2.1 Hyper-Kähler manifolds and their cohomology

Let $X$ be a hyper-Kähler manifold of complex dimension $2n$, i.e. a simply connected compact Kähler manifold such that $\mathrm {H}^0(X,\Omega _X^2)$ is generated by an everywhere non-degenerate holomorphic two-form. The second cohomology $\mathrm {H}^2(X,{\mathbb {Z}})$ possesses an integral primitive quadratic form $b$ called the BBF form. It is characterised up to sign by the property that there exists a constant $c_X$, called the Fujiki constant, that only depends on the deformation type of $X$ such that

\[ \int_X \omega^{2n}=c_X \frac{(2n)!}{2^nn!}b(\omega, \omega)^n \]

for all $\omega \in \mathrm {H}^2(X,{\mathbb {Z}})$. For the known examples of hyper-Kähler manifolds we have

\[ c_X=\begin{cases} 1 & \mathrm{K3}^{[n]} \rm{ or } \mathrm{OG}^{10}\rm{-type,}\\ n+1 & \mathrm{Kum}^n \rm{ or } \mathrm{OG}^6\rm{-type.}\end{cases} \]

For the following, see [Reference Gross, Huybrechts and JoyceGHJ03, Corollary 23.17].

Proposition 2.1 Let $X$ be a hyper-Kähler manifold of dimension $2n$ and consider a class $\mu \in \mathrm {H}^{4p}(X,{\mathbb {R}})$ which is of type $(2p,2p)$ on all small deformations of $X$. Then there exists a constant $C(\mu )\in {\mathbb {R}}$ such that

\[ \int_X \mu \omega^{2n-2p}=C(\mu)b(\omega,\omega)^{n-p} \]

for all $\omega \in \mathrm {H}^2(X,{\mathbb {R}})$.

Using Rozansky–Witten theory, Nieper-Wißkirchen [Reference Nieper-WißkirchenNie03] established results on characteristic classes and Riemann–Roch formulae for hyper-Kähler manifolds.

Definition 2.2 For $\omega \in \mathrm {H}^2(X,{\mathbb {R}})$ define its characteristic value as

\[ \lambda(\omega):= \frac{(2n)!12c_X}{2^nn!(2n-1)C(\mathrm{c}_2(X))}b(\omega,\omega). \]

Using the Fujiki relations, one can check that this definition agrees with [Reference Nieper-WißkirchenNie03, Definition 17]. For the formula of the square root of the Todd class, we need the following result, cf. [Reference Nieper-WißkirchenNie03, p. 738].

Proposition 2.3 For $X$ a hyper-Kähler manifold of dimension $2n$ and arbitrary $\omega \in \mathrm {H}^2(X,{\mathbb {R}})$, the following holds:

\[ \int_X \mathsf{td}^{1/2} \exp(\omega)=(1+\lambda(\omega))^n\int_X\mathsf{td}^{1/2}. \]

2.2 Verbitsky component

Denote by $(\tilde {\rm {H}}(X,\mathbb {Q}),\tilde {b})$ the rational quadratic vector space defined by

\[ {\mathbb{Q}} \alpha \oplus \mathrm{H}^2(X,\mathbb{Q}) \oplus {\mathbb{Q}} \beta. \]

The quadratic form $\tilde {b}$ on $\tilde {\mathrm {H}}(X,{\mathbb {Q}})$ restricts to the BBF form $b$ on $\mathrm {H}^2(X,{\mathbb {Q}})$ and the two classes $\alpha$ and $\beta$ are orthogonal to $\mathrm {H}^2(X,{\mathbb {Q}})$ and satisfy $\tilde {b}(\alpha,\beta )=-1$ as well as $\tilde {b}(\alpha, \alpha )= \tilde {b}(\beta,\beta )=0$. Although it is not integral, we call $\tilde {\mathrm {H}}(X,{\mathbb {Q}})$ the extended Mukai lattice of $X$.

Furthermore, we define on $\tilde {\mathrm {H}}(X,{\mathbb {Q}})$ a grading by declaring $\alpha$ to be of degree zero, $\mathrm {H}^2(X,{\mathbb {Q}})$ remains in degree two and $\beta$ is of degree four. Finally, the extended Mukai lattice is equipped with a weight-two Hodge structure:

\begin{align*} (\tilde{\mathrm{H}}(X,{\mathbb{Q}}) \otimes_{\mathbb{Q}} {\mathbb{C}})^{2,0} &:= \mathrm{H}^{2,0}(X),\\ (\tilde{\mathrm{H}}(X,{\mathbb{Q}}) \otimes_{\mathbb{Q}} {\mathbb{C}})^{0,2} &:= \mathrm{H}^{0,2}(X),\\ (\tilde{\mathrm{H}}(X,{\mathbb{Q}}) \otimes_{\mathbb{Q}} {\mathbb{C}})^{1,1} &:= \mathrm{H}^{1,1}(X) \oplus {\mathbb{C}} \alpha \oplus {\mathbb{C}} \beta. \end{align*}

Let $\rm {SH}(X,\mathbb {Q})$ be the Verbitsky component, i.e. the graded subalgebra of $\rm {H}^*(X,\mathbb {Q})$ generated by $\rm {H}^2(X,\mathbb {Q})$. Verbitsky [Reference BogomolovBog96, Reference VerbitskyVer96] proved the existence of a graded morphism $\psi \colon \mathrm {SH}(X,{\mathbb {Q}}) \to \mathrm {Sym}^n(\tilde {\mathrm {H}}(X,{\mathbb {Q}}))$ sitting in a short exact sequence

\[ 0 \rightarrow \rm{SH}(X,\mathbb{Q}) \xrightarrow{\psi} \rm{Sym}^n(\tilde{\rm{H}}(X,\mathbb{Q})) \xrightarrow{\Delta} \rm{Sym}^{n-2}(\tilde{\mathrm{H}}(X,\mathbb{Q}))\rightarrow 0. \]

Here, the map $\Delta$ is the Laplacian operator defined on pure tensors via

\[ v_1 \cdots v_n \mapsto \sum_{i< j} \tilde{b}(v_i,v_j) v_1 \cdots \hat{v_i} \cdots \hat{v_j} \cdots v_n. \]

The map $\psi$ is uniquely determined (up to scaling) by the condition that it is a morphism of $\mathfrak {g}(X)$-modules, where $\mathfrak {g}(X)$ denotes the LLV algebra, see [Reference Looijenga and LuntsLL97] or [Reference Green, Kim, Laza and ColleenGKLC22]. Recall that ${\mathfrak {g}}(X)$ is the Lie algebra generated by all ${\mathfrak {s}}{\mathfrak {l}}_2$-triples $(e_{\omega },h,f_\omega )$ with $e_\omega$ the Lefschetz operator for $\omega \in \mathrm {H}^2(X,{\mathbb {Q}})$ satisfying the Hard Lefschetz property, $h$ the grading operator and $f_\omega$ the dual Lefschetz operator.

The ${\mathfrak {g}}(X)$-structure of $\tilde {\mathrm {H}}(X,{\mathbb {Q}})$ is defined by the conditions $e_\omega (\alpha )=\omega$, $e_\omega (\mu )=b(\omega,\mu )\beta$ and $e_\omega (\beta )=0$ for all classes $\omega, \mu \in \mathrm {H}^2(X,{\mathbb {Q}})$. The $n$th symmetric power $\mathrm {Sym}^n(\tilde {\mathrm {H}}(X,{\mathbb {Q}}))$ then inherits the structure of a ${\mathfrak {g}}(X)$-module by letting ${\mathfrak {g}}(X)$ act by derivations. The inclusion realises $\rm {SH}(X,\mathbb {Q})$ as an irreducible Lefschetz module [Reference VerbitskyVer96]. We fix once and for all a choice of $\psi$ by setting $\psi (1)=\alpha ^n/n!$.

Taelman [Reference TaelmanTae19, § 3] showed that the map $\psi$ is an isometry with respect to the Mukai pairing

\[ b_{\rm{SH}}(\omega_1 \cdots \omega_m, \mu_1 \cdots \mu_{2n-m})=(-1)^m \int_X \omega_1 \cdots \omega_m \mu_1 \cdots \mu_{2n-m} \]

on $\rm {SH}(X,\mathbb {Q})$ and the pairing

\[ b_{[n]}(x_1 \cdots x_n, y_1 \cdots y_n)=(-1)^n c_X \sum_{\sigma \in \mathfrak{S}_n}\prod_{i=1}^n \tilde{b}(x_i,y_{\sigma(i)}) \]

on $\rm {Sym}^n(\tilde {\mathrm {H}}(X,{\mathbb {Q}}))$. Note that our definition of $b_{[n]}$ differs from Taelman's definition by the Fujiki constant. Ours has the advantage that $\psi$ is always an isometry. The orthogonal projection onto the subspace $\mathrm {SH}(X,{\mathbb {Q}})$ will be denoted by

\[ T \colon \mathrm{Sym}^n(\tilde{\mathrm{H}}(X,{\mathbb{Q}}))\to \mathrm{SH}(X,{\mathbb{Q}}). \]

Bogomolov and Verbitsky [Reference BogomolovBog96, Reference VerbitskyVer96] showed that the Verbitsky component can also be described via

(2.1)\begin{equation} \mathrm{SH}(X,{\mathbb{C}})\cong\mathrm{Sym}^{\bullet}\mathrm{H}^2(X, {\mathbb{C}})/\langle \mu^{n+1}\mid b(\mu,\mu)=0 \rangle. \end{equation}

2.3 Derived equivalences

The following is [Reference TaelmanTae19, Theorem A].

Theorem 2.5 (Taelman)

Let $X$ and $Y$ be projective hyper-Kähler manifolds together with an equivalence $\Phi \colon \mathrm {D}^{\rm {b}}(X)\cong \mathrm {D}^{\rm {b}}(Y)$. Then $\Phi$ induces a canonical Lie algebra isomorphism

\[ \Phi^{{\mathfrak{g}}}\colon {\mathfrak{g}}(X) \cong {\mathfrak{g}}(Y), \]

which is equivariant for the induced isometry $\Phi ^{\mathrm {H}}\colon \mathrm {H}^\ast (X,{\mathbb {Q}}) \cong \mathrm {H}^\ast (Y,{\mathbb {Q}})$.

