1 Introduction
Given a polarized abelian surface $(A, H)$ defined over an arbitrary field k, we may study moduli spaces of geometrically Hstable sheaves on A with a fixed Mukai vector $v=(r,l,s)$ ; that is, fixed rank, NéronSeveri class of the determinