on ${\rm H}(X,{\mathbf{C}})$, normalised to be symmetric along the central part ${\rm H}^{d,d}$. Note that $h=h_{p}+h_{q}$. The action of ${\rm HH}^{n}(X)$ on ${\rm H}(X,{\mathbf{C}})$ has degree $n$ for the grading defined by $h^{\prime}=h_{q}-h_{p}$.

Lemma 2.4 and Hard Lefschetz imply:

Corollary 2.5**.**

For a Zariski dense-collection of $\mu\in{\rm HH}^{2}(X)$ the action by $\mu$

[TABLE]

has the hard Lefschetz property with respect to the grading defined by $h^{\prime}$. ∎

In particular, for every such $\mu$ we have a complex subalgebra ${\mathfrak{g}}_{\mu}\subset\operatorname{End}({\rm H}(X,{\mathbf{C}}))$ isomorphic to $\operatorname{\mathfrak{sl}}_{2}$, and the collection of such algebras generates a Lie algebra which we denote by ${\mathfrak{g}}^{\prime}(X)\subset\operatorname{End}({\rm H}(X,{\mathbf{C}}))$. From Lemma 2.4 we also obtain:

Corollary 2.6**.**

The complex Lie algebras ${\mathfrak{g}}^{\prime}(X)$and ${\mathfrak{g}}(X)\otimes_{\mathbf{Q}}{\mathbf{C}}$ are isomorphic.∎

In the next section, we will show something stronger: that ${\mathfrak{g}}^{\prime}(X)$ and ${\mathfrak{g}}(X)\otimes_{\mathbf{Q}}{\mathbf{C}}$ coincide as sub-Lie algebras of $\operatorname{End}({\rm H}(X,{\mathbf{C}}))$. 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 $\Phi\colon{\mathcal{D}}X_{1}\overset{\sim}{\to}{\mathcal{D}}X_{2}$ there exists a canonical isomorphism of complex Lie algebras

[TABLE]

It has the property that the map $\Phi^{\rm H}\colon{\rm H}(X_{1},{\mathbf{C}})\overset{\sim}{\to}{\rm H}(X_{2},{\mathbf{C}})$ is equivariant with respect to $\Phi^{{\mathfrak{g}}^{\prime}}$.

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.

Proposition 2.8**.**

If $X$ is holomorphic symplectic, then ${\mathfrak{g}}(X)\otimes_{\mathbf{Q}}{\mathbf{C}}={\mathfrak{g}}^{\prime}(X)$ as sub-Lie algebras of $\operatorname{End}({\rm H}(X,{\mathbf{C}}))$.

Let $X$ be holomorphic symplectic. If ${\mathcal{F}}$ is a coherent ${\mathcal{O}}_{X}$-module then we will simply write ${\rm H}^{i}({\mathcal{F}})$ for ${\rm H}^{i}(X,{\mathcal{F}})$. We have decompositions

[TABLE]

and

[TABLE]

We will use the same symbol $\lambda$ to denote an element $\lambda\in{\rm H}^{2}(X,{\mathbf{C}})$ and the endomorphism of $\operatorname{End}({\rm H}(X,{\mathbf{C}}))$ given by cup product with $\lambda$. Note that we have $\lambda\in{\mathfrak{g}}(X)\otimes_{\mathbf{Q}}{\mathbf{C}}$ by construction. Similarly, we will use the same symbol for $\mu\in{\rm HH}^{2}(X)$ and the resulting $\mu\in\operatorname{End}({\rm H}(X,{\mathbf{C}}))$, given by contraction with $\mu$. We have $\mu\in{\mathfrak{g}}^{\prime}(X)$.

For a symplectic form $\sigma\in{\rm H}^{0}(\Omega^{2}_{X})$ we denote by $\check{\sigma}\in{\rm H}^{0}(\wedge^{2}T_{X})$ the image of the form $\sigma\in{\rm H}^{0}(\Omega^{2}_{X})$ under the isomorphism $\Omega^{2}_{X}\to\wedge^{2}T_{X}$ defined by $\sigma$. In suitable local coordinates, we have

[TABLE]

and

[TABLE]

Lemma 2.9**.**

If $\sigma$ is a nowhere degenerate symplectic form then $(\sigma,h_{p},\check{\sigma})$ is an $\operatorname{\mathfrak{sl}}_{2}$-triple in $\operatorname{End}({\rm H}(X,{\mathbf{C}}))$.

Proof.

Clearly $\sigma$ has degree $2$ and $\check{\sigma}$ has degree $-2$ for the grading given by $h_{p}$, so that $[h_{p},\sigma]=2\sigma$ and $[h_{p},\check{\sigma}]=-2\check{\sigma}$.

We need to show that $[\sigma,\check{\sigma}]=h_{p}$. This follows immediately from a local computation: in the above local coordinates, one verifies that on the standard basis of $\Omega^{p}$ the commutator $[\sigma,\check{\sigma}]$ acts as $p-d$. ∎

Note that the existence of one nowhere degenerate $\sigma$ implies that a Zariski dense collection of $\sigma\in{\rm H}^{0}(\Omega^{2}_{X})$ is nowhere degenerate.

Lemma 2.10**.**

For a Zariski-dense collection $\alpha\in{\rm H}^{2}(X,{\mathcal{O}}_{X})$ there is a $\check{\alpha}\in\operatorname{End}({\rm H}(X,{\mathbf{C}}))$ so that $(\alpha,h_{q},\check{\alpha})$ is an $\operatorname{\mathfrak{sl}}_{2}$-triple.

Proof.

This follows from Lemma 2.9 and Hodge symmetry. ∎

Lemma 2.11**.**

For all $\tau\in{\rm H}^{0}(X,\wedge^{2}T_{X})$ the endomorphism $\tau$ lies in ${\mathfrak{g}}(X)\otimes_{\mathbf{Q}}{\mathbf{C}}$.

Proof.

It suffices to show that this holds for a Zariski dense collection of $\tau$, hence we may assume without loss of generality that $\tau=\check{\sigma}$ with $\sigma$ and $\check{\sigma}$ as in Lemma 2.9. Let $\alpha$ and $\check{\alpha}$ be as in Lemma 2.10. Because $\sigma$ and $h_{p}$ commute with both $\alpha$ and $h_{q}$, we have that every element of the $\operatorname{\mathfrak{sl}}_{2}$-triple $(\sigma,h_{p},\check{\sigma})$ commutes with every element of the $\operatorname{\mathfrak{sl}}_{2}$-triple $(\alpha,h_{q},\check{\alpha})$. From this, it follows that

[TABLE]

are $\operatorname{\mathfrak{sl}}_{2}$-triples. Since the elements $\alpha\pm\sigma$ lie in ${\rm H}^{2}(X,{\mathbf{C}})$, and apparently have the hard Lefschetz property, we conclude that the endomorphisms $\check{\alpha}\pm\check{\sigma}$ lie in ${\mathfrak{g}}(X)\otimes_{\mathbf{Q}}{\mathbf{C}}$, hence also $\tau=\check{\sigma}$ lies in ${\mathfrak{g}}(X)\otimes_{\mathbf{Q}}{\mathbf{C}}$. ∎

Corollary 2.12**.**

$h_{p}$* and $h_{q}$ lie in ${\mathfrak{g}}(X)\otimes_{\mathbf{Q}}{\mathbf{C}}$.*

Proof.

By Lemma 2.9 we have $h_{p}=[\sigma,\check{\sigma}]$, which by Lemma 2.11 lies in ${\mathfrak{g}}(X)\otimes_{\mathbf{Q}}{\mathbf{C}}$. Since $h_{q}=h-h_{p}$ we also have that $h_{q}$ lies in ${\mathfrak{g}}(X)\otimes_{\mathbf{Q}}{\mathbf{C}}$. ∎

Fix a $\tau\in{\rm H}^{0}(X,\wedge^{2}T_{X})$ that is nowhere degenerate as an alternating form on $\Omega^{1}_{X}$. This defines isomorphisms $c_{\tau}\colon\Omega^{1}_{X}\to T_{X}$ and $c_{\tau}\colon{\rm H}^{1}(\Omega^{1}_{X})\to{\rm H}^{1}(T_{X})$ given by contracting sections of $\Omega^{1}_{X}$ with $\tau$.

Lemma 2.13**.**

For all $\eta\in{\rm H}^{1}(\Omega^{1}_{X})$ we have $[\tau,\eta]=c_{\tau}(\eta)$ in $\operatorname{End}({\rm H}(X,{\mathbf{C}}))$.

Proof.

This is again a local computation. If $\eta$ is a local section of $\Omega^{1}_{X}$, then a computation on a local basis shows that $[\tau,\eta]=c_{\tau}(\eta)$ as maps $\Omega^{p}_{X}\to\Omega^{p-1}_{X}$. ∎

Corollary 2.14**.**

Every element $\eta^{\prime}$ of ${\rm H}^{1}(X,T_{X})$ lies in ${\mathfrak{g}}(X)\otimes_{\mathbf{Q}}{\mathbf{C}}$.

Proof.

Every such $\eta^{\prime}$ is of the form $c_{\tau}(\eta)$ for a unique $\eta\in{\rm H}^{1}(\Omega^{1}_{X})$, and hence the corollary follows from Lemma 2.13, Lemma 2.11, and the fact that $\eta$ lies in ${\mathfrak{g}}(X)\otimes_{\mathbf{Q}}{\mathbf{C}}$. ∎

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 ${\mathfrak{g}}^{\prime}(X)$ is contained in ${\mathfrak{g}}(X)\otimes_{\mathbf{Q}}{\mathbf{C}}$. By Proposition 2.1 it suffices to show that $h^{\prime}$ is contained in ${\mathfrak{g}}(X)\otimes_{\mathbf{Q}}{\mathbf{C}}$, and that for almost every $a\in{\rm HH}^{2}(X)$ we have that the action of $a$ on ${\rm H}(X,{\mathbf{C}})$ is contained in ${\mathfrak{g}}(X)\otimes_{\mathbf{Q}}{\mathbf{C}}$. This follows from Lemma 2.11, Corollary 2.12, and Corollary 2.14, and the fact that the action of any $\alpha\in{\rm H}^{2}({\mathcal{O}}_{X})$ lies in ${\mathfrak{g}}(X)\otimes_{\mathbf{Q}}{\mathbf{C}}$. ∎

Together with Proposition 2.7, this proves Theorem A.