We reproduce some consequences of this result from [Reference TaelmanTae19, § 4] needed in the following. This theorem implies that given an auto-equivalence $\Phi \in \operatorname {Aut}(\mathrm {D}^{\rm {b}}(X))$, the induced action on cohomology

\[ \Phi^{\mathrm{H}} \colon \mathrm{H}^\ast(X,{\mathbb{Q}}) \cong \mathrm{H}^{\ast}(X,{\mathbb{Q}}) \]

restricts to a Hodge isometry

\[ \Phi^{\mathrm{SH}} \colon \mathrm{SH}(X,{\mathbb{Q}}) \cong \mathrm{SH}(X,{\mathbb{Q}}), \]

which is equivariant with respect to $\Phi ^{{\mathfrak {g}}}$. This yields a representation

\[ \rho^{\mathrm{SH}} \colon \operatorname{Aut}(\mathrm{D}^{\rm{b}}(X))\to \mathrm{O}(\mathrm{SH}(X,{\mathbb{Q}})). \]

Moreover, this representation $\rho ^{\mathrm {SH}}$ factors over a representation

\[ \rho^{\tilde{\mathrm{H}}} \colon \operatorname{Aut}(\mathrm{D}^{\rm{b}}(X))\to \mathrm{O}(\tilde{\mathrm{H}}(X,{\mathbb{Q}})) \]

under the assumption that $n$ is odd, or having $n$ even and $b_2(X)$ odd. Note that all known examples of hyper-Kähler manifolds satisfy one of the two conditions.

More precisely, for odd $n$ every Hodge isometry $\Phi ^{\mathrm {SH}}$ of $\rm {SH}(X,\mathbb {Q})$ is induced by an isometry of $\rm {Sym}^n(\tilde {\rm {H}}(X,\mathbb {Q}))$ which comes from a unique Hodge isometry $\Phi ^{\tilde {\mathrm {H}}}$ of $\tilde {\rm {H}}(X,\mathbb {Q})$ (see [Reference TaelmanTae19, Proposition 4.1]), i.e. the following diagram commutes.

For even $n$ and odd $b_2(X)$, there is an extra sign $\det (\Phi ^{\tilde {\mathrm {H}}})=\epsilon (\Phi ^{\tilde {\mathrm {H}}})\in \{\pm 1\}$ such that the following diagram commutes.

(2.2)

We refer to [Reference TaelmanTae19, § 4] for more details and proofs.

The process of associating to $\Phi ^\mathrm {SH}$ the isometry $\Phi ^{\tilde {\mathrm {H}}}$ is non-trivial. Given an equivalence we cannot say directly how it will act on $\tilde {\rm {H}}(X,\mathbb {Q})$, e.g. there is no obvious cycle associated to the kernel of the equivalence. We circumvent this obstacle by using the extended Mukai vector.

4.1 Square $-2r_X$

Let $X$ be a hyper-Kähler manifold of dimension $2n$. A line bundle ${\mathcal {L}} \in \mathop {\rm Pic}\nolimits (X)$ naturally induces an auto-equivalence

\[ \mathsf{M}_{{\mathcal{L}}} := \_ \otimes {\mathcal{L}} \in \operatorname{Aut}(\mathrm{D}^{\rm{b}}(X)). \]

Its action $\mathsf {M}_{\mathcal {L}}^{\mathrm {H}}$ on singular cohomology is given by multiplication with the Chern character of ${\mathcal {L}}$, i.e.

\[ \mathsf{M}_{\mathcal{L}}^\mathrm{H}=\_ \cdot \exp(\lambda) \in \operatorname{End}(\mathrm{H}^\ast(X,{\mathbb{Q}})), \]

where $\lambda =\mathrm {c}_1({\mathcal {L}})\in \mathrm {H}^2(X,{\mathbb {Z}})$. Furthermore, denote by $B_\lambda \in \mathrm {O}(\tilde {\rm {H}}(X,\mathbb {Q}))$ the isometry defined by

\[ B_{\lambda}(r\alpha + \mu + s\beta)=r\alpha + \mu +r\lambda +\biggl(s + b(\lambda,\mu) + r\frac{b(\lambda,\lambda)}{2} \biggr) \beta. \]

As checked in [Reference TaelmanTae19, Proposition 3.2] we have $\mathsf {M}_{\mathcal {L}}^{\tilde {\mathrm {H}}}=B_{\lambda }$, i.e. if we restrict $\mathsf {M}_{\mathcal {L}}^{\mathrm {H}}$ to $\mathrm {SH}(X,{\mathbb {Q}}) \subset \mathrm {H}^{\ast }(X,{\mathbb {Q}})$, then it is given by the natural action of $B_\lambda$ on $\mathrm {Sym}^n(\tilde {\mathrm {H}}(X,{\mathbb {Q}}))$ via the diagram

(4.1)

(where we set $\epsilon (\Phi ^{\tilde {\mathrm {H}}})=1$ for all equivalences if $n$ is odd). By definition,

\[ \alpha^n=\psi(n!\mathsf{1})=\psi(n!{\mathrm{ch}}({\mathcal{O}}_X))\in \mathrm{Sym}^n(\tilde{\mathrm{H}}(X,{\mathbb{Q}})), \]

which yields

\[ \psi({\mathrm{ch}}({\mathcal{L}}))=\frac{B_{\lambda}(\alpha)^n}{n!}= \frac{(\alpha + \lambda + (({b(\lambda, \lambda)})/{2}) \beta )^n}{n!} \in \mathrm{Sym}^n(\tilde{\mathrm{H}}(X,{\mathbb{Q}})). \]

We will upgrade this to the Mukai vector. An immediate consequence from Proposition 3.4 is that for the trivial line bundle ${\mathcal {O}}_X \in \mathop {\rm Pic}\nolimits (X)$ and its Mukai vector $v({\mathcal {O}}_X)$ one has

\[ \overline{v({\mathcal{O}}_X)}=T\biggl(\frac{(\alpha+r_X\beta)^n}{n!}\biggr) \in \mathrm{SH}(X,{\mathbb{Q}}), \]

where $\overline {(\_)}$ denotes again the projection from the cohomology $\mathrm {H}^\ast (X,{\mathbb {Q}})$ to the Verbitsky component $\mathrm {SH}(X,{\mathbb {Q}})$. Note that (2.2) also induces a commutative diagram

(4.2)

for all equivalences $\Phi \in \operatorname {Aut}(\mathrm {D}^{\rm {b}}(X))$. As $\mathsf {M}_{\mathcal {L}}({\mathcal {O}}_X)= {\mathcal {L}}$ and by the compatibility of the cohomological Fourier–Mukai transform $\mathsf {M}_{\mathcal {L}}^{\mathrm {H}}(v({\mathcal {O}}_X))=v({\mathcal {L}})$ we infer that

(4.3)\begin{equation} \overline{v({\mathcal{L}})}=T\left(\frac{(\alpha+\lambda+(r_X+{b(\lambda,\lambda)}/{2})\beta)^n}{n!}\right) \in \mathrm{SH}(X,{\mathbb{Q}}) \end{equation}

for all line bundles ${\mathcal {L}}\in \mathop {\rm Pic}\nolimits (X)$.

Definition 4.1 For ${\mathcal {L}}\in \mathop {\rm Pic}\nolimits (X)$ we define the extended Mukai vector of ${\mathcal {L}}$ with $\mathrm {c}_1({\mathcal {L}})=\lambda$ as

\[ \tilde{v}({\mathcal{L}})=\alpha + \lambda + \biggl( r_X + \frac{b(\lambda,\lambda)}{2} \biggr)\beta \in \tilde{\mathrm{H}}(X,{\mathbb{Q}}). \]

With this definition, we have

(4.4)\begin{equation} \overline{v({\mathcal{L}})}=T\biggl( \frac{\tilde{v}({\mathcal{L}})^n}{n!}\biggr) \in \mathrm{SH}(X,{\mathbb{Q}}). \end{equation}

Formula (4.4) is a helpful tool to deduce properties of $\Phi ^{\tilde {\mathrm {H}}}$ and compute its action on $\tilde {\mathrm {H}}(X,{\mathbb {Q}})$ for an auto-equivalence $\Phi \in \operatorname {Aut}(\mathrm {D}^{\rm {b}}(X))$. Here is one example.

If $n$ is even, then the functoriality of the representation $\rho ^{\tilde {\mathrm {H}}}\colon \operatorname {Aut}(\mathrm {D}^{\rm {b}}(X))\to \mathrm {O}(\tilde {\mathrm {H}}(X,{\mathbb {Q}}))$ depends on the determinant of the isometry of $\tilde {\mathrm {H}}(X,{\mathbb {Q}})$. A useful criterion to calculate $\epsilon (\Phi ^{\tilde {\mathrm {H}}})$ for an auto-equivalence $\Phi \in \operatorname {Aut}(\mathrm {D}^{\rm {b}}(X))$ is the following.

Instead of twists with line bundles, we can also use other auto-equivalences to define an extended Mukai vector for a larger set of objects.

Definition 4.3 Let $X$ be a projective hyper-Kähler manifold of dimension $2n$ with $n$ odd and ${\mathcal {E}}\in \mathrm {D}^{\rm {b}}(X)$ an object such that there exists an auto-equivalence $\Phi \in \operatorname {Aut}(\mathrm {D}^{\rm {b}}(X))$ with $\Phi ({\mathcal {O}}_X)\cong {\mathcal {E}}$. We define the extended Mukai vector of ${\mathcal {E}}$ as

\[ \tilde{v}({\mathcal{E}}) := \Phi^{\tilde{\mathrm{H}}}(\tilde{v}({\mathcal{O}}_X))\in \tilde{\mathrm{H}}(X,{\mathbb{Q}}). \]

This definition does not depend on the chosen equivalence. For such an object ${\mathcal {E}}$ we have the equality

(4.5)\begin{equation} \overline{v({\mathcal{E}})}=T\biggl(\frac{\tilde{v}({\mathcal{E}})^n}{n!}\biggr). \end{equation}

We say that such objects are in the ${\mathcal {O}}_X$-orbit. With this terminology, objects in the ${\mathcal {O}}_X$-orbit are cohomologically linearisable as in (4.5), which means that they admit an extended Mukai vector in the extended Mukai lattice, see also Definition 4.15.

Let $X$ be a projective hyper-Kähler manifold of dimension $2n$ with $n$ even and $b_2(X)$ odd and choose once and for all a very general Kähler class $\omega \in \mathrm {H}^2(X,{\mathbb {R}})$. Consider an object ${\mathcal {E}} \in \mathrm {D}^{\rm {b}}(X)$ such that there is an equivalence $\Phi \in \operatorname {Aut}(\mathrm {D}^{\rm {b}}(X))$ satisfying

\[ \Phi({\mathcal{O}}_X)\cong {\mathcal{E}}. \]

If the rank of ${\mathcal {E}}$ is strictly positive, then Lemma 4.2 forces $\epsilon (\Phi ^{\tilde {\mathrm {H}}})=1$ and for negative rank we obtain $\epsilon (\Phi ^{\tilde {\mathrm {H}}})=-1$. This motivates the following.

This definition is well-defined, i.e. it is independent of the chosen equivalence $\Phi$. Keeping the same notation, let us denote

\[ v=\Phi^{\tilde{\mathrm{H}}}(\tilde{v}({\mathcal{O}}_X))=r\alpha + \lambda + s\beta. \]

We define the signum $\mathrm {sgn}(v)\in \{\pm 1\}$ of the vector $v$. For $r\neq 0$ we set $\mathrm {sgn}(v):= \mathrm {sgn}(r)$ as the signum of the number $r$. If $r=0$, then the self-pairing of $\tilde {v}({\mathcal {O}}_X)$ forces $\lambda \neq 0$ and $c=b(\omega,\lambda )\neq 0$ as the Kähler class $\omega$ was assumed to be very general. We define in this case $\mathrm {sgn}(v):= \mathrm {sgn}(c)$.

Definition 4.6 The extended Mukai vector of ${\mathcal {E}}$ is

\[ \tilde{v}({\mathcal{E}}):= \epsilon(\Phi^{\tilde{\mathrm{H}}})\mathrm{sgn}(v)v. \]

We also say that such objects are in the ${\mathcal {O}}_X$-orbit. The extended Mukai vector $\tilde {v}({\mathcal {E}})$ satisfies a version of (4.5) namely

(4.6)\begin{equation} \overline{v({\mathcal{E}})}=\epsilon(\Phi^{\tilde{\mathrm{H}}})T\biggl( \frac{\tilde{v}({\mathcal{E}})^n}{n!} \biggr). \end{equation}

That is, objects in the ${\mathcal {O}}_X$-orbit are cohomologically linearisable, but for negative objects we have to add an extra sign.

Definition 4.6 for the case of even $n$ is up to sign compatible with derived equivalences, i.e. for ${\mathcal {E}}$ and ${\mathcal {F}}$ two objects in the ${\mathcal {O}}_X$-orbit and a derived equivalence $\Phi \in \operatorname {Aut}(\mathrm {D}^{\rm {b}}(X))$ with $\Phi ({\mathcal {E}})\cong {\mathcal {F}}$ we have

(4.7)\begin{equation} \Phi^{\tilde{\mathrm{H}}}(\tilde{v}({\mathcal{E}}))=\pm \tilde{v}({\mathcal{F}}) \in \tilde{\mathrm{H}}(X,{\mathbb{Q}}), \end{equation}

respectively,

(4.8)\begin{equation} (\Phi^{\tilde{\mathrm{H}}}(\tilde{v}({\mathcal{E}})))^n= (\tilde{v}({\mathcal{F}}))^n \in \mathrm{Sym}^n(\tilde{\mathrm{H}}(X,{\mathbb{Q}})). \end{equation}

We list some easy properties.

Proof. For the notion of ${\mathbb {P}}^n$-object and their properties see [Reference Huybrechts and ThomasHT06]. The pairing in property (ii) is the generalised Mukai product introduced in [Reference CăldăraruCăl05].

The first four points follow easily from the definitions and the fact that for an equivalence $\Phi \in \operatorname {Aut}(\mathrm {D}^{\rm {b}}(X))$ the induced isomorphism $\Phi ^{\tilde {\mathrm {H}}}$ of $\tilde {\mathrm {H}}(X,{\mathbb {Q}})$ is an isometry. Let $r=a^n$ be the rank of ${\mathcal {E}}$ and $\lambda =\mathrm {c}_1({\mathcal {E}})$ be its determinant. Equation (4.5) implies that we only have to determine the extended Mukai vector of ${\mathcal {E}}$. Using that the orthogonal projection $T$ is injective in degrees $0$ and $2$ we deduce that up to sign

\[ \tilde{v}({\mathcal{E}})=a\alpha + \frac{\lambda}{a^{n-1}} + c \beta\in \tilde{\mathrm{H}}(X,{\mathbb{Q}}) \]

for $c\in {\mathbb {Q}}$ uniquely determined by the property $\tilde {b}(\tilde {v}({\mathcal {E}}), \tilde {v}({\mathcal {E}}))=-2r_X$.

The lemma implies that the Chern classes of such objects are severely restricted.

4.2 Square $0$

There is another class of objects for which one can naturally define an extended Mukai vector. This does not involve Proposition 3.4.

Lemma 3.5 yields that the element $\beta \in \tilde {\mathrm {H}}(X,{\mathbb {Q}})$ has the property

\[ \beta^n=\psi(c_X {\mathsf{p}})\in \mathrm{SH}(X,{\mathbb{Q}}). \]

As we have, for a point $x \in X$,

\[ v(k(x))={\mathrm{ch}}(k(x))={\mathsf{p}} \in \mathrm{H}^{4n}(X,{\mathbb{Q}}), \]

we obtain the relation

\[ \psi(v(k(x)))=\frac{\beta^n}{c_X} \in \mathrm{Sym}^n(\tilde{\mathrm{H}}(X,{\mathbb{Q}})), \]

respectively,

(4.9)\begin{equation} v(k(x))=T \biggl( \frac{\beta^n}{c_X} \biggr) \in \mathrm{SH}(X,{\mathbb{Q}}). \end{equation}

Definition 4.10 For a point $x\in X$ and the associated skyscraper sheaf $k(x)$ we define its extended Mukai vector as

\[ \tilde{v}(k(x)):= \beta. \]

As in the case of objects in the ${\mathcal {O}}_X$-orbit this definition can be extended using derived equivalences.

Definition 4.11 Let $X$ be a projective hyper-Kähler manifold of dimension $2n$ with $n$ odd, ${\mathcal {E}}\in \mathrm {D}^{\rm {b}}(X)$ an object and $\Phi \in \operatorname {Aut}(\mathrm {D}^{\rm {b}}(X))$ such that $\Phi (k(x))\cong {\mathcal {E}}$ for some $x\in X$. We define the extended Mukai vector of ${\mathcal {E}}$ as

\[ \tilde{v}({\mathcal{E}}) := \Phi^{\tilde{\mathrm{H}}}(\beta) \in \tilde{\mathrm{H}}(X,{\mathbb{Q}}). \]

We say that such objects are in the $k(x)$-orbit. The analogous relation to (4.9) for objects in the $k(x)$-orbit reads

(4.10)\begin{equation} v({\mathcal{E}})=T\biggl( \frac{\tilde{v}({\mathcal{E}})^n}{c_X} \biggr) \in \mathrm{SH}(X,{\mathbb{Q}}). \end{equation}

Again in the case $n$ even and $b_2(X)$ odd one has to be more careful. Let ${\mathcal {E}}\in \mathrm {D}^{\rm {b}}(X)$ be such that there exists $\Phi \in \operatorname {Aut}(\mathrm {D}^{\rm {b}}(X))$ with $\Phi (k(x))\cong {\mathcal {E}}$ for some $x\in X$. Let us again write

\[ v=\Phi^{\tilde{\mathrm{H}}}(\beta)=r\alpha+\lambda+s\beta. \]

We define again the signum $\mathrm {sgn}(v)$ of the vector $v$. As before, for $r\neq 0$ we set $\mathrm {sgn}(v) := \mathrm {sgn}(r)$. In the case $r=0$ and $\lambda \neq 0$ the Hodge index theorem asserts that $c=b(\lambda,\omega )\neq 0$ for all Kähler classes $\omega$. We assign $\mathrm {sgn}(v) := \mathrm {sgn}(c)$. Finally, for $r=\lambda =0$ we define $\mathrm {sgn}(v) := \mathrm {sgn}(s)$.

Definition 4.12 The extended Mukai vector of ${\mathcal {E}}$ is defined as

\[ \tilde{v}({\mathcal{E}}):= \epsilon(\Phi^{\tilde{\mathrm{H}}})\mathrm{sgn}(v)v. \]

We also say that such objects are in the $k(x)$-orbit. As before, we have

(4.11)\begin{equation} v({\mathcal{E}})=\epsilon(\Phi^{\tilde{\mathrm{H}}})T\biggl( \frac{\tilde{v}({\mathcal{E}})^n}{c_X} \biggr)\in \mathrm{SH}(X,{\mathbb{Q}}) \end{equation}

and the formation of the extended Mukai vector is as in (4.7) and (4.8) functorial for derived equivalences.

Proof. The first three points follow easily from the definition and the fourth point is analogous to Lemma 4.8.