3. 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

[TABLE]

its Beauville–Bogomolov–Fujiki, and by $c_{X}$ its Fujiki constant. These are related by

[TABLE]

for $\lambda\in{\rm H}^{2}(X,{\mathbf{Q}})$.

We extend $b$ to a bilinear form on

[TABLE]

by declaring $\alpha$ and $\beta$ to be orthogonal to ${\rm H}^{2}(X,{\mathbf{Q}})$, and setting $b(\alpha,\beta)=-1$, $b(\alpha,\alpha)=0$ and $b(\beta,\beta)=0$. We equip $\widetilde{\rm H}(X,{\mathbf{Q}})$ with a grading satisfying $\deg\alpha=-2$, $\deg\beta=2$, and for which ${\rm H}^{2}(X,{\mathbf{Q}})$ sits in degree [math]. This induces a grading on the Lie algebra $\operatorname{\mathfrak{so}}(\widetilde{\rm H}(X,{\mathbf{Q}}))$.

For $\lambda\in{\rm H}^{2}(X,{\mathbf{Q}})$ we consider the endomorphism $e_{\lambda}\in\operatorname{\mathfrak{so}}(\widetilde{\rm H}(X,{\mathbf{Q}}))$ given by $e_{\lambda}(\alpha)=\lambda$, $e_{\lambda}(\mu)=b(\lambda,\mu)\beta$ for all $\mu\in{\rm H}^{2}(X,{\mathbf{Q}})$, and $e_{\lambda}(\beta)=0$.

Theorem 3.1** (Looijenga–Lunts, Verbitsky).**

There is a unique isomorphism of graded Lie algebras

[TABLE]

that maps $e_{\lambda}$ to $e_{\lambda}$ for every $\lambda\in{\rm H}^{2}(X,{\mathbf{Q}})$.

Proof.

See [34, Prop. 4.5] or [46, Thm. 1.4] for the theorem over the real numbers. This readily descends to ${\mathbf{Q}}$, see [43, Prop. 2.9] for more details. ∎

The representation of $\operatorname{\mathfrak{so}}(\widetilde{\rm H}(X,{\mathbf{Q}}))$ on ${\rm H}(X,{\mathbf{Q}})$ integrates to a representation of $\operatorname{Spin}(\widetilde{\rm H}(X,{\mathbf{Q}}))$ on ${\rm H}(X,{\mathbf{Q}})$. Let $\lambda\in{\rm H}^{2}(X,{\mathbf{Q}})$. Then $e_{\lambda}$ is nilpotent, and hence $B_{\lambda}:=\exp e_{\lambda}$ is an element of $\operatorname{Spin}(\widetilde{\rm H}(X,{\mathbf{Q}}))$. It acts on $\widetilde{\rm H}(X,{\mathbf{Q}})$ as follows:

[TABLE]

for all $r,s\in{\mathbf{Q}}$ and $\mu\in{\rm H}^{2}(X,{\mathbf{Q}})$. The action on the total cohomology of $X$ is given by:

Proposition 3.2**.**

$B_{\lambda}$* acts as multiplication by $\operatorname{ch}(\lambda)$ on ${\rm H}(X,{\mathbf{Q}})$. ∎*

In particular, if ${\mathcal{L}}$ is a line bundle on $X$ and $\Phi\colon{\mathcal{D}}X\to{\mathcal{D}}X$ is the equivalence that maps ${\mathcal{F}}$ to ${\mathcal{F}}\otimes{\mathcal{L}}$, then $\Phi^{\rm H}=B_{c_{1}({\mathcal{L}})}$.

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 ${\mathbf{Q}}$-algebra

[TABLE]

and the Verbitsky component of the cohomology of $X$ as the sub-${\mathbf{Q}}$-algebra ${\mathrm{SH}}(X,{\mathbf{Q}})$ of ${\rm H}^{\rm ev}(X,{\mathbf{Q}})$ generated by ${\rm H}^{2}(X,{\mathbf{Q}})$. Clearly, ${\mathrm{SH}}(X,{\mathbf{Q}})[2d]$ is a sub-Lefschetz-module of ${\rm H}^{\rm ev}(X,{\mathbf{Q}})[2d]$ for ${\rm H}^{2}(X,{\mathbf{Q}})$.

Lemma 3.3** (Verbitsky [45, 8]).**

The kernel of the ${\mathbf{Q}}$-algebra homomorphism

[TABLE]

is generated by the elements $\lambda^{d+1}$ with $\lambda\in{\rm H}^{2}(X,{\mathbf{Q}})$ satisfying $b(\lambda,\lambda)=0$.∎

Lemma 3.4** (Verbitsky).**

${\mathrm{SH}}(X,{\mathbf{Q}})[2d]$* is an irreducible Lefschetz module.*

Proof.

It is the smallest sub-Lefschetz module of ${\rm H}^{\rm ev}(X,{\mathbf{Q}})[2d]$ having a non-trivial component of degree $-2d$. ∎

Verbitsky also describes the space ${\mathrm{SH}}(X,{\mathbf{Q}})$ explicitly. Below we normalise this description, and use it to compute the Mukai pairing on ${\mathrm{SH}}(X,{\mathbf{Q}})$.

Proposition 3.5**.**

There is a unique map

[TABLE]

satisfying

1. (i)

$\Psi$* is morphism of Lefschetz modules* 2. (ii)

$\Psi(1)=\alpha^{d}/d!$**

Note that the Lefschetz module structure on $\operatorname{Sym}^{d}\widetilde{\rm H}(X,{\mathbf{Q}})$ is given by the Leibniz rule

[TABLE]

Proof of Proposition 3.5.

Uniqueness is clear. For existence, consider the map

[TABLE]

given by

[TABLE]

This map is well-defined since the $e_{\lambda_{i}}$ commute, and by construction it is a morphism of Lefschetz modules satisfying $\Psi(1)=\alpha^{d}/d!$. It suffices to show that $\widetilde{\Psi}$ vanishes on the ideal generated by the $\lambda^{d+1}$ for $\lambda\in{\rm H}^{2}(X,{\mathbf{Q}})$ satisfying $b(\lambda,\lambda)=0$. Equivalently, it suffices to show that for every $x\in\operatorname{Sym}^{d}\widetilde{\rm H}(X,{\mathbf{Q}})$ and for every $\lambda\in{\rm H}^{2}(X,{\mathbf{Q}})$ with $b(\lambda,\lambda)=0$ we have $e_{\lambda}^{d+1}(x)=0$.

Without loss of generality, we may assume that $x$ is a monomial of the form

[TABLE]

By degree reasons, we have $e_{\lambda}^{k}(\beta^{j}\lambda_{1}\cdots\lambda_{m})=0$ for $k>m$. Moreover, it follows from $b(\lambda,\lambda)=0$ that $e_{\lambda}^{k}(\alpha^{i})=0$ for $k>i$. Combining these, one concludes that $e_{\lambda}^{d+1}(x)=0$, which is what we had to prove. ∎

Lemma 3.6**.**

$\Psi({\rm pt}_{X})=\beta^{d}/c_{X}$.

Proof.

Choose $\lambda\in{\rm H}^{2}(X,{\mathbf{Q}})$ with $b(\lambda,\lambda)\neq 0$. Then we have

[TABLE]

Dividing by (2) gives the claimed identity. ∎

Consider the contraction (or Laplacian) operator

[TABLE]

given by

[TABLE]

This is a morphism of Lefschetz modules, or equivalently of $\operatorname{\mathfrak{so}}(\widetilde{{\rm H}}(X,{\mathbf{Q}}))$-modules.

Lemma 3.7**.**

The sequence of Lefschetz modules

[TABLE]

is exact.

Proof.

Since $\Delta\Psi(1)=0$, we have $\Delta\circ\Psi=0$. The map $\Delta$ is well-known to be a surjective map of $\operatorname{\mathfrak{so}}(\widetilde{\rm H}(X,{\mathbf{Q}}))$-modules with irreducible kernel. Since $\Psi$ is non-zero and ${\mathrm{SH}}(X,{\mathbf{Q}})$ is irreducible, it follows that the sequence is exact. ∎

The Mukai pairing [14] on ${\rm H}^{\rm ev}(X,{\mathbf{Q}})$ restricts to a pairing $b_{\mathrm{SH}}$ on ${\mathrm{SH}}(X,{\mathbf{Q}})$. It pairs elements of degree $m$ with elements of degree $2d-m$, according to the formula

[TABLE]

Note that $b_{\mathrm{SH}}(e_{\lambda}x,y)+b_{\mathrm{SH}}(x,e_{\lambda}y)=0$ for all $x,y\in{\mathrm{SH}}(X,{\mathbf{Q}})$ and $\lambda\in{\rm H}^{2}(X,{\mathbf{Q}})$, so that $b_{\mathrm{SH}}$ is $\operatorname{\mathfrak{so}}(\widetilde{\rm H}(X,{\mathbf{Q}}))$-invariant.

The pairing on $\widetilde{\rm H}(X,{\mathbf{Q}})$ induces a pairing on $\operatorname{Sym}^{d}\widetilde{\rm H}(X,{\mathbf{Q}})$ defined by

[TABLE]

By construction, $b_{[d]}$ is $\operatorname{\mathfrak{so}}(\widetilde{\rm H}(X,{\mathbf{Q}}))$-invariant. The map $\Psi$ is almost an isometry, in the following sense.

Proposition 3.8**.**

For all $x,y\in{\mathrm{SH}}(X,{\mathbf{Q}})$ we have

[TABLE]

Proof.

Both the Mukai form on ${\mathrm{SH}}(X,{\mathbf{Q}})[2d]$ and the pairing on $\operatorname{Sym}^{d}\widetilde{\rm H}(X,{\mathbf{Q}})$ are $\operatorname{\mathfrak{so}}(\widetilde{\rm H}(X,{\mathbf{Q}}))$-invariant. Since ${\mathrm{SH}}(X,{\mathbf{Q}})$ is an irreducible $\operatorname{\mathfrak{so}}(\widetilde{\rm H}(X,{\mathbf{Q}}))$-module, it suffices to verify the identity for some $x,y\in{\mathrm{SH}}(X,{\mathbf{Q}})$ with $b_{{\mathrm{SH}}}(x,y)\neq 0$.