Suppose that the rank of ${\mathcal {E}}$ is zero and write

\[ \tilde{v}({\mathcal{E}})=\lambda + s\beta \in \tilde{\mathrm{H}}(X,{\mathbb{Q}}) \]

with $\lambda \in \mathrm {H}^{1,1}(X,{\mathbb {Q}})$ and $s\in {\mathbb {Q}}$. We will assume that $\lambda \neq 0$, the other case being trivial. As

\[ \tilde{b}(\tilde{v}({\mathcal{E}}),\tilde{v}({\mathcal{E}}))=0 \]

we infer that $b(\lambda,\lambda )=0$. Equation (4.10) gives

(4.12)\begin{equation} v({\mathcal{E}})=T\biggl( \frac{(\lambda + s \beta)^n}{c_X} \biggr) \in \mathrm{SH}(X,{\mathbb{Q}}), \end{equation}

which is supported in cohomological degrees ranging from $2n$ to $4n$. Up to a constant, the degree $2n$ component of $v({\mathcal {E}})$ equals the $n$th Chern character ${\mathrm {ch}}_n({\mathcal {E}})$, which again up to a constant equals the Chern class $\mathrm {c}_n({\mathcal {E}})$. From (4.12), we obtain

\[ \mathrm{c}_n({\mathcal{E}})=dT(\lambda^n) \in \mathrm{SH}^{2n}(X,{\mathbb{Q}}) \]

for some $d\in {\mathbb {Q}}$. As $\psi$ is a morphism of ${\mathfrak {g}}(X)$-modules, we obtain

\[ \psi(\lambda^n)=\psi(e_{\lambda}^n(\mathsf{1}))=e_{\lambda}^n \biggl( \frac{\alpha^n}{n!} \biggr) =\lambda^n \in \mathrm{Sym}^n(\tilde{\mathrm{H}}(X,{\mathbb{Q}})), \]

where the last equality used that $b(\lambda,\lambda )=0$. Therefore $T(\lambda ^n)=\lambda ^n\in \mathrm {SH}(X,{\mathbb {Q}})$ and so, in particular,

\[ \mathrm{c}_n({\mathcal{E}})=d \lambda^n. \]

The assertion of the lemma is therefore that $\sigma \mathrm {c}_n({\mathcal {E}})=d\sigma \lambda ^n=0$. We have

(4.13)\begin{equation} b(\lambda,\lambda)=b(\sigma,\sigma)=b(\lambda,\sigma)=0, \end{equation}

which shows that

(4.14)\begin{equation} (\lambda + \sigma)^{n+1} \in \mathrm{SH}^{2n+2}(X,{\mathbb{C}}) \end{equation}

must vanish by using (2.1). As each summand in (4.14) lies in a different piece of the Hodge decomposition, we deduce that $\lambda ^n\sigma =0$.

For $k > n$, induction on $k$ shows that $\sigma \mathrm {c}_k({\mathcal {E}}) = 0$ if and only if $\sigma v({\mathcal {E}})_{2k} = 0$ where $v({\mathcal {E}})_{2k}$ denotes the cohomological degree $2k$ part of the Mukai vector. Equation (4.12) gives that the degree $2k$ part of $v({\mathcal {E}})$ equals

\[ v({\mathcal{E}})_{2k}=\frac{s^{k-n}}{c_X}{n\choose k-n}T( \lambda^{2n-k}\beta^{k-n}) \in \mathrm{SH}^{2k}(X,{\mathbb{Q}}). \]

To determine the image of $\lambda ^{2n-k}\beta ^{k-n}$ under the orthogonal projection note that $T$ is as well a morphism of ${\mathfrak {g}}(X)$-modules. Similarly, one may show that

\[ \lambda^{2n-k}\beta^{k-n}=e_{\lambda}^{2n-k}\biggl( \frac{\alpha^{2n-k}\beta^{k-n}}{(2n-k)!} \biggr) \in \mathrm{Sym}^n(\tilde{\mathrm{H}}(X,{\mathbb{Q}})) \]

using again that $b(\lambda,\lambda )=0$. Lemma 3.5 implies that we have

\[ T(\alpha^{2n-k}\beta^{k-n})= (2n-k)!\mathsf{q}_{2k-2n}, \]

which yields

\[ v({\mathcal{E}})_{2k}=\frac{s^{k-n}}{c_X}{n \choose k-n}\lambda^{2n-k} \mathsf{q}_{2k-2n} \in \mathrm{SH}(X,{\mathbb{Q}}). \]

Ignoring constants we have to show that $\lambda ^{2n-k} \mathsf {q}_{2k-2n}\sigma \in \mathrm {SH}(X,{\mathbb {C}})$ vanishes, which is equivalent to

\[ \lambda^{2n-k} \mathsf{q}_{2k-2n} \sigma \mu^{2n-k-1}=0\in \mathrm{SH}^{4n}(X,{\mathbb{C}}) \]

for all $\mu \in \mathrm {H}^2(X,{\mathbb {Q}})$. This follows similarly using again (4.13) and the polarised version of the Fujiki relations, see Proposition 2.1.

Note that the number from Lemma 4.13(iii)

\[ \frac{a^nn!}{c_X} \]

must, in particular, be an integer. For all known examples of hyper-Kähler manifolds $c_X\in {\mathbb {Z}}$ and, therefore, we must already have $a\in {\mathbb {Z}}$ using Legendre's or de Polignac's formula.

4.3 Structural result

The discussion of the two previous subsections enables us to prove the following general structural result for derived equivalences between hyper-Kähler manifolds.

Proof. Let us first assume that either $n$ is odd or that $n$ is even and $b_2(X)$ odd.

We distinguish three cases depending on the image vector

\[ v=a\alpha + \lambda + s\beta \in \tilde{\mathrm{H}}(X,{\mathbb{Q}}) \]

of $\beta$ under $\Phi ^{\tilde {\mathrm {H}}}$. For $a \neq 0$, the assertion on the rank follows from Lemma 4.13.

In the case that $a=0$, but $\lambda \neq 0$, we consider the $n$th Chern class $\mathrm {c}_n({\mathcal {E}})\in \mathsf {A}^n(X\times Y)$ in the Chow ring with rational coefficients. The compatibility of derived equivalences with the induced maps between Chow and cohomology groups shows that for all $x\in X$ and all $y \in Y$ the cycles $\mathrm {c}_n({\mathcal {E}})|_{x\times Y}\in \mathsf {A}^n(Y)$ and $\mathrm {c}_n({\mathcal {E}})|_{X\times y}\in \mathsf {A}^n(X)$, respectively, are non-zero. Indeed the cohomological degree $2n$ component of $v({\mathcal {E}}_x)$ is equal to $\mathrm {c}_n({\mathcal {E}}_x)$ considered in cohomology which, by assumption, equals $\lambda ^n\in \mathrm {SH}(Y,{\mathbb {Q}})$ and similarly for $X$. This shows that viewing the cycle $\mathrm {c}_n({\mathcal {E}})$ as a family of cycles on $Y$ parametrised by points $x\in X$ these cycles cover $Y$ in the sense that for each $y\in Y$ the family of cycles $\mathrm {c}_n({\mathcal {E}})$ restricts non-trivially to the subvariety $X\times y$. As being isotropic is a cohomological property, the assertion follows now from Lemma 4.13(v).

Lastly, we assume that $v=s\beta$ for $s\in {\mathbb {Q}}$. This assumption implies that the element ${\mathsf {p}}\in \mathrm {H}^{4n}(X,{\mathbb {Q}})$ gets sent to $\pm s^n {\mathsf {p}} \in \mathrm {H}^{4n}(Y,{\mathbb {Q}})$ under the induced cohomological Fourier–Mukai transform $\Phi ^{\mathrm {H}}$. We can view the image of topological $K$-theory under the Mukai vector map $v(K_{\rm {top}}(X))$ as a lattice inside the full cohomology $\mathrm {H}^{\ast }(X,{\mathbb {Q}})$ equipped with the generalised Mukai pairing. We refer to [Reference Addington and ThomasAT14] for a recollection of topological $K$-theory and its relationship to derived categories and Fourier–Mukai transforms. The isomorphism $\Phi ^{\mathrm {H}}$ then induces an isometry between the lattices $v(K_{\rm {top}}(X))$ and $v(K_{\rm {top}}(Y))$. As ${\mathsf {p}} \in v(K_{\rm {top}}(X))$ is a primitive element, the same must be true for $\Phi ^{\mathrm {H}}({\mathsf {p}})=\pm s^n{\mathsf {p}}$. Therefore, $s\in \{\pm 1\}$.

We may assume without loss of generality that $s=1$. As $\Phi ^{\tilde {\mathrm {H}}}$ is an isometry, we infer that

\[ \Phi^{\tilde{\mathrm{H}}}(\alpha)= \alpha + \lambda + t\beta \in \tilde{\mathrm{H}}(Y,{\mathbb{Q}}). \]

We claim that already $\lambda \in \mathrm {H}^2(Y,{\mathbb {Z}})$. To see this, consider $\tilde {v}({\mathcal {O}}_X)=\alpha + r_X \beta$ and its image

\[ \Phi^{\tilde{\mathrm{H}}}(\tilde{v}({\mathcal{O}}_X))=\alpha + \lambda + (t+r_X)\beta \in \tilde{\mathrm{H}}(Y,{\mathbb{Q}}). \]

As previously, the element $v({\mathcal {O}}_X)$ belongs to $v(K_{\rm {top}}(X))$ and therefore $\Phi ^{\mathrm {H}}(v({\mathcal {O}}_X))$ must be contained in $v(K_{\rm {top}}(Y))$. Using that the projection of $v(K_{\rm {top}}(Y))$ to its degree-two component lands inside $\mathrm {H}^2(Y,{\mathbb {Z}})$ and the compatibility (4.2), we infer that also $\lambda$ belongs to $\mathrm {H}^2(Y,{\mathbb {Z}})$.

Employing that $\Phi ^{\tilde {\mathrm {H}}}$ is a Hodge isometry we furthermore conclude that $\lambda \in \mathrm {H}^{1,1}(Y,{\mathbb {Z}})$. Hence, there exists a line bundle ${\mathcal {L}}\in \mathop {\rm Pic}\nolimits (Y)$ with first Chern class $-\lambda$. Changing $\Phi$ by postcomposing it with $\mathsf {M}_{\mathcal {L}}$ we obtain a derived equivalence $\mathrm {D}^{\rm {b}}(X)\cong \mathrm {D}^{\rm {b}}(Y)$ still denoted by $\Phi$ which satisfies $\Phi ^{\tilde {\mathrm {H}}}(\alpha )=\alpha$ and $\Phi ^{\tilde {\mathrm {H}}}(\beta )=\beta$. Thus, $\Phi ^{\tilde {\mathrm {H}}}$ restricts to a Hodge isometry

\[ \mathrm{H}^2(X,{\mathbb{Q}}) \cong \mathrm{H}^2(Y,{\mathbb{Q}}). \]

It remains to show that this isometry sends $\mathrm {H}^2(X,{\mathbb {Z}})$ to $\mathrm {H}^2(Y,{\mathbb {Z}})$. We employ the same strategy as before. For $\lambda \in \mathrm {H}^2(X,{\mathbb {Z}})$ the vector

\[ \tilde{v}({\mathcal{L}})=\alpha + \lambda + \biggl( r_X + \frac{b(\lambda,\lambda)}{2} \biggr) \beta \]

can be viewed as the extended Mukai vector of the (topological) line bundle ${\mathcal {L}}$ with first Chern class $\lambda$. The Mukai vector $v({\mathcal {L}})$ lies inside $v(K_{\rm {top}}(X))$ and $\tilde {v}({\mathcal {L}})$ is mapped under $\Phi ^{\tilde {\mathrm {H}}}$ to

\[ \alpha + \Phi^{\tilde{\mathrm{H}}}(\lambda) + \biggl( r_X + \frac{b(\lambda,\lambda)}{2} \biggr)\beta. \]

As before, the compatibility with topological $K$-theory forces $\Phi ^{\tilde {\mathrm {H}}}(\lambda )$ to lie inside $\mathrm {H}^2(Y,{\mathbb {Z}})$. This finishes the proof in the case $n$ odd or $n$ even and $b_2(X)$ odd.

If we now assume that $n$ as well as $b_2(X)$ are odd, then we cannot apply the results from [Reference TaelmanTae19] as explained in § 2.3 directly. That is, given a derived equivalence $\Phi \colon \mathrm {D}^{\rm {b}}(X) \cong \mathrm {D}^{\rm {b}}(Y)$ the induced cohomological Fourier–Mukai transform $\Phi ^{\mathrm {H}} \colon \mathrm {H}^\ast (X,{\mathbb {Q}}) \cong \mathrm {H}^\ast (Y,{\mathbb {Q}})$ still restricts to a Hodge isometry

\[ \Phi^{\mathrm{SH}} \colon \mathrm{SH}(X,{\mathbb{Q}}) \cong \mathrm{SH}(Y,{\mathbb{Q}}), \]

but there may not exist a Hodge isometry $\varphi \in \mathrm {O}(\tilde {\mathrm {H}}(X,{\mathbb {Q}}))$ such that (2.2) commutes. However, inspecting [Reference TaelmanTae19, Proposition 4.1] and its proof we see that there exists a Hodge isometry $\varphi \in \mathrm {O}(\tilde {\mathrm {H}}(X,{\mathbb {Q}}))$ unique up to sign such that via (2.2) either $\Phi ^{\mathrm {SH}}$ or $-\Phi ^{\mathrm {SH}}$ agrees with $\varphi ^n$. Reinspecting the proof, we see that this sign discrepancy does not affect the arguments and the proof remains valid also in the case $n$ even and $b_2(X)$ even.

4.4 Concluding remarks and further examples

For general objects ${\mathcal {E}} \in \mathrm {D}^{\rm {b}}(X)$, we make the following definition.

Definition 4.15 An object ${\mathcal {E}} \in \mathrm {D}^{\rm {b}}(X)$ admits an extended Mukai vector $\tilde {v}({\mathcal {E}})\in \tilde {\mathrm {H}}(X,{\mathbb {Q}})$ if there exists $c\in {\mathbb {Q}}$ such that

\[ \overline{v({\mathcal{E}})}=cT(\tilde{v}({\mathcal{E}})^n) \in \mathrm{SH}(X,{\mathbb{Q}}). \]

With this definition, the vector $\tilde {v}({\mathcal {E}})$ is not uniquely defined. One rather considers a one-dimensional subspace $V\subset \tilde {\mathrm {H}}(X,{\mathbb {Q}})$ and demands that $\overline {v({\mathcal {E}})}$ lies in the one-dimensional subspace $T(V^n)$. In the two series of examples we considered a certain natural choice of $c\in {\mathbb {Q}}$ which then enabled us to define the extended Mukai vector as a uniquely determined vector in $\tilde {\mathrm {H}}(X,{\mathbb {Q}})$.

We give two observations of how one can generate new examples of objects admitting an extended Mukai vector from known ones.

• • If $\Phi \colon \mathrm {D}^{\rm {b}}(X)\cong \mathrm {D}^{\rm {b}}(Y)$ is an equivalence, then ${\mathcal {E}} \in \mathrm {D}^{\rm {b}}(X)$ admits an extended Mukai vector if and only if $\Phi ({\mathcal {E}})\in \mathrm {D}^{\rm {b}}(Y)$ admits an extended Mukai vector.

• • Let $\pi \colon {\mathcal {X}} \to B$ be a smooth and projective morphism with hyper-Kähler manifolds as fibres and ${\mathcal {E}}$ on ${\mathcal {X}}$ a $B$-flat sheaf or a $B$-perfect complex. For two points $b,b'\in B$ we have that ${\mathcal {E}}|_{{\mathcal {X}}_{b}}$ admits an extended Mukai vector if and only if ${\mathcal {E}}|_{{\mathcal {X}}_{b'}}$ admits an extended Mukai vector.

One can prove similar results as in Lemmas 4.8 and 4.13 for objects ${\mathcal {E}}\in \mathrm {D}^{\rm {b}}(X)$ satisfying Definition 4.15 for a fixed $c\in {\mathbb {Q}}$. We just mention that if ${\mathcal {E}}$ has zero rank, then the projections of all Chern classes of ${\mathcal {E}}$ to $\mathrm {SH}(X,{\mathbb {Q}})$ are isotropic. To see this one writes $\tilde {v}({\mathcal {E}})=\lambda +s\beta$ for $\lambda \in \mathrm {H}^2(X,{\mathbb {Q}})$ and $s\in {\mathbb {Q}}$ and uses that $e_{\sigma }(\tilde {v}({\mathcal {E}})) = e_{\bar {\sigma }}(\tilde {v}({\mathcal {E}}))=0$ for $\sigma$ and $\bar {\sigma }$ the (anti-)holomorphic two-form.

An important class of cohomologically linearisable objects are line bundles and skyscraper sheaves. We give further examples.

Example 4.16 Let $S$ be a projective K3 surface and ${\mathbb {P}}^1\cong C\subset S$ a smooth rational curve with class $l=[C]\in \mathrm {H}^2(S,{\mathbb {Z}})$. This yields a Lagrangian projective space ${\mathbb {P}}^n\cong C^{[n]}\subset S^{[n]}$ inside the Hilbert scheme of $n$ points. The proof of Proposition 7.2 will imply that the structure sheaf ${\mathcal {O}}_{{\mathbb {P}}^n}$ is in the ${\mathcal {O}}_{S^{[n]}}$-orbit. If we write $\mathrm {H}^2(S^{[n]},{\mathbb {Z}})=\mathrm {H}^2(S,{\mathbb {Z}}) \oplus {\mathbb {Z}} \delta$ where $2\delta$ is the class of the exceptional divisor, one has

\[ \tilde{v}({\mathcal{O}}_{{\mathbb{P}}^n})=l+\frac{\delta}{2} + \frac{n+1}{2}\beta \in \tilde{\mathrm{H}}(S^{[n]},{\mathbb{Q}}). \]

In particular, the projection $\overline {[{\mathbb {P}}^n]}$ of the class $[{\mathbb {P}}^n]$ to $\mathrm {SH}^{2n}(S^{[n]},{\mathbb {Q}})$ equals

\[ \overline{[{\mathbb{P}}^n]}=T\biggl(\frac{(l + {\delta}/{2})^n}{n!} \biggr) \in \mathrm{SH}^{2n}(S^{[n]},{\mathbb{Q}}). \]

This yields a partial answer to a question posed by Bakker [Reference BakkerBak17, Q. 29]. For more on this example, we refer to Proposition 7.2 and Remark 7.3.

Example 4.17 For a very general projective K3 surface $S$ of degree $2g-2$ we study in § 10.2 the case of the moduli space of stable sheaves $M=M^S_H(0,1,d+1-g)$ which admits naturally a Lagrangian fibration $\pi \colon M\to {\mathbb {P}}^g=|H|$. The general fibre $A$ is a smooth abelian variety and a degree-zero line bundle ${\mathcal {L}}$ supported on $A$ is an example of an object in the $k(x)$-orbit with

\[ \tilde{v}({\mathcal{L}})=f\in \tilde{\mathrm{H}}(M,{\mathbb{Q}}), \]

where $f\in \mathrm {H}^2(M,{\mathbb {Z}})$ is the image of the ample generator of $\mathop {\rm Pic}\nolimits ({\mathbb {P}}^g)$ under pullback via $\pi$. For $d=0$ the section ${\mathbb {P}}^g\subset M$ again yields an object ${\mathcal {O}}_{{\mathbb {P}}^g}\in \mathrm {D}^{\rm {b}}(M)$ in the ${\mathcal {O}}_M$-orbit.