Let $\lambda\in{\rm H}^{2}(X,{\mathbf{Q}})$ with $b(\lambda,\lambda)\neq 0$. We have

[TABLE]

By (3) we have

[TABLE]

and hence

[TABLE]

which agrees with the identity claimed in the proposition. ∎

Remark 3.9**.**

If $X$ is of type $\mathrm{K3}^{[d]}$ then $c_{X}=(2d)!/(2^{d}d!)$ and $\Psi$ is an isometry.

4. 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

[TABLE]

The Lie algebra $\operatorname{\mathfrak{so}}(V)$ acts faithfully on $S_{[d]}V$, inducing an inclusion $\operatorname{\mathfrak{so}}(V)\subset\operatorname{End}(S_{d}V)$. Consider the normalizer of $\operatorname{\mathfrak{so}}(V)$ in $\operatorname{GL}(S_{[d]}V)$, that is, the group

[TABLE]

Proposition 4.1**.**

There is an exact sequence

[TABLE]

where the inclusion maps $\epsilon$ to $(\epsilon,\epsilon^{d})$ and the surjection maps $(\varphi,\lambda)$ to $\lambda S_{[d]}(\varphi)$.

Proof.

The only non-trivial part is surjectivity of $\operatorname{O}(V)\times K^{\times}\to N(V,d)$. Denote by $\varphi\colon\operatorname{O}(V)\to N(V,d)$ the restriction of this map to the first component.

The representation $S_{[d]}V$ of $\operatorname{\mathfrak{so}}(V)$ is irreducible, so by Schur’s lemma the centralizer of $\operatorname{\mathfrak{so}}(V)$ in $\operatorname{GL}(S_{[d]}V)$ is $K^{\times}$, and we have an exact sequence

[TABLE]

It therefore suffices to show that the image of $\psi$ equals the image of $\psi\circ\varphi$.

The adjoint group of $\operatorname{\mathfrak{so}}(V)$ is $\operatorname{SO}(V)$, and we have a short exact sequence

[TABLE]

where $\operatorname{Out}(\operatorname{\mathfrak{so}}(V))$ coincides with the group of symmetries of the Dynkin diagram.

If $\dim V=2n+1$, then we have $\operatorname{O}(V)=\{\pm 1\}\times\operatorname{SO}(V)$. The Dynkin diagram (of type $B_{n}$) has no non-trivial automorphisms, so $\operatorname{Aut}(\operatorname{\mathfrak{so}}(V,b))=\operatorname{SO}(V)$. The composition $\psi\circ\varphi$ maps $\operatorname{SO}(V)$ identically to $\operatorname{SO}(V)$, and we conclude that the image of $\psi$ is the image of $\psi\circ\varphi$.

If $m=2n$, then we have $\operatorname{O}(V)/\{\pm 1\}\subset\operatorname{Aut}\operatorname{\mathfrak{so}}(V)$, with elements of determinant $-1$ inducing the reflection in the horizontal axis in the Dynkin diagram (of type $D_{n}$). For $n\neq 4$, this inclusion is an equality, while for $n=4$ ‘triality’ gives extra automorphisms. However, expressed on simple roots the highest weight of the representation $S_{[d]}V$ of $\operatorname{\mathfrak{so}}(V)$ is

[TABLE]

so that for $n=4$ the extra automorphisms of $\operatorname{\mathfrak{so}}(V)$ do not lift to automorphisms of $S_{[d]}V$. We conclude that the image of $\psi$ is contained in $\operatorname{O}(V)/\{\pm 1\}$ and that the composition $\psi\circ\varphi$ is the natural map $\operatorname{O}(V)\to\operatorname{O}(V)/\{\pm 1\}$, so that also in this case the image of $\psi$ coincides with the image of $\psi\circ\varphi$. ∎

Proposition 4.2**.**

Let $V_{1}$ and $V_{2}$ be non-zero quadratic spaces. Assume that there is a linear map $f\colon S_{[d]}V_{1}\to S_{[d]}V_{2}$ such that $f\operatorname{\mathfrak{so}}(V_{1})f^{-1}=\operatorname{\mathfrak{so}}(V_{2})$ as subspaces of $\operatorname{End}(V_{2})$. Then there exists a $\mu\in K^{\times}$ and a similitude $\varphi\colon V_{1}\to V_{2}$ such that $f=\mu S_{[d]}(\varphi)$.

Proof.

Let $\bar{K}$ be a separable closure of $K$. After base change to $\bar{K}$ the quadratic spaces $V_{1}$ and $V_{2}$ become isometric, hence $V_{2}$ determines a class $\gamma$ in the Galois cohomology group ${\rm H}^{1}(\operatorname{Gal}(\bar{K}/K),\operatorname{O}(V_{\bar{K}})$. The existence of $f$ shows that $\gamma$ is mapped to the trivial element under the natural map

[TABLE]

By Proposition 4.1 (and Hilbert 90), this shows that $\gamma$ is in the image of the map

[TABLE]

induced by $\{\pm 1\}\to\operatorname{O}(V_{1,\bar{K}})$. But his means $(V_{1},\lambda b_{1})$ and $(V_{2},b_{2})$ are isomorphic for any representative $\lambda$ of a class in $K^{\times}/(K^{\times})^{2}$ mapping to $\gamma$.

In particular, there exists a similitude $\varphi_{0}\colon V_{1}\to V_{2}$. The map $f^{-1}\circ S_{[d]}(\varphi_{0})$ is an element of $N(V_{1},d)$. Now let $(\psi,\mu)\in\operatorname{O}(V_{1})\times K^{\times}$ be a pre-image of this element under the map from Proposition 4.1, and set $\varphi:=\varphi_{0}\circ\psi^{-1}$. Then the pair $(\mu,\varphi)$ satisfies the requirements. ∎

The bilinear form $b$ on $V$ induces a bilinear form $b_{[d]}$ on $S_{[d]}V$ defined as

[TABLE]

Consider the group

[TABLE]

of isometries of $S_{[d]}V$ that preserve the subspace $\operatorname{\mathfrak{so}}(V)$ of $\operatorname{End}S_{[d]}V$.

Proposition 4.3**.**

Let $V=(V,b)$ be an object of ${\mathcal{B}}$. We have

[TABLE]

Proof.

This follows immediately from Proposition 4.1. ∎

4.2. The Verbitsky component

Theorem 4.4**.**

Let $X_{1}$ and $X_{2}$ be hyperkähler varieties and $\Phi\colon{\mathcal{D}}X_{1}\to{\mathcal{D}}X_{2}$ an equivalence. Then the induced isomorphism $\Phi^{{\rm H}}\colon{\rm H}(X_{1},{\mathbf{Q}})\to{\rm H}(X_{2},{\mathbf{Q}})$ restricts to an isomorphism $\Phi^{{\mathrm{SH}}}\colon{\mathrm{SH}}(X_{1},{\mathbf{Q}})\to{\mathrm{SH}}(X_{2},{\mathbf{Q}})$. Moreover

1. (i)

$\Phi^{{\mathrm{SH}}}$* is an isometry with respect to the Mukai pairings* 2. (ii)

$\Phi^{{\mathrm{SH}}}{\mathfrak{g}}(X_{1})(\Phi^{{\mathrm{SH}}})^{-1}={\mathfrak{g}}(X_{2})$* in $\operatorname{End}({\mathrm{SH}}(X_{2},{\mathbf{Q}}))$.*

Proof.

Note that ${\mathrm{SH}}(X,{\mathbf{Q}})$ can be characterized as the minimal sub-${\mathfrak{g}}(X)$-module of ${\rm H}(X,{\mathbf{Q}})$ whose Hodge structure attains the maximal possible level (width). It then follows from Theorem A and from Lemma 3.4 that $\Phi^{{\rm H}}$ restricts to an isomorphism

[TABLE]

respecting the Lie algebras ${\mathfrak{g}}(X_{1})$ and ${\mathfrak{g}}(X_{2})$. By [14], the map $\Phi^{\rm H}$ respects the Mukai pairings, and the theorem follows. ∎

Definition 4.5**.**

For a complex hyperkähler variety we equip ${\mathrm{SH}}(X,{\mathbf{Q}})$ and $\widetilde{\rm H}(X,{\mathbf{Q}})$ with Hodge structures of weight [math], given by

[TABLE]

and

[TABLE]

Lemma 4.6**.**

Let $X$ be a hyperkähler variety of dimension $2d$. Then the map

[TABLE]

of Proposition 3.5 is a morphism of Hodge structures of weight [math].

Proof.

This is clear from the proof of Proposition 3.5: the map $\widetilde{\Psi}$ is a morphism of Hodge structures, and so is the quotient map $\operatorname{Sym}^{\bullet}{\rm H}(X,{\mathbf{Q}}(1))\to{\mathrm{SH}}(X,{\mathbf{Q}})$. ∎

Proposition 4.7**.**

Let $X_{1}$ and $X_{2}$ be derived equivalent hyperkähler varieties. Then there exists a Hodge similitude $\varphi\colon\widetilde{\rm H}(X_{1},{\mathbf{Q}})\overset{\sim}{\to}\widetilde{\rm H}(X_{2},{\mathbf{Q}})$ and a scalar $\lambda\in K^{\times}$ so that the square

[TABLE]

commutes.

Proof.

By Theorem 4.4 and Proposition 4.2 there exists a similitude $\varphi$ and a scalar $\lambda$ that make the square commute.

It remains to check that $\varphi$ respects the Hodge structures. The Hodge structure on $\widetilde{\rm H}(X_{i},{\mathbf{Q}})$ is given by a morphism $h_{i}\colon{\mathbf{C}}^{\times}\to\operatorname{O}({\rm H}(X_{i},{\mathbf{R}}))$, and the preceding lemma implies that the Hodge structure on ${\mathrm{SH}}(X_{i},{\mathbf{Q}})$ is given by composing $h_{i}$ with the injective map $\operatorname{O}({\rm H}(X_{i},{\mathbf{R}}))\to\operatorname{GL}({\mathrm{SH}}(X_{i},{\mathbf{R}}))$. Since $\varphi$ maps the Hodge structure on ${\mathrm{SH}}(X_{1},{\mathbf{Q}})$ to the Hodge structure on ${\mathrm{SH}}(X_{2},{\mathbf{Q}})$, we conclude that $\varphi$ maps $h_{1}$ to $h_{2}$. ∎

Theorem 4.8** ($d$ odd).**

Assume that $d$ is odd, and that $X_{1}$ and $X_{2}$ are deformation-equivalent hyperkähler varieties of dimension $2d$. Let $\Phi\colon{\mathcal{D}}X_{1}\overset{\sim}{\to}{\mathcal{D}}X_{2}$ be an equivalence. Then there is a unique Hodge isometry $\Phi^{\widetilde{\rm H}}$ making the square

[TABLE]

commute. The formation of $\Phi^{\widetilde{\rm H}}$ is functorial in $\Phi$.

Proof.

Since $X_{1}$ and $X_{2}$ are deformation equivalent, we can choose an isometry $\varphi\colon\widetilde{\rm H}(X_{1},{\mathbf{Q}})\overset{\sim}{\to}\widetilde{\rm H}(X_{2},{\mathbf{Q}})$. Moreover, $X_{1}$ and $X_{2}$ have the same Fujiki constant, so $\operatorname{Sym}^{d}\varphi$ restricts to an isometry between the images of $\Psi$. Then by Theorem 4.4 and Proposition 4.3, there is a unique isometry $\psi\in\operatorname{O}(\widetilde{\rm H}(X_{2},{\mathbf{Q}}))$ such that $\Phi^{\widetilde{\rm H}}:=\psi\varphi$ makes the square commute. Uniqueness forces its formation to be functorial.

That $\Phi^{\widetilde{\rm H}}$ 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

[TABLE]

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 $\operatorname{SO}\times\{\pm 1\}\cong\operatorname{O}$ to salvage the situation a bit.

Define an orientation on $X$ to be the choice of a generator of $\det{\rm H}^{2}(X,{\mathbf{R}})$, up to ${\mathbf{R}}^{\times}_{>0}$. Equivalently, an orientation is the choice of generator of $\det\widetilde{\rm H}(X,{\mathbf{R}})$ up to ${\mathbf{R}}^{\times}_{>0}$. Define the sign $\epsilon(\varphi)$ of a Hodge isometry $\varphi\colon\widetilde{\rm H}(X_{1},{\mathbf{Q}})\overset{\sim}{\to}\widetilde{\rm H}(X_{2},{\mathbf{Q}})$ as $\epsilon(\varphi)=1$ if $\varphi$ respects the orientations and $\epsilon(\varphi)=-1$ 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 $\Phi\colon{\mathcal{D}}X_{1}\overset{\sim}{\to}{\mathcal{D}}X_{2}$ 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 ${\rm H}^{2}(X_{1},{\mathbf{Q}})$ and ${\rm H}^{2}(X_{2},{\mathbf{Q}})$ are isometric. Then there exists a unique Hodge isometry $\Phi^{\widetilde{\rm H}}$ making the square

[TABLE]

commute. Morever, the formation of $\Phi^{\widetilde{\rm H}}$ 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 $\Phi\colon{\mathcal{D}}X_{1}\overset{\sim}{\to}{\mathcal{D}}X_{2}$ 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 ${\rm H}^{2}$. 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:

Theorem 4.11**.**

Let $X$ be a hyperkähler variety of dimension $2d$. Assume that either $d$ is odd or that $d$ is even and $b_{2}(X)$ is odd. Then the representation $\rho^{\mathrm{SH}}\colon\operatorname{Aut}{\mathcal{D}}(X)\to\operatorname{GL}({\mathrm{SH}}(X,{\mathbf{Q}}))$ factors over a map $\rho^{\widetilde{\rm H}}\colon\operatorname{Aut}{\mathcal{D}}(X)\to\operatorname{O}(\widetilde{\rm H}(X,{\mathbf{Q}}))$.

Remark 4.12**.**

For $d$ odd, the implicit map $\operatorname{O}(\widetilde{\rm H}(X,{\mathbf{Q}}))\to\operatorname{GL}({\mathrm{SH}}(X,{\mathbf{Q}}))$ is the natural map coming from the isomorphism ${\mathrm{SH}}(X,{\mathbf{Q}})\cong S_{[d]}\widetilde{\rm H}(X,{\mathbf{Q}}))$. For $d$ even (and $b_{2}$ odd), it is the twist of the natural map with the determinant character $\operatorname{O}(\widetilde{\rm H}(X,{\mathbf{Q}}))\to\{\pm 1\}$.

5. 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 $\operatorname{\mathbf{SO}}(V)$, $\operatorname{\mathbf{Spin}}(V)$, and $\operatorname{\mathbf{GSpin}}(V)$ over $\operatorname{Spec}{\mathbf{Q}}$. These groups sit in exact sequences

[TABLE]

We will write $\operatorname{SO}(V)$, $\operatorname{Spin}(V)$, and $\operatorname{GSpin}(V)$ for the groups of ${\mathbf{Q}}$-points of these algebraic groups.

Lemma 5.1**.**

Let $X$ be a hyperkähler variety. There exists a unique action of $\operatorname{\mathbf{GSpin}}(\widetilde{\rm H}(X,{\mathbf{Q}}))$ on ${\rm H}(X,{\mathbf{Q}})$ such that

1. (i)

the action of $\operatorname{\mathbf{Spin}}(\widetilde{\rm H}(X,{\mathbf{Q}}))\subset\operatorname{\mathbf{GSpin}}(\widetilde{\rm H}(X,{\mathbf{Q}}))$ integrates the action of ${\mathfrak{g}}(X)=\operatorname{\mathfrak{so}}(\widetilde{\rm H}(X,{\mathbf{Q}}))$ 2. (ii)

a section $\lambda\in{\mathbf{G}}_{m}\subset\operatorname{\mathbf{GSpin}}(\widetilde{\rm H}(X,{\mathbf{Q}}))$ acts as $\lambda^{i-2d}$ on ${\rm H}^{i}(X,{\mathbf{Q}})$.

Proof.

The action of $\operatorname{\mathfrak{so}}(\widetilde{\rm H}(X,{\mathbf{Q}}))$ integrates to an action of the simply connected group $\operatorname{\mathbf{Spin}}(\widetilde{\rm H}(X,{\mathbf{Q}}))$. By (6) it suffices to verify that $-1\in\operatorname{Spin}(\widetilde{\rm H}(X,{\mathbf{Q}}))$ acts as $(-1)^{i}$ on ${\rm H}^{i}(X,{\mathbf{Q}})$. Any $\operatorname{\mathfrak{sl}}_{2}$-triple $(e_{\lambda},h,f_{\lambda})$ in ${\mathfrak{g}}(X)$ induces an action of $\operatorname{SL}_{2}$ on ${\rm H}(X,{\mathbf{Q}})$, with the property that $\operatorname{diag}(\lambda,\lambda^{-1})\in\operatorname{SL}_{2}({\mathbf{Q}})$ acts as $\lambda^{i}$ on ${\rm H}^{2d+i}(X,{\mathbf{Q}})$. Since the inclusion $\operatorname{\mathbf{SL}}_{2}\subset\operatorname{\mathbf{Spin}}$ maps $\operatorname{diag}(1,-1)$ to $-1$, the lemma follows. ∎

Recall from Definition 4.5 that we have equipped $\widetilde{\rm H}(X,{\mathbf{Q}})$ and ${\rm H}^{\rm ev}(X,{\mathbf{Q}})$ with Hodge structures of weight [math]. Similarly, we equip the odd cohomology of $X$ with a Hodge structure of weight $1$ as follows

[TABLE]

Lemma 5.2**.**

Let $g\in\operatorname{GSpin}(\widetilde{\rm H}(X,{\mathbf{Q}}))$. If the action of $g$ on $\widetilde{\rm H}(X,{\mathbf{Q}})$ respects the Hodge structure, then so does its action on ${\rm H}^{\rm ev}(X,{\mathbf{Q}})$ and on ${\rm H}^{\rm odd}(X,{\mathbf{Q}})$.

Proof.

This follows immediately from the fact that the Hodge structure is determined by the action of $h^{\prime}\in{\mathfrak{g}}(X)\otimes_{\mathbf{Q}}{\mathbf{C}}$ (see § 2.3), and from the faithfulness of the ${\mathfrak{g}}(X)$-module $\widetilde{\rm H}(X,{\mathbf{Q}})$. ∎

Theorem 5.3**.**

Let $X_{1}$ and $X_{2}$ be hyperkähler varieties, and let $\Phi\colon{\mathcal{D}}X_{1}\overset{\sim}{\to}{\mathcal{D}}X_{2}$ be an equivalence. Then for every $i$ the ${\mathbf{Q}}$-Hodge structures ${\rm H}^{i}(X_{1},{\mathbf{Q}})$ and ${\rm H}^{i}(X_{2},{\mathbf{Q}})$ are isomorphic.

Proof.

Consider the Lie algebra isomorphism $\Phi^{\mathfrak{g}}\colon{\mathfrak{g}}(X_{1})\overset{\sim}{\to}{\mathfrak{g}}(X_{2})$ from Theorem A. By Proposition 4.7, there exists a Hodge similitude $\phi\colon\widetilde{\rm H}(X_{1},{\mathbf{Q}})\overset{\sim}{\to}\widetilde{\rm H}(X_{2},{\mathbf{Q}})$ so that the square

[TABLE]

commutes. Here the vertical maps are the isomorphisms from Theorem 3.1.