Example 4.18 To the universal ideal sheaf ${\mathcal {I}}$ on $S\times S^{[2]}$ one associates the Fourier–Mukai kernel [Reference AddingtonAdd16a, Reference Markman and MehrotraMM15]

\[ \mathcal{E}^1 := {\mathcal {Ext}}^1_{\pi_{13}}(\pi_{12}^*(\mathcal{I}), \pi_{23}^*(\mathcal{I})) \in \mathrm{Coh}(S^{[2]} \times S^{[2]}), \]

where $\pi _{ij}$ denote the projections from $S^{[2]}\times S \times S^{[2]}$. Consider a point $p\in S^{[2]}$ parametrising two distinct points $x,y\in S$ and denote by $Z_x$ and $Z_y$ the subvarieties of $S^{[2]}$ parametrising subschemes whose support contains $x$ and $y$, respectively. The derived equivalence $\mathsf {FM}_{{\mathcal {E}}^1}$ sends $k(p)$ to the sheaf ${\mathcal {E}}^1_{p\times S^{[2]}}$ which sits in a short exact sequence

\[ 0 \to {\mathcal{O}}(-\delta) \to {\mathcal{E}}^1_{p\times S^{[2]}} \to I_{Z_x\cup Z_y}\to 0 \]

and is an example of an object in the $k(x)$-orbit with extended Mukai vector

\[ \tilde{v}({\mathcal{E}}^1_{p\times S^{[2]}})=\alpha -\frac{\delta}{2}-\frac{1}{4}\beta. \]

For more on this example, see § 10.1.

Remark 4.19 Ideally, one would like to define a vector $\tilde {w}$ for all elements ${\mathcal {E}}\in \mathrm {D}^{\rm {b}}(X)$ in a coherent way. By this, we mean, for example, that its formation should factor through the $K$-group $K(X)$ and is compatible with derived equivalences, i.e. the following diagram should commute.

However, this is too much to ask for.

Indeed, consider, for example, the case $X=S^{[2]}$ of the Hilbert scheme of two points on a projective K3 surface $S$. Using the Koszul resolution, one can check that the structure sheaf of a complete intersection of divisors of codimension larger than two must have trivial image under $\tilde {w}$. In particular, all sheaves supported on a zero-dimensional subscheme must have trivial image under $\tilde {w}$. Therefore, all previously defined objects in the $k(x)$-orbit must map to zero under $\tilde {w}$. Hence, the vector $\tilde {w}$ vanishes for ${\mathcal {E}}^1_{p\times S^{[2]}}\otimes {\mathcal {L}}$ for all line bundles ${\mathcal {L}} \in \mathop {\rm Pic}\nolimits (X)$. It therefore also has to vanish on all divisors.

The same argument as before also shows that any vector

\[ K^0_{\rm{top}}(S^{[2]})\xrightarrow{\tilde{w}}\tilde{\mathrm{H}}(S^{[2]},{\mathbb{Q}}) \]

compatible with derived equivalences as before must vanish on all topological line bundles.

5.1 Lattices

In this section, we want to discuss the (potential) integral lattices inside the extended Mukai lattice that appear and set up notation which will be used throughout the rest of the paper. We will fix for $X$ once and for all an isometry

(5.1)\begin{equation} \mathrm{H}^2(X,{\mathbb{Z}})\cong \mathrm{H}^2(S^{[n]},{\mathbb{Z}}) \cong \mathrm{H}^2(S,{\mathbb{Z}}) \oplus {\mathbb{Z}} \delta, \end{equation}

where $S$ is a K3 surface, $S^{[n]}$ is the $n$th Hilbert scheme with $2\delta$ the class of the exceptional divisor of the Hilbert–Chow morphism and the second isometry is given by (6.1) (for $X=S^{[n]}$, we choose the first isometry to be the identity).

Let us first quickly review the case of a K3 surface $S$. There, the full integral cohomology $\mathrm {H}^{\ast }(S,{\mathbb {Z}})=\tilde {\mathrm {H}}(S,{\mathbb {Z}})$ with the Mukai pairing is governing the derived category [Reference OrlovOrl97, Reference MukaiMuk87]. That is, an equivalence of K3 surfaces yields a Hodge isometry between the full integral cohomologies and two K3 surfaces are derived equivalent if and only if their integral cohomologies are Hodge isometric. Moreover, the lattice spanned by the Mukai vectors of topological line bundles equals the full integral cohomology.

In higher dimensions, the situation changes. There are several relevant lattices.

Definition 5.1 We define the integral extended Mukai lattice as the lattice

\[ \tilde{\mathrm{H}}(X,{\mathbb{Z}}) := {\mathbb{Z}} \alpha \oplus \mathrm{H}^2(X,{\mathbb{Z}}) \oplus {\mathbb{Z}} \beta \subset \tilde{\mathrm{H}}(X,{\mathbb{Q}}). \]

Several facts suggest that in higher dimensions this is the wrong lattice to look at. First, Definition 4.1 and Examples 4.16 and 4.18 suggest that one should allow certain denominators. Second, we show in Proposition 7.2 that derived equivalences do not send integral elements to integral elements.

Definition 5.2 For $\delta \in \mathrm {H}^2(X,{\mathbb {Z}})$ as above we define the $\mathrm {K3}^{[n]}$ lattice as

\[ \Lambda := B_{-\delta/2}(\tilde{\mathrm{H}}(X,{\mathbb{Z}}))\subset \tilde{\mathrm{H}}(X,{\mathbb{Q}}). \]

This is independent of our fixed choice of $\delta$, i.e. for any class $\gamma \in \mathrm {H}^2(X,{\mathbb {Z}})$ of square $2-2n$ and divisibility $2n-2$ one has

\[ \Lambda=B_{-\gamma/2}(\tilde{\mathrm{H}}(X,{\mathbb{Z}}))\subset \tilde{\mathrm{H}}(X,{\mathbb{Q}}). \]

In dimension $4$, this lattice was also considered by Taelman [Reference TaelmanTae19, Theorem E].

We introduce the notation

\[ \tilde{\alpha}:= B_{-\delta/2}(\alpha)=\alpha- \frac{\delta}{2}+ \frac{1-n}{4}\beta, \quad \tilde{\delta}:= B_{-\delta/2}(\delta)= \delta + (n-1)\beta. \]

With it one can define equivalently the $\mathrm {K3}^{[n]}$ lattice as the lattice

(5.2)\begin{equation} \Lambda = {\mathbb{Z}} \tilde{\alpha} \oplus \mathrm{H}^2(S,{\mathbb{Z}}) \oplus {\mathbb{Z}}\tilde{\delta} \oplus {\mathbb{Z}}\beta = \Lambda_S \oplus {\mathbb{Z}} \tilde{\delta}, \end{equation}

where

\[ \Lambda_S={\mathbb{Z}} \tilde{\alpha} \oplus \mathrm{H}^2(S,{\mathbb{Z}}) \oplus {\mathbb{Z}} \beta. \]

Note that $\tilde {\alpha }$ and $\beta$ still generate an integral hyperbolic plane and that the decomposition $\Lambda _S\oplus {\mathbb {Z}} \tilde {\delta }$ is orthogonal. The integral extended Mukai lattice and the $\mathrm {K3}^{[n]}$ lattice are isometric as abstract lattices and neither is included in the other when seen inside $\tilde {\mathrm {H}}(X,{\mathbb {Q}})$.

Definition 5.3 The geometric lattice $\Lambda _g$ is defined as

\[ \Lambda_g:= \Lambda_S \oplus {\mathbb{Z}} \frac{\tilde{\delta}}{2}\subset \tilde{\mathrm{H}}(X,{\mathbb{Q}}). \]

Be aware that the quadratic form of $\Lambda _g$ inherited from $\tilde {\mathrm {H}}(X,{\mathbb {Q}})$ may not be integral.

We want to motivate this definition. Recall that $r_X=({n+3})/{4}$ and let us look at the lattice generated by all extended Mukai vectors of topological line bundles, i.e.

\[ \Lambda_{\rm LB}:= \biggl\langle \biggl\{\tilde{v}(\lambda):= \alpha + \lambda + \biggl( \frac{n+3}{4} + \frac{b(\lambda,\lambda)}{2} \biggr)\beta \; \biggl|\; \lambda \in \mathrm{H}^2(X,{\mathbb{Z}}) \biggr\} \biggr\rangle. \]

Note that one can write the generators equivalently as

\[ \tilde{v}(\lambda)= \tilde{\alpha} + \frac{\tilde{\delta}}{2} + \lambda + \biggl(1+\frac{b(\lambda,\lambda)}{2}\biggr)\beta. \]

If one ignores the term ${\tilde {\delta }}/{2}$ for a moment, then the expression resembles the Mukai vector on a K3 surface (where $\mathsf {td}^{1/2}=\mathsf {1} + {\mathsf {p}}$). One can check that as an abstract lattice $\Lambda _{\rm LB}$ is isometric to $\tilde {\mathrm {H}}(X,{\mathbb {Z}})$.

In § 4.4 we saw that there are more objects than line bundles and skyscraper sheaves of points for which we can define an extended Mukai vector. We have

\[ \tilde{v}({\mathcal{O}}_{{\mathbb{P}}^n})=l+\frac{\tilde{\delta}}{2}+\beta,\quad \tilde{v}({\mathcal{E}}^1_{p\times S^{[n]}})=\tilde{\alpha}. \]

5.2 Hodge structures

All the previously defined lattices carry a weight-two Hodge structure from their inclusion into $\tilde {\mathrm {H}}(X,{\mathbb {Q}})$.