The K3-type Hodge structure $\widetilde{\rm H}(X_{2},{\mathbf{Q}})$ decomposes as $N\oplus T$, with $N$ and $T$ its algebraic and transcendental parts, respectively. The Hodge similitude $\phi$ maps the distinguished elements $\alpha_{1}$ and $\beta_{1}$ of $\widetilde{\rm H}(X_{1},{\mathbf{Q}})$ to ${N}$. By Witt cancellation, there exists a $\psi_{N}\in\operatorname{SO}(N)$ and $\lambda,\mu\in{\mathbf{Q}}^{\times}$ such that $\psi_{N}\phi(\alpha_{1})=\lambda\alpha_{2}$ and $\psi_{N}\phi(\beta_{1})=\mu\beta_{2}$. Extending by the identity, we find a Hodge isometry $\psi\in\operatorname{SO}(\widetilde{\rm H}(X_{2},{\mathbf{Q}}))$ such that $\psi\phi\colon\widetilde{\rm H}(X_{1},{\mathbf{Q}})\overset{\sim}{\to}\widetilde{\rm H}(X_{2},{\mathbf{Q}})$ is a graded Hodge similitude. In particular, the induced map $\psi\phi\colon{\mathfrak{g}}(X_{1})\overset{\sim}{\to}{\mathfrak{g}}(X_{2})$ is graded, and $\psi\phi$ maps the grading element $h_{1}\in{\mathfrak{g}}(X_{1})$ to the grading element $h_{2}\in{\mathfrak{g}}(X_{2})$.

By the exact sequence (5) and by Hilbert 90, the element $\psi$ lifts to an element $\widetilde{\psi}\in\operatorname{GSpin}(\widetilde{\rm H}(X_{2},{\mathbf{Q}}))$. By Lemma 5.1 and Lemma 5.2, it induces automorphisms of the Hodge structures ${\rm H}^{\rm ev}(X_{2},{\mathbf{Q}})$ and ${\rm H}^{\rm odd}(X_{2},{\mathbf{Q}})$. Now by construction, the composition $\widetilde{\psi}\circ\Phi^{\rm H}$ defines isomorphisms

[TABLE]

which respect both the grading and the Hodge structure, so that they induce isomorphisms of Hodge structures ${\rm H}^{i}(X_{1},{\mathbf{Q}})\overset{\sim}{\to}{\rm H}^{i}(X_{2},{\mathbf{Q}})$, 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 ${\mathbf{C}}$ we have a ${\mathbf{Z}}/2{\mathbf{Z}}$-graded abelian group

[TABLE]

This is functorial for pull-back and proper pushforward, and carries a product structure. The group ${\mathrm{K}}^{0}_{{\rm top}}(X)$ is the Grothendieck group of topological vector bundles on the differentiable manifold $X^{\rm an}$. 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 ${\mathbf{Z}}/2{\mathbf{Z}}$-graded map

[TABLE]

which in even degree is given by $v_{X}^{\rm top}({\mathcal{F}})=\sqrt{{\rm Td}_{X}}\cdot\operatorname{ch}({\mathcal{F}})$. The image of $v_{X}^{\rm top}$ is a ${\mathbf{Z}}$-lattice of full rank.

There is a ‘forgetful map’ ${\mathrm{K}}^{0}(X)\to{\mathrm{K}}_{\rm top}(X)$ from the Grothendieck group of algebraic vector bundles (or equivalently of the triangulated category ${\mathcal{D}}X$). This is compatible with pull back, multiplication, and proper pushfoward. The mukai vector $v_{X}\colon{\mathrm{K}}^{0}(X)\to{\rm H}(X,{\mathbf{Q}})$ factors over $v_{X}^{\rm top}$.

If ${\mathcal{P}}$ is an object in ${\mathcal{D}}(X\times Y)$ then convolution with its class in ${\mathrm{K}}_{\rm top}^{0}(X\times Y)$ defines a map $\Phi^{\mathrm{K}}_{\mathcal{P}}\colon{\mathrm{K}}_{\rm top}(X)\to{\mathrm{K}}_{\rm top}(Y)$, in such a way that the diagram

[TABLE]

commutes.

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 ${\mathrm{K}}^{0}_{G}(X)$ the Grothendieck group of $G$-equivariant algebraic vector bundles on $X$, or equivalently the Grothendieck group of the bounded derived category ${\mathcal{D}}_{G}X$ of $G$-equivariant coherent ${\mathcal{O}}_{X}$-modules. This is functorial for pull-back along $G$-equivariant maps and push-forward along $G$-equivariant proper maps.

Similarly, we have the $G$-equivariant topological $K$-theory

[TABLE]

where ${\mathrm{K}}_{{\rm top},G}^{0}(X)$ is the Grothendieck group of topological $G$-equivariant vector bundles.

There is a natural map ${\mathrm{K}}^{0}_{G}(X)\to{\mathrm{K}}^{0}_{{\rm top},G}(X)$ compatible with pull-back and tensor product. If $f\colon X\to Y$ is proper and $G$-equivariant, then we have a push-forward map $f_{\ast}\colon{\mathrm{K}}_{{\rm top},G}(X)\to{\mathrm{K}}_{{\rm top},G}(Y)$. There is a Riemann–Roch theorem [4, 29], stating that the square

[TABLE]

commutes.

Now assume that we have a finite group $G$ acting on $X$, and a finite group $H$ acting on $Y$. If ${\mathcal{P}}$ is an object in ${\mathcal{D}}_{G\times H}(X\times Y)$, then convolution with ${\mathcal{P}}$ induces a functor $\Phi_{\mathcal{P}}\colon{\mathcal{D}}_{G}X\to{\mathcal{D}}_{H}Y$, see [41] for more details. Similarly, convolution with the class of ${\mathcal{P}}$ in ${\mathrm{K}}^{0}_{{\rm top},G\times H}(X\times Y)$ induces a map $\Phi_{\mathcal{P}}^{\mathrm{K}}\colon{\mathrm{K}}_{{\rm top},G}(X)\to{\mathrm{K}}_{{\rm top},H}(Y)$. These satisfy the usual Fourier–Mukai calculus, and moreover they are compatible, in the sense that the square

[TABLE]

commutes.

7. 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.

7.1. Line bundles on the Hilbert square

Let $G=\{1,\sigma\}$ be the group of order two, acting on $S\times S$ by permuting the factors. The Hilbert square $X$ sits in a diagram

[TABLE]

where $p\colon Z\to S\times S$ is the blow-up along the diagonal, and where $q\colon Z\to X$ is the quotient map for the natural action of $G$ on $Z$. Denote by $R\subset Z$ the exceptional divisor of $p$. Then $R$ equals the ramification locus of $q$. We have $q_{\ast}{\mathcal{O}}_{Z}={\mathcal{O}}_{X}\oplus{\mathcal{E}}$ for some line bundle ${\mathcal{E}}$, and $q^{\ast}{\mathcal{E}}\cong{\mathcal{O}}_{Z}(-R)$.

If ${\mathcal{L}}$ is a line bundle on $S$ then

[TABLE]

is a line bundle on $X$. The map

[TABLE]

is an isomorphism.

7.2. Cohomology of the Hilbert square

There is an isomorphism

[TABLE]

with the property that $c_{1}({\mathcal{L}})$ is mapped to $c_{1}({\mathcal{L}}_{2})$, and $\delta$ is mapped to $c_{1}({\mathcal{E}})$. We will use this isomorphism to identify ${\rm H}^{2}(S,{\mathbf{Z}})\oplus{\mathbf{Z}}\delta$ with ${\rm H}^{2}(X,{\mathbf{Z}})$. The Beauville–Bogomolov form on ${\rm H}^{2}(X,{\mathbf{Z}})$ satisfies

[TABLE]

for all $\lambda\in{\rm H}^{2}(S,{\mathbf{Z}})$.

Cup product defines an isomorphism $\operatorname{Sym}^{2}{\rm H}^{2}(X,{\mathbf{Q}})\overset{\sim}{\to}{\rm H}^{4}(X,{\mathbf{Q}})$. By Poincaré duality, there is a unique $q_{X}\in{\rm H}^{4}(X,{\mathbf{Q}})$ representing the Beauville–Bogomolov form, in the sense that

[TABLE]

for all $\lambda_{1},\lambda_{2}\in{\rm H}^{2}(X,{\mathbf{Z}})$. Multiplication by $q_{X}$ defines an isomorphism ${\rm H}^{2}(X,{\mathbf{Q}})\to{\rm H}^{6}(X,{\mathbf{Q}})$ and for all $\lambda_{1},\lambda_{2},\lambda_{3}$ in ${\rm H}^{2}(X,{\mathbf{Q}})$ we have

[TABLE]

in ${\rm H}^{6}(X,{\mathbf{Q}})$. Finally, for all $\lambda\in{\rm H}^{2}(X,{\mathbf{Q}})$ the Fujiki relation

[TABLE]

holds.

7.3. Todd class of the Hilbert square

Proposition 7.1**.**

${\rm Td}_{X}=1+\frac{5}{2}q_{X}+3[{\rm pt}]$.

Proof.

See also [23, §23.4]. Since the Todd class is invariant under the monodromy group of $X$, we necessarily have

[TABLE]

for some $s,t\in{\mathbf{Q}}$. By Hirzebruch–Riemann–Roch, for every line bundle $L$ on $S$ with $c_{1}(L)=\lambda$ we have

[TABLE]

By the relations (9) and (7) the right hand side reduces to

[TABLE]

By [23, §23.4] or [18, 5.1] the left hand side computes to

[TABLE]

Comparing the two expressions yields the result. ∎

8. 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 $p\colon Z\to S\times S$ and $q\colon Z\to X$, and the group $G=\{1,\sigma\}$ acting on $S\times S$ and $Z$.

The derived McKay correspondence [11] is the triangulated functor ${\rm BKR}\colon{\mathcal{D}}^{b}(X)\to{\mathcal{D}}^{b}_{G}(S\times S)$ given as the composition

[TABLE]

where the first functor maps ${\mathcal{F}}$ to $q^{\ast}{\mathcal{F}}$ equipped with the trivial $G$-linearization. By [11, 1.1] and [21, 5.1] the functor ${\rm BKR}$ is an equivalence of categories. Its inverse is given by:

Proposition 8.1**.**

The inverse equivalence of ${\rm BKR}$ is given by

[TABLE]

for all ${\mathcal{F}}$ in ${\mathcal{D}}_{G}(S\times S)$.

Proof.

This follows from combining [33, 4.1] with the projection formula for $q$ and the fact that $q^{\ast}{\mathcal{E}}\cong{\mathcal{O}}_{Z}(-R)$. ∎

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 $\Phi\colon{\mathcal{D}}S_{1}\overset{\sim}{\to}{\mathcal{D}}S_{2}$ induces an equivalence ${\mathcal{D}}_{G}(S_{1}\times S_{2})\overset{\sim}{\to}{\mathcal{D}}_{G}(S_{2}\times S_{2})$, and hence via the derived McKay correspondence an equivalence $\Phi^{[2]}\colon{\mathcal{D}}X_{1}\overset{\sim}{\to}{\mathcal{D}}X_{2}$.

8.2. Topological $K$-theory of the Hilbert square

Theorem 8.2**.**

The composition

[TABLE]

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 ${\rm BKR}$ and its inverse are given by kernels ${\mathcal{P}}\in{\mathcal{D}}_{G}(X\times S\times S)$ and ${\mathcal{Q}}\in{\mathcal{D}}_{G}(S\times S\times X)$. The map ${\rm BKR}_{\rm top}$ is given by convolution with the class of ${\mathcal{P}}$ in ${\mathrm{K}}^{0}_{{\rm top},G}(X\times S\times S)$. The identities in ${\mathrm{K}}^{0}(X\times X)$ and ${\mathrm{K}}^{0}_{G\times G}(S\times S\times S\times S)$ witnessing that ${\mathcal{P}}$ and ${\mathcal{Q}}$ are mutually inverse equivalences induce analogous identities in ${\mathrm{K}}^{0}_{\rm top}$. These show that convolution with the class of ${\mathcal{Q}}$ defines a two-sided inverse to ${\rm BKR}_{\rm top}$. ∎

Consider the map

[TABLE]

obtained as the composition of ${\rm BKR}_{\rm top}$ and the forgetful map from ${\mathrm{K}}^{0}_{{\rm top},G}(S\times S)$ to ${\mathrm{K}}^{0}_{\rm top}(S\times S)$. Also, consider the map

[TABLE]

where $[{\mathcal{F}}\boxtimes{\mathcal{F}},\pm 1]$ denotes the class of the topological vector bundle ${\mathcal{F}}\boxtimes{\mathcal{F}}$ equipped with $\pm$ the natural $G$-linearisation.

By construction, these maps are ‘functorial’ in ${\mathcal{D}}S$, in the following sense:

Proposition 8.3**.**

If $\Phi\colon{\mathcal{D}}S_{1}\overset{\sim}{\to}{\mathcal{D}}S_{2}$ is a derived equivalence between K3 surfaces, and $\Phi^{[2]}\colon{\mathcal{D}}X_{1}\overset{\sim}{\to}{\mathcal{D}}X_{2}$ the induced equivalence between their Hilbert squares, then the squares

[TABLE]

commute.∎

Proposition 8.4**.**

The sequence

[TABLE]

is exact.

Proof.

In the proof, we will implicitly identify ${\mathrm{K}}_{{\rm top},G}(S\times S)$ and ${\mathrm{K}}_{\rm top}(X)$.

Note that the map $\theta^{\mathrm{K}}$ is additive. Indeed, let ${\mathcal{F}}_{1}$ and ${\mathcal{F}}_{2}$ be (topological) vector bundles on $S$. Then the cross term $\theta^{{\mathrm{K}}}[{\mathcal{F}}_{1}\oplus{\mathcal{F}}_{2}]-\theta^{{\mathrm{K}}}[{\mathcal{F}}_{1}]-\theta^{{\mathrm{K}}}[{\mathcal{F}}_{2}]$ computes to

[TABLE]

which vanishes because the matrices $\left(\begin{smallmatrix}0&1\\ 1&0\end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}0&-1\\ -1&0\end{smallmatrix}\right)$ are conjugated over ${\mathbf{Z}}$.

The composition $\psi^{\mathrm{K}}\theta^{\mathrm{K}}$ vanishes, and since the Schur multiplier of $G$ is trivial, the map $\psi^{\mathrm{K}}$ is surjective. Computing the ${\mathbf{Q}}$-dimensions one sees that it suffices to show that $\theta^{\mathrm{K}}$ is injective.

Pulling back to the diagonal and taking invariants defines a map

[TABLE]

This composition computes to

[TABLE]

This coincides with the second Adams operation, which is injective on ${\mathrm{K}}^{0}_{\rm top}(S)\otimes_{\mathbf{Z}}{\mathbf{Q}}$, since it has eigenvalues $1$, $2$, and $4$. We conclude that $\theta^{\mathrm{K}}$ is injective, and the proposition follows. ∎

Remark 8.5**.**

One can show that the sequence

[TABLE]

with integral coefficients is exact.

8.3. 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.

Consider the map $\theta^{\rm H}\colon{\rm H}(S,{\mathbf{Q}})\to{\rm H}(X,{\mathbf{Q}})$ given by

[TABLE]

for all $s,t\in{\mathbf{Q}}$ and $\lambda\in{\rm H}^{2}(S,{\mathbf{Q}})$. See § 7.2 for the definition of $\delta\in{\rm H}^{2}(X,{\mathbf{Q}})$ and $q_{X}\in{\rm H}^{4}(X,{\mathbf{Q}})$.

Proposition 8.6**.**

The square

[TABLE]

commutes.

Proof.

Since ${\mathrm{K}}^{0}_{\rm top}(S)\otimes_{\mathbf{Z}}{\mathbf{Q}}$ is additively generated by line bundles, it suffices to show

[TABLE]

for a topological line bundle ${\mathcal{L}}$ with $\lambda=c_{1}({\mathcal{L}})$. Deforming $S$ if necessary, we may assume that ${\mathcal{L}}$ is algebraic.

Using Proposition 8.1 and the fact that the natural map

[TABLE]

is an isomorphism of ${\mathcal{O}}_{X}$-modules we find

[TABLE]

We conclude that $\theta^{\mathrm{K}}$ maps ${\mathcal{L}}$ to $[{\mathcal{L}}_{2}](1-[{\mathcal{E}}^{-1}])$ in ${\mathrm{K}}^{0}(X)$.

We compute its image under $v_{X}$. Using the formula for the Todd class from Proposition 7.1 we find

[TABLE]

Since $1-e^{-\delta}$ has no term in degree [math], the degree $8$ part of the square root of the Todd class is irrelevant, so we have

[TABLE]

By the Fujiki relation (9) from § 7.2, we have $\lambda^{3}\delta=0$, so the above can be rewritten as

[TABLE]

Since we have $q_{X}\delta\lambda=b(\delta,\lambda)=0$, we can rewrite this further as

[TABLE]

Comparing this with the right-hand-side of (11) we see that it suffices to show

[TABLE]

in ${\rm H}(X,{\mathbf{Q}})$. This boils down to the identities

[TABLE]

in ${\rm H}^{6}(X,{\mathbf{Q}})$ and ${\rm H}^{8}(X,{\mathbf{Q}})$ 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 $X^{\prime}$, a proper smooth family ${\mathcal{X}}\to B$ , a path $\gamma\colon[0,1]\to{\mathcal{X}}$ and isomorphisms $X\overset{\sim}{\to}{\mathcal{X}}_{\gamma(0)}$ and $X^{\prime}\overset{\sim}{\to}{\mathcal{X}}_{\gamma(1)}$. We will informally say that $X^{\prime}$ as a deformation of $X$, the other data being implicitly understood. Parallel transport along $\gamma$ defines an isomorphism ${\rm H}(X,{\mathbf{Q}})\overset{\sim}{\to}{\rm H}(X^{\prime},{\mathbf{Q}})$.

If $X^{\prime}$ and $X^{\prime\prime}$ are deformations of $X$, and if $\phi\colon X^{\prime}\to X^{\prime\prime}$ is an isomorphism of projective varieties, then we obtain a composite isomorphism

[TABLE]

We call such isomorphism a monodromy operator for $X$, and denote by $\operatorname{Mon}(X)$ the subgroup of $\operatorname{GL}({\rm H}(X,{\mathbf{Q}}))$ generated by all monodromy operators.

If $X^{\prime}$ and $X^{\prime\prime}$ are deformations of $X$, and if $\Phi\colon{\mathcal{D}}X^{\prime}\overset{\sim}{\to}{\mathcal{D}}X^{\prime\prime}$ is an equivalence, then we obtain an isomorphism

[TABLE]

We call such isomorphism a derived monodromy operator for $X$, and denote by $\operatorname{DMon}(X)$ the subgroup of $\operatorname{GL}({\rm H}(X,{\mathbf{Q}}))$ 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 $\rho_{X}$, and we have a commutative square of groups

[TABLE]

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 $X$.

Proposition 9.2**.**

If $S$ is a K3 surface, then $\operatorname{DMon}(S)=\operatorname{O}^{+}(\widetilde{\rm H}(S,{\mathbf{Z}}))$.

Proof.

Indeed, if $\Phi\colon{\mathcal{D}}S_{1}\to{\mathcal{D}}S_{2}$ is an equivalence, then

[TABLE]

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 $\operatorname{O}^{+}(\widetilde{\rm H}(S,{\mathbf{Z}}))$.

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 $\operatorname{O}^{+}(\widetilde{\rm H}(S,{\mathbf{Z}}))$ is generated by relfections in $-2$ vectors $\delta$. By the Torelli theorem, any such $-2$-vector will become algebraic on a suitable deformation $S^{\prime}$ of $S$, and by [32] there exists a spherical object ${\mathcal{E}}$ on $S^{\prime}$ with Mukai vector $v({\mathcal{E}})=\delta$. The spherical twist in ${\mathcal{E}}$ then shows that reflection in $\delta$ is indeed a derived monodromy operator. ∎

9.2. Action of $\operatorname{DMon}(S)$ on ${\rm H}(X,{\mathbf{Q}})$

By the derived McKay correspondence, any derived equivalence $\Phi_{S}\colon{\mathcal{D}}S_{1}\overset{\sim}{\to}{\mathcal{D}}S_{2}$ between K3 surfaces induces a derived equivalence $\Phi_{X}\colon{\mathcal{D}}X_{1}\overset{\sim}{\to}{\mathcal{D}}X_{2}$ between the corresponding Hilbert squares. By Propositions 8.3 and 8.4, the induced map $\Phi_{X}^{H}$ only depends on $\Phi_{S}^{H}$. Since any deformation of a K3 surface $S$ induces a deformation of $X=S^{[2]}$, we conclude that we have a natural homomorphism

[TABLE]

and hence an action of $\operatorname{DMon}(S)$ on ${\rm H}(X,{\mathbf{Q}})$. In this paragraph, we will explicitly compute this action. As a first approximation, we determine the $\operatorname{DMon}(S)$-module structure of ${\rm H}(X,{\mathbf{Q}})$, up to isomorphism.