Definition 5.6 For a lattice $\Gamma \subset \tilde {\mathrm {H}}(X,{\mathbb {Q}})$ we define its algebraic part as

\[ \Gamma_{\rm{alg}} := \Gamma \cap \tilde{\mathrm{H}}^{1,1}(X,{\mathbb{C}}) \subset \tilde{\mathrm{H}}(X,{\mathbb{Q}}) \]

and its transcendental part as

\[ \Gamma_{\rm{tr}} := \Gamma_{\rm{alg}}^{\perp}\cap \Gamma \subset \tilde{\mathrm{H}}(X,{\mathbb{Q}}). \]

With this definition the transcendental part of the integral extended Mukai lattice equals the transcendental lattice of the hyper-Kähler manifold $X$, i.e.

\[ \mathrm{H}^2(X,{\mathbb{Z}})_{\rm{tr}} := \mathrm{H}^2(X,{\mathbb{Z}})\cap \rm{NS(X)}^{\perp}=\tilde{\mathrm{H}}(X,{\mathbb{Z}})_{\rm{tr}} \subset \mathrm{H}^2(X,{\mathbb{Z}}). \]

7.1 Sign equivalence

Denote by $F \in \operatorname {Aut}(\textrm {D}^{\rm {b}}_{\mathfrak {S}_n}(S^n))$ the auto-equivalence given by tensoring with the sign-representation. It is the image of the generator of ${\mathbb {Z}}/2{\mathbb {Z}}$ under $\phi _{[n]}$. We will also denote by $F$ the auto-equivalence of $\mathrm {D}^{\rm {b}}(S^{[n]})$ induced via the equivalence $\Psi$. For a vector $v \in \tilde {\mathrm {H}}^{1,1}(S^{[n]},{\mathbb {Q}})$, we denote by

\[ s_v \in \mathrm{O}(\tilde{\mathrm{H}}(S^{[n]},{\mathbb{Q}})), \quad x\mapsto x-2 \frac{\tilde{b}(x,v)}{\tilde{b}(v,v)}v \]

the Hodge isometry given by reflection along $v$.

Proof. For all topological line bundles ${\mathcal {L}} \in K_{\rm {top}}^0(S)$ the involution $F$ exchanges the equivariant objects $({\mathcal {L}}^{\boxtimes n},1)$ and $({\mathcal {L}}^{\boxtimes n},-1)$ viewed as elements in equivariant topological $K$-theory $K_{\rm {top}, \mathfrak {S}_n}^0(S^n)$. Thus, by (6.2) the induced isometry $F^{\tilde {\mathrm {H}}}$ on the extended Mukai lattice exchanges $\tilde {v}({\mathcal {L}}_{n})$ and $\tilde {v}({\mathcal {L}}_n\otimes {\mathcal {O}}_{S^{[n]}}(-\delta ))$.

If $n$ is odd, then we conclude from the above that for all $\lambda \in \mathrm {H}^2(S,{\mathbb {Z}})\subset \mathrm {H}^2(S^{[n]},{\mathbb {Z}})$ the action on the extended Mukai lattice $F^{\tilde {\mathrm {H}}}$ satisfies

\[ \tilde{\alpha} + \frac{\tilde{\delta}}{2} + \lambda + \biggl( 1+\frac{\tilde{b}(\lambda,\lambda)}{2} \biggr)\beta \mapsto \tilde{\alpha} - \frac{\tilde{\delta}}{2} + \lambda + \biggl( 1+\frac{\tilde{b}(\lambda,\lambda)}{2} \biggr)\beta. \]

This property completely characterises $F^{\tilde {\mathrm {H}}}$.

If $n$ is even, Lemma 4.2 implies that the determinant of $F^{\tilde {\mathrm {H}}}$ must be one, because $F$ preserves the rank of objects. The result then follows as for $n$ odd.

7.2 Spherical twist

An object ${\mathcal {E}} \in \mathrm {D}^{\rm {b}}(S)$ is called spherical if its $\operatorname {Ext}$-algebra satisfies $\operatorname {Ext}^{\ast }({\mathcal {E}},{\mathcal {E}})\cong \mathrm {H}^{\ast }(S^2,{\mathbb {C}})$. The auto-equivalence $\mathsf {ST}_{{\mathcal {E}}}$ given by the Fourier–Mukai functor $\mathsf {FM}_{{\mathcal {G}}}$ with Fourier–Mukai kernel defined via the distinguished triangle

\[ {\mathcal{E}}^{\vee}\boxtimes {\mathcal{E}}\to {\mathcal{O}}_{\Delta}\to {\mathcal{G}} \]

in $\mathrm {D}^{\rm {b}}(S\times S)$ is called the spherical twist [Reference Seidel and ThomasST01]. Its action on the Mukai lattice $\tilde {\mathrm {H}}(S,{\mathbb {Z}})$ is given by the reflection $s_{v({\mathcal {E}})}$.

An important example is the spherical twist $\mathsf {ST}_{{\mathcal {O}}_S}$ along the structure sheaf ${\mathcal {O}}_S$. It induces on cohomology the reflection along the vector $\mathsf {1} + {\mathsf {p}}$. The morphism $\phi _{[n]}$ yields an equivalence $P \in \operatorname {Aut}(\mathrm {D}^{\rm {b}}(S^{[n]}))$.

Proof. We want to understand the images of line bundles under $P$. The spherical twist $\mathsf {ST}_{{\mathcal {O}}_S}$ sends the structure sheaf ${\mathcal {O}}_S$ to ${\mathcal {O}}_S[-1]$. Applying [Reference Ploog and SosnaPS14, Proposition 2.3] we see that $P({\mathcal {O}}_{S^n},1)=({\mathcal {O}}_{S^n},-1)[-n]$ and $P({\mathcal {O}}_{S^n},-1)=({\mathcal {O}}_{S^n},1)[-n]$.Footnote ³ Lemma 4.2 shows that $\epsilon (P^{\tilde {\mathrm {H}}})=1$ if $n$ is even.

We first consider the case when $n$ is odd. Assume there exists a smooth rational curve $C\subset S$ and let ${\mathcal {L}}={\mathcal {O}}_S(C)$ be the corresponding line bundle with first Chern class $l:= \mathrm {c}_1({\mathcal {L}})$. Then by Riemann–Roch ${\mathcal {L}}$ has a unique section up to scaling and the higher cohomologies of ${\mathcal {L}}$ vanish. The auto-equivalence $\mathsf {ST}_{{\mathcal {O}}_S}$ sends the line bundle ${\mathcal {L}}$ to ${\mathcal {L}}|_C$. We infer that the equivariant object $({\mathcal {L}}^{\boxtimes n},-1)$ is being sent to $(({\mathcal {L}}|_C)^{\boxtimes n},-1)$ under the auto-equivalence $P$. We want to transfer this identity to the Hilbert scheme via $\Psi$. From (6.2), we know that $\Psi (({\mathcal {L}}^{\boxtimes n}),-1) \cong {\mathcal {L}}_n \otimes {\mathcal {O}}_{S^{[n]}}(-\delta )$.

It is left to compute $\Psi (({\mathcal {L}}|_C)^{\boxtimes n},-1)$.Footnote ⁴ We claim $\Psi (({\mathcal {L}}|_C)^{\boxtimes n},-1) \cong \iota _\ast \omega _{Z}$ for $\iota \colon Z = C^{[n]} \cong {\mathbb {P}}^n \subset S^{[n]}$. We sketch the arguments, see also [Reference OberdieckObe22, § 3.2] for a thorough computation of this identity.

The sheaf ${\mathcal {O}}_C$ admits the resolution ${\mathcal {L}}^\vee \to {\mathcal {O}}_X$. Taking the $n$th box product, we obtain

\[ \biggl[ W^n({\mathcal{L}}^\vee) \to W^{n-1}({\mathcal{L}}^\vee) \to \dots \to W^1({\mathcal{L}}^\vee) \to W^0({\mathcal{L}}^\vee) \biggr] \cong ({\mathcal{O}}_C^{\boxtimes n},1), \]

where we used the notation as in [Reference KrugKru18, Definition 3.4]. Tensoring with the sign representation and invoking [Reference OberdieckObe22, Lemma 3.3]

(7.1)\begin{equation} \left[ W^0({\mathcal{L}}) \to W^{1}({\mathcal{L}}) \to \dots \to W^{n-1}({\mathcal{L}}) \to W^n({\mathcal{L}}) \right] \cong (({\mathcal{L}}|_C)^{\boxtimes n},-1). \end{equation}

Applying $\Psi$ to (7.1) and using [Reference KrugKru18, Theorem 1.1], we find

(7.2)\begin{equation} \left[ {\mathcal{O}}_{S^{[n]}} \to {\mathcal{L}}^{[n]} \to \dots \to \bigwedge^{n-1}{\mathcal{L}}^{[n]} \to \det({\mathcal{L}}^{[n]}) \right] \cong \Psi(({\mathcal{L}}|_C)^{\boxtimes n},-1). \end{equation}

In particular, the derived dual of $\Psi ({\mathcal {O}}_X^{\boxtimes n},-1)$ is via (7.2) identified with the Koszul resolution of a regular section of the bundle ${\mathcal {L}}^{[n]}$ shifted by $[n]$. As the zero locus of this section is exactly $Z = C^{[n]}$ the claim follows from Grothendieck–Verdier duality.

Taking extended Mukai vectors and using Lemma 4.8, we see that $P^{\tilde {\mathrm {H}}}$ sends the vector $\tilde {v}({\mathcal {L}}_n \otimes {\mathcal {O}}_{S^n}(-\delta ))=\tilde {\alpha } -{\tilde {\delta }}/{2}+l$ to $w=\lambda +c\beta$ with $\lambda \in \mathrm {H}^2(S^{[n]},{\mathbb {Q}})$, because $\iota _\ast \omega _Z$ has rank zero. We already know $P^{\tilde {\mathrm {H}}}(\tilde {v}({\mathcal {O}}_{S^{[n]}}(-\delta )))=-\tilde {v}({\mathcal {O}}_{S^{[n]}})$ (we assume $n$ odd) and because $P^{\tilde {\mathrm {H}}}$ is an isometry, we conclude that

\[ c=\tilde{b}(w,-\tilde{v}({\mathcal{O}}_{S^{[n]}}))=\tilde{b}(\tilde{v}({\mathcal{L}}_n \otimes {\mathcal{O}}_{S^{[n]}}(-\delta)),\tilde{v}({\mathcal{O}}_{S^{[n]}}(-\delta)))=\frac{-1-n}{2}. \]

Similarly, we can use that $P^{\tilde {\mathrm {H}}}(\tilde {v}({\mathcal {O}}_{S^{[n]}}))=-\tilde {v}({\mathcal {O}}_{S^{[n]}}(-\delta ))$ to infer

\[ w=\lambda' - \frac{\delta}{2} + \frac{-1-n}{2}\beta=\lambda' -\frac{\tilde{\delta}}{2} -\beta \]

with $\lambda '\in \mathrm {H}^2(S,{\mathbb {Q}}) \subset \mathrm {H}^2(S^{[n]},{\mathbb {Q}})$.