Proposition 9.3**.**

We have ${\rm H}(X,{\mathbf{Q}})\cong\widetilde{{\rm H}}(S,{\mathbf{Q}})\oplus\operatorname{Sym}^{2}\widetilde{{\rm H}}(S,{\mathbf{Q}})$ as representations of $\operatorname{DMon}(S)=\operatorname{O}^{+}(\widetilde{{\rm H}}(S,{\mathbf{Z}}))$.

Proof.

This follows from Propositions 8.3 and 8.4. ∎

Since ${\mathfrak{g}}(X)$ is a purely topological invariant, it is preserved under deformations. In particular, Theorem 4.11 implies that we have an inclusion $\operatorname{DMon}(X)\subset\operatorname{O}(\widetilde{\rm H}(X,{\mathbf{Q}}))$. We conclude there exists a unique map of algebraic groups $h$ making the square

[TABLE]

commute.

Theorem 9.4**.**

The map $h$ in the square (12) is given by

[TABLE]

with $\iota\colon\operatorname{O}(\widetilde{\rm H}(S,{\mathbf{Q}}))\to\operatorname{O}(\widetilde{\rm H}(X,{\mathbf{Q}}))$ the natural inclusion.

The proof of this theorem will occupy the remainder of this paragraph.

Consider the unique homomorphism of Lie algebras $\iota\colon{\mathfrak{g}}(S)\to{\mathfrak{g}}(X)$ that respects the grading and maps $e_{\lambda}$ to $e_{\lambda}$ for all $\lambda\in{\rm H}^{2}(S,{\mathbf{Q}})\subset{\rm H}^{2}(X,{\mathbf{Q}})$. Under the isomorphism of Theorem 3.1 this corresponds to the map $\operatorname{\mathfrak{so}}(\widetilde{\rm H}(S,{\mathbf{Q}}))\to\operatorname{\mathfrak{so}}(\widetilde{\rm H}(X,{\mathbf{Q}}))$ induced by the inclusion of quadratic spaces $\widetilde{\rm H}(S,{\mathbf{Q}})\subset\widetilde{\rm H}(X,{\mathbf{Q}})$.

Recall from Section 8.3 the map $\theta^{\rm H}\colon{\rm H}(S,{\mathbf{Q}})\to{\rm H}(X,{\mathbf{Q}})$.

Lemma 9.5**.**

The map $\theta^{\rm H}\colon{\rm H}(S,{\mathbf{Q}})\to{\rm H}(X,{\mathbf{Q}})$ is equivariant with respect to

[TABLE]

Proof.

We have $\theta^{\rm H}=e^{-\delta/2}\cdot\theta^{\rm H}_{0}$, with

[TABLE]

The map $\theta^{\rm H}_{0}$ respects the grading, and we claim that for every $\mu\in{\rm H}^{2}(S,{\mathbf{Q}})$ the diagram

[TABLE]

commutes. Indeed, we have

[TABLE]

One verifies easily that these agree, using identities (8) and (7) from § 7.2, and the fact that $b(\lambda,\delta)=b(\mu,\delta)=0$. This shows that the left-hand square commutes. The right-hand square commutes trivially, so the outer rectangle commutes, which shows that $\theta^{\rm H}=e^{-\delta/2}\cdot\theta^{\rm H}_{0}$ is indeed equivariant with respect to $\theta^{\mathfrak{g}}$. ∎

Lemma 9.6**.**

There is an isomorphism

[TABLE]

of representations of $G=\operatorname{O}(\widetilde{\rm H}(X,{\mathbf{Q}}))$.

Proof.

This follows from Lemma 3.7, Theorem 4.11 and Remark 4.12. ∎

We are now ready to prove the main result of this paragraph.

Proof of Theorem 9.4.

By Proposition 8.6 the map $\theta^{\rm H}$ is equivariant for the action of $\operatorname{DMon}(S)$. Lemma 9.5 then implies that

[TABLE]

for all $g\in\operatorname{SO}(\widetilde{\rm H}(S,{\mathbf{Q}}))$. We have an orthogonal decomposition

[TABLE]

with $C$ of rank $1$. Since $\operatorname{SO}(\widetilde{\rm H}(S,{\mathbf{Q}}))$ is normal in $\operatorname{O}(\widetilde{\rm H}(S,{\mathbf{Q}}))$, the action of $\operatorname{O}(\widetilde{\rm H}(S,{\mathbf{Q}}))$ (via $h$) must preserve this decomposition. With respect to this decomposition $h$ must then be given by

[TABLE]

where the $\epsilon_{i}(g)\colon\operatorname{O}(\widetilde{\rm H}(S,{\mathbf{Q}}))\to\{\pm 1\}$ are quadratic characters. This leaves four possibilities for $h$. One verifies that $\epsilon_{1}=\epsilon_{2}=\det g$ 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 $b\colon L\times L\to{\mathbf{Z}}$ be an even non-degenerate lattice. Let $U$ be a hyperbolic plane with basis consisting of isotropic vectors $\alpha$, $\beta$ satisfying $b(\alpha,\beta)=-1$.

As before, to a $\lambda\in L$ we associate the isometry $B_{\lambda}\in\operatorname{O}(U\oplus L)$ defined as

[TABLE]

for all $r,s\in{\mathbf{Z}}$ and $\mu\in L$. Let $\gamma$ be the isometry of $U\oplus L$ given by $\gamma(\alpha)=\beta$, $\gamma(\beta)=\alpha$, and $\gamma(\lambda)=-\lambda$ for all $\lambda\in L$.

Lemma 9.7**.**

Let $L$ be an even lattice containing a hyperbolic plane. Let $G\subset\operatorname{O}(U\oplus L)$ be the subgroup generated by $\gamma$ and by the $B_{\lambda}$ for all $\lambda\in L$. Then for all $\delta\in U\oplus L$ with $\delta^{2}=-2$ and for all $g\in\operatorname{O}(U\oplus L)$ there exists a $g^{\prime}\in G$ such that $g^{\prime}g$ fixes $\delta$.

Proof.

This follows from classical results of Eichler. A convenient modern source is [20, §3], whose notation we adopt. The isometry $B_{\lambda}$ coincides with the Eichler transvection $t(\beta,-\lambda)$. The conjugate $\gamma B_{\lambda}\gamma^{-1}$ is the Eicher transvection $t(\alpha,\lambda)$. Hence $G$ contains the subgroup $E_{U}(L)\subset\operatorname{O}(U\oplus L)$ of unimodular transvections with respect to $U$. By [20, Prop. 3.3], there exists a $g^{\prime}\in E_{U}(L)$ mapping $g\delta$ to $\delta$. ∎

9.4. Proof of Theorem E

Let $X$ be a hyperkähler variety of type $\mathrm{K3}^{[2]}$. Let $\delta\in{\rm H}^{2}(X,{\mathbf{Z}})$ be any class satisfying $\delta^{2}=-2$ and $b(\delta,\lambda)\in 2{\mathbf{Z}}$ for all $\lambda\in{\rm H}^{2}(X,{\mathbf{Z}})$. For example, if $X=S^{[2]}$, we may take $\delta=c_{1}({\mathcal{E}})$ as in § 7.2. Consider the integral lattice

[TABLE]

The sub-group $\Lambda\subset\widetilde{\rm H}(X,{\mathbf{Q}})$ does not depend on the choice of $\delta$. In this section, we will prove Theorem E. More precisely, we will show:

Theorem 9.8**.**

$\operatorname{O}^{+}(\Lambda)\subset\operatorname{DMon}(X)\subset\operatorname{O}(\Lambda)$.

We start with the lower bound.

Proposition 9.9**.**

We have $\operatorname{O}^{+}(\Lambda)\subset\operatorname{DMon}(X)$ as subgroups of $\operatorname{O}(\widetilde{\rm H}(X,{\mathbf{Q}}))$.

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 $\delta=c_{1}({\mathcal{E}})$ as in § 7.2.

The shift functor $[1]$ on ${\mathcal{D}}X$ acts as $-1$ on ${\rm H}(X,{\mathbf{Q}})$, which coincides with the action of $-1\in\operatorname{O}(\widetilde{\rm H}(X,{\mathbf{Q}}))$. In particular, $-1\in\operatorname{O}^{+}(\Lambda)$ lies in $\operatorname{DMon}(X)$, so it suffices to show that $\operatorname{SO}^{+}(\Lambda)$ is contained in $\operatorname{DMon}(X)$.

Consider the isometry $\gamma\in\operatorname{O}^{+}(\widetilde{\rm H}(S,{\mathbf{Q}}))$ given by $\gamma(\alpha)=-\beta$, $\gamma(\beta)=-\alpha$ and $\gamma(\lambda)=\lambda$ for all $\lambda\in{\rm H}^{2}(S,{\mathbf{Q}})$. Then $\det(\gamma)=-1$ and by Theorem 9.4 its image $h(\gamma)$ interchanges $B_{\delta/2}\alpha$ and $B_{\delta/2}\beta$ and acts by $-1$ on $B_{\delta/2}{\rm H}^{2}(X,{\mathbf{Z}})$. Since $\gamma$ lies in $\operatorname{DMon}(S)\subset\operatorname{O}(\widetilde{\rm H}(S,{\mathbf{Q}}))$, we have that $h(\gamma)$ lies in $\operatorname{DMon}(X)\subset\operatorname{O}(\widetilde{\rm H}(X,{\mathbf{Q}}))$.

Let $G\subset\operatorname{O}(\widetilde{\rm H}(X,{\mathbf{Q}}))$ be the subgroup generated by $h(\gamma)$ and the isometries $B_{\lambda}$ for $\lambda\in{\rm H}^{2}(X,{\mathbf{Z}})$. Clearly $G$ is contained in $\operatorname{DMon}(X)$.

Let $g$ be an element of $\operatorname{SO}^{+}(\Lambda)$, and consider the image $gB_{\delta/2}\delta$ of $B_{\delta/2}\delta$. By Lemma 9.7 there exists a $g^{\prime}\in G\subset\operatorname{DMon}(X)$ so that $g^{\prime}g$ fixes $B_{\delta/2}\delta$. But then $g^{\prime}g$ acts on

[TABLE]

with determinant $1$, and preserving the orientation of a maximal positive subspace. In particular, $g^{\prime}g$ lies in the image of $\operatorname{DMon}(S)\to\operatorname{DMon}(X)$, and we conclude that $g$ lies in $\operatorname{DMon}(X)$. ∎

The proof of the upper bound is now almost purely group-theoretical.

Proposition 9.10**.**

$\operatorname{SO}(\Lambda)$* is the unique maximal arithmetic subgroup of $\operatorname{SO}(\Lambda\otimes_{\mathbf{Z}}{\mathbf{Q}})$ containing $\operatorname{SO}^{+}(\Lambda)$.*

Proof.

More generally, this holds for any even lattice $\Lambda$ with the property that the quadratic form $q(x)=b(x,x)/2$ on the ${\mathbf{Z}}$-module $\Lambda$ is semi-regular [30, § IV.3].

For such $\Lambda$, the group schemes $\operatorname{\mathbf{Spin}}(\Lambda)$ and $\operatorname{\mathbf{SO}}(\Lambda)$ are smooth over $\operatorname{Spec}{\mathbf{Z}}$, see e.g. [28]. In particular, for every prime $p$ the subgroups $\operatorname{Spin}(\Lambda\otimes{\mathbf{Z}}_{p})$ and $\operatorname{SO}(\Lambda\otimes{\mathbf{Z}}_{p})$ of $\operatorname{Spin}(\Lambda\otimes{\mathbf{Q}}_{p})$ resp. $\operatorname{SO}(\Lambda\otimes{\mathbf{Q}}_{p})$ are maximal compact subgroups. It follows that the groups

[TABLE]

and

[TABLE]

are maximal arithmetic subgroups of $\operatorname{Spin}(\Lambda\otimes{\mathbf{Q}})$ resp. $\operatorname{SO}(\Lambda\otimes{\mathbf{Q}})$.

The subgroup $\operatorname{SO}^{+}(\Lambda)\subset\operatorname{SO}(\Lambda)$ is the kernel of the spinor norm, and the short exact sequence $1\to\mu_{2}\to\operatorname{\mathbf{Spin}}\to\operatorname{\mathbf{SO}}\to 1$ of fppf sheaves on $\operatorname{Spec}{\mathbf{Z}}$ induces an exact sequence of groups

[TABLE]

Let $\Gamma\subset\operatorname{SO}(\Lambda\otimes{\mathbf{Q}})$ be a maximal arithmetic subgroup containing $\operatorname{SO}^{+}(\Lambda)$. Let $\widetilde{\Gamma}$ be its inverse image in $\operatorname{Spin}(\Lambda\otimes{\mathbf{Q}})$, so that we have an exact sequence

[TABLE]

Since the group $\widetilde{\Gamma}$ is arithmetic and contains $\operatorname{Spin}(\Lambda)$, we have $\widetilde{\Gamma}=\operatorname{Spin}(\Lambda)$. Moreover, $\Gamma$ normalizes $\operatorname{SO}^{+}(\Lambda)=\ker(\Gamma\to{\mathbf{Q}}^{\times}/2)$, and as the normalizer of an arithmetic subgroup of $\operatorname{SO}(\Lambda\otimes{\mathbf{Q}})$ is again arithmetic, we have that $\Gamma$ must equal the normalizer of $\operatorname{SO}^{+}(\Lambda)$. But then $\Gamma$ contains $\operatorname{SO}(\Lambda)$, and we conclude $\Gamma=\operatorname{SO}(\Lambda)$. ∎

Corollary 9.11**.**

$\operatorname{DMon}(X)\subset\operatorname{O}(\Lambda)$.

Proof.

$\operatorname{DMon}(X)$ preserves the integral lattice ${\mathrm{K}}_{\rm top}(X)$ in the representation ${\rm H}(X,{\mathbf{Q}})$ of $\operatorname{O}(\widetilde{\rm H}(X,{\mathbf{Q}}))$, and hence is contained in an arithmetic subgroup of $\operatorname{O}(\widetilde{\rm H}(X,{\mathbf{Q}}))=\operatorname{SO}(\widetilde{\rm H}(X,{\mathbf{Q}}))\times\{\pm 1\}$. By Proposition 9.9 it contains $\operatorname{SO}^{+}(\Lambda)\times\{\pm 1\}$, so we conclude from the preceding proposition that $\operatorname{DMon}(X)$ must be contained in $\operatorname{O}(\Lambda)$. ∎

Together with Proposition 9.9 this proves Theorem 9.8.

10. The image of $\operatorname{Aut}({\mathcal{D}}X)$ on ${\rm H}(X,{\mathbf{Q}})$

10.1. Upper bound for the image of $\rho_{X}$

We continue with the notation of the previous section. In particular, we denote by $X$ a hyperkähler variety of type $\mathrm{K3}^{[2]}$, and by $\Lambda\subset\widetilde{\rm H}(X,{\mathbf{Q}})$ the lattice defined in § 9.4. We equip $\widetilde{\rm H}(X,{\mathbf{Q}})$ with the weight [math] Hodge structure

[TABLE]

We denote by $\operatorname{Aut}(\Lambda)\subset\operatorname{O}(\Lambda)$ the group of isometries of $\Lambda$ that preserve this Hodge structure.

Proposition 10.1**.**

$\operatorname{im}(\rho_{X})\subset\operatorname{Aut}(\Lambda)$.

Proof.

By Theorem 9.8 we have $\operatorname{im}(\rho_{X})\subset\operatorname{O}(\Lambda)$. The Hodge structure on

[TABLE]

induces a Hodge structure on ${\mathfrak{g}}(X)\subset\operatorname{End}({\rm H}(X,{\mathbf{Q}}))$, which agrees with the Hodge structure on $\operatorname{\mathfrak{so}}(\widetilde{\rm H}(X,{\mathbf{Q}}))$ induces by the Hodge structure on $\widetilde{\rm H}(X,{\mathbf{Q}})$. If $\Phi\colon{\mathcal{D}}X\overset{\sim}{\to}{\mathcal{D}}X$ is an equivalence, then $\Phi^{\rm H}\colon{\rm H}(X,{\mathbf{Q}})\overset{\sim}{\to}{\rm H}(X,{\mathbf{Q}})$ and $\Phi^{\mathfrak{g}}\colon{\mathfrak{g}}(X)\overset{\sim}{\to}{\mathfrak{g}}(X)$ are isomorphisms of ${\mathbf{Q}}$-Hodge structures, from which it follows that $\Phi^{\rm H}$ must land in $\operatorname{Aut}(\Lambda)\subset\operatorname{O}(\Lambda)$. ∎

10.2. Lower bound for the image of $\rho_{X}$

We write $\operatorname{Aut}^{+}(\Lambda)$ for the index $2$ subgroup $\operatorname{Aut}(\Lambda)\cap\operatorname{O}^{+}(\Lambda)$ of $\operatorname{Aut}(\Lambda)$.

Theorem 10.2**.**

Let $S$ be a K3 surface and let $X$ be the Hilbert square of $S$. Assume that ${\rm NS}(X)$ contains a hyperbolic plane. Then $\operatorname{Aut}^{+}(\Lambda)\subset\operatorname{im}\rho_{X}\subset\operatorname{Aut}(\Lambda)$.

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

[TABLE]

The shift functor $[1]\in\operatorname{Aut}({\mathcal{D}}X)$ maps to $-1\in\operatorname{Aut}^{+}(\Lambda)$, so it suffices to show that $\operatorname{Aut}^{+}(\Lambda)\cap\operatorname{SO}(\Lambda)$ is contained in $\operatorname{im}\rho_{X}$.

Let $\gamma_{S}\in\operatorname{Aut}({\mathcal{D}}S)$ be the composition of the spherical twist in ${\mathcal{O}}_{S}$ with the shift $[1]$. On the Mukai lattice $\widetilde{\rm H}(S,{\mathbf{Z}})={\mathbf{Z}}\alpha\oplus{\rm H}^{2}(X,{\mathbf{Z}}(1))\oplus{\mathbf{Z}}\beta$ this equivalence maps $\alpha$ to $-\beta$ and $\beta$ to $-\alpha$ and is the identity on ${\rm H}^{2}(S,{\mathbf{Z}})$. Under the derived McKay correspondence this induces an autoequivalence $\gamma_{X}\in\operatorname{Aut}{\mathcal{D}}X$. By Theorem 9.4, the automorphism $\rho_{X}(\gamma_{X})\in\operatorname{Aut}(\Lambda)$ interchanges $B_{\delta/2}\alpha$ and $B_{\delta/2}\beta$ and acts by $-1$ on $B_{\delta/2}{\rm H}^{2}(X,{\mathbf{Z}})$.

Denote by $G\subset\operatorname{Aut}(\Lambda)$ the subgroup generated by $\rho_{X}(\gamma_{X})$ and the isometries $B_{\lambda}=\rho_{X}(-\otimes{\mathcal{L}})$ with ${\mathcal{L}}$ a line bundle of class $\lambda\in{\rm NS}(X)$. Clearly $G$ is contained in the image of $\rho_{X}$. Note that $G$ acts on the lattice

[TABLE]

and that by our assumption ${\rm NS}(X)$ contains a hyperbolic plane.

Let $g\in\operatorname{Aut}^{+}(\Lambda)$. By Lemma 9.7 applied to $L={\rm NS}(X)$, there exists a $g^{\prime}\in G$ such that $g^{\prime}g$ fixes $B_{\delta/2}\delta$. But then $g^{\prime}g$ acts on

[TABLE]

with determinant $1$, and preserving the Hodge structure and the orientation of a maximal positive subspace. In particular, $g^{\prime}g$ lies in the image of $\operatorname{Aut}({\mathcal{D}}S)$, and we conclude that $g$ lies in $\operatorname{im}\rho_{X}$. ∎