As $Z\cong {\mathbb {P}}^n$, all curve classes on $Z$ are multiples of each other. A line in $Z$ is known to have homology class $l+ (n-1)\delta ^{\vee }\in \mathrm {H}_2(X,{\mathbb {Z}})$ (see [Reference Hassett and TschinkelHT10, Example 4.11]), where $\delta ^{\vee }$ is the dual class to $\delta$ satisfying $\int _{S^{[n]}} \delta \delta ^\vee = 1$. Denoting $s=l-{\delta }/{2}$, the cohomology class $s^{2n-1}\in \mathrm {SH}^{4n-2}(S^{[n]},{\mathbb {Q}})$ is Poincaré dual to a multiple of the homology class $l+(n-1)\delta ^{\vee }$. Therefore, the degree-$4n-2$ part in $\mathrm {SH}(S^{[n]},{\mathbb {Q}})$ of $\overline {v(\iota _\ast \omega _Z)}$ must be a multiple of $s^{2n-1}$. As for $\mu \in \mathrm {H}^2(S^{[n]},{\mathbb {Q}})$ we have

\[ \psi(\mu^{2n-1})=\psi(e_{\mu}^{2n-1}(\mathsf{1}))=e_{\mu}^{2n-1}(\psi(\mathsf{1}))=b(\mu,\mu)^{n-1}\frac{(2n)!}{2^nn!}\mu\beta^{n-1}\in \mathrm{Sym}^n(\tilde{\mathrm{H}}(X,{\mathbb{Q}})), \]

we conclude that $\lambda '=l$ and

\[ \tilde{\alpha} + \frac{\tilde{\delta}}{2}+l \mapsto \frac{\tilde{\delta}}{2}+l -\beta. \]

In general, for a class $l\in \mathrm {H}^2(S,{\mathbb {Z}})$ of square $-2$ there exists a deformation $S'$ of $S$ such that either $l$ or $-l$ is the class of a smooth rational curve $C'\subset S'$. Using Proposition 6.3 we can assume that the topological line bundle ${\mathcal {L}}$ on $S$ with first Chern class $l$ is algebraic and that ${\mathcal {L}}\cong {\mathcal {O}}_S(C)$, where $C\subset S$ is a smooth rational curve. By the conclusion above, we therefore know the image of $\tilde {v}({\mathcal {L}}_n \otimes {\mathcal {O}}_{S^{[n]}}(-\delta ))$ under $P^{\tilde {\mathrm {H}}}$. As the vectors $\tilde {v}({\mathcal {L}}_n \otimes {\mathcal {O}}_{S^{[n]}}(-\delta ))$ for ${\mathcal {L}}$ a topological line bundle on $S$ whose first Chern class has self-intersection $-2$ together with $\tilde {v}({\mathcal {O}}_{S^{[n]}})$ and $\tilde {v}({\mathcal {O}}_{S^{[n]}}(-\delta ))$ generate the vector space $\tilde {\mathrm {H}}(S^{[n]},{\mathbb {Q}})$, we have proven the assertion in the case that $n$ is odd.

If $n$ is even, then the above shows that $P^{\tilde {\mathrm {H}}}$ must be either $s_v$ or $-s_v$. Using that $\epsilon (P^{\tilde {\mathrm {H}}})=\det (P^{\tilde {\mathrm {H}}})=1$ yields the assertion.

Remark 7.3 We give an observation from the proof which might help in understanding the extended Mukai lattice $\tilde {\mathrm {H}}(S^{[n]},{\mathbb {Q}})$.

Given a smooth rational curve $C\subset S$ inside a K3 surface and the corresponding line bundle ${\mathcal {L}}={\mathcal {O}}_S(C) \in \mathop {\rm Pic}\nolimits (S)$, we have associated to it a line bundle ${\mathcal {L}}_n\in \mathop {\rm Pic}\nolimits (S^{[n]})$. Its Mukai vector $v({\mathcal {L}}_n)$ has self-pairing $n+1$ under the generalised Mukai pairing. We also associate to ${\mathcal {L}}_n$ the class $\tilde {v}({\mathcal {L}}_n)\in \tilde {\mathrm {H}}(S^{[n]},{\mathbb {Q}})$. This class has self-pairing $-(n+3)/2$.

The auto-equivalence $P$ induced from the spherical twist $\mathsf {ST}_{{\mathcal {O}}_S}$ via Ploog's map $\phi _{[n]}$ sends the line bundle ${\mathcal {L}}_n \otimes {\mathcal {O}}_{S^{[n]}}(-\delta )$ to $\iota _\ast \omega _Z$. This is compatible with the pairings because the self-intersection of the projective space ${\mathbb {P}}^n\cong C^{[n]}\subset S^{[n]}$ is $(-1)^n (n+1)$. The image of $\tilde {v}({\mathcal {L}}_n \otimes {\mathcal {O}}_{S^{[n]}}(-\delta ))$ under $P^{\tilde {\mathrm {H}}}$ is

\[ l-\frac{\delta}{2} +\frac{-1-n}{2}\beta=l-\frac{\tilde{\delta}}{2} - \beta. \]

Its self-intersection is equal to $-(n+3)/2$, which is exactly the value of $b(\ell,\ell )$, where $\ell$ is the class of a line in the projective space $C^{[n]}$ and we view a curve class as an element in $\mathrm {H}^2(S^{[n]},{\mathbb {Q}})$ via Poincaré duality.

In the proof we have calculated $\Psi ({\mathcal {O}}_C^{\boxtimes n},-1)$. One can also consider the image of $({\mathcal {O}}_C^{\boxtimes n},1)$ under $\Psi$, i.e. with the canonical linearisation. One can show that this is ${\mathcal {O}}_{Y_C}$, where $Y_C \subset S^{[n]}$ is the reducible subscheme which is the preimage of $C^{(n)} \subset S^{(n)}$ under the Hilbert–Chow morphism.

7.3 From K3 surfaces to Hilbert schemes

We can now describe the homomorphism $d_n$ from Proposition 6.3. Consider the natural inclusion $\tilde {\mathrm {H}}(S,{\mathbb {Q}}) \hookrightarrow \tilde {\mathrm {H}}(S,{\mathbb {Q}}) \oplus {\mathbb {Q}} \delta = \tilde {\mathrm {H}}(S^{[n]},{\mathbb {Q}})$ of quadratic spaces. For $g\in \mathrm {O}(\tilde {\mathrm {H}}(S,{\mathbb {Q}}))$ we define $\iota (g) \in \mathrm {O}(\tilde {\mathrm {H}}(S^{[n]},{\mathbb {Q}}))$ via $\iota (g)(\lambda ) =g(\lambda )$ for $\lambda \in \tilde {\mathrm {H}}(S,{\mathbb {Q}}) \subset \tilde {\mathrm {H}}(S^{[n]},{\mathbb {Q}})$ and $\iota (g)(\delta )=\delta$. This yields a group homomorphism

\[ \iota \colon \mathrm{O}(\tilde{\mathrm{H}}(S,{\mathbb{Q}})) \to \mathrm{O}(\tilde{\mathrm{H}}(S^{[n]},{\mathbb{Q}})). \]

Theorem 7.4 The homomorphism $d_n\colon \mathrm {DMon}(S)\to \mathrm {DMon}(S^{[n]})$ is given by

\[ g\mapsto \det(g)^{n+1} B_{-\delta/2} \circ \iota(g) \circ B_{\delta/2}. \]

8.1 Realising orthogonal transformations as derived equivalences

The first inclusion follows easily from the results of the previous sections.

Proposition 8.2 There is an inclusion

\[ \hat{\mathrm{O}}^+(\Lambda)\subset \mathrm{DMon}(X). \]

Proof. The shift $[1]$ acts on the extended Mukai lattice by $-\mathrm {id}$ and, therefore, acts non-trivially on the discriminant lattice and has determinant $-1$. Proposition 7.2 endows us with an isometry whose action on the discriminant lattice is trivial if and only if its determinant is non-trivial and vice versa. Hence, it suffices to show that $\widetilde {\mathrm {SO}}^+(\Lambda )$, i.e. the group of all isometries with spinor norm and determinant 1 acting trivially on the discriminant, is contained in $\mathrm {DMon}(X)$. For this, we use the notion of Eichler transvections; for details and notation, see [Reference Gritsenko, Hulek and SankaranGHS09, § 3].

Let us orthogonally decompose

\[ \Lambda = U \oplus \Lambda', \]

where the hyperbolic plane $U$ is spanned by $\tilde {\alpha }$ and $-\beta$. The group $\widetilde {\mathrm {SO}}^+(\Lambda )$ equals the group $E_U(\Lambda ')$ of unimodular transvections [Reference Gritsenko, Hulek and SankaranGHS09, Proposition 3.4]. For $\lambda \in \Lambda '$ the Eichler transvection $t(-\beta,\lambda )$ equals $B_{\lambda }$ (note that for $\tilde {\delta } \in \Lambda '$ the transvection $t(-\beta,\tilde {\delta })$ also equals $B_{\delta }$). Using tensoring with line bundles we see that all these isometries are contained in $\mathrm {DMon}(X)$. Furthermore, we infer from Proposition 7.2 that the reflection $s_v$ along the vector $v=\tilde {\alpha } + \beta$ lies in $\mathrm {DMon}(X)$. This involution exchanges $\tilde {\alpha }$ and $-\beta$ and acts trivially on $\Lambda '$. Using [Reference Gritsenko, Hulek and SankaranGHS09, Equation (6)] we deduce that the transvections $t(\tilde {\alpha },\lambda )$ for $\lambda \in \Lambda '$ are contained in $\mathrm {DMon}(X)$. By [Reference Gritsenko, Hulek and SankaranGHS09, Proposition 3.4] these isometries generate $\widetilde {\mathrm {SO}}^+(\Lambda )$ yielding the assertion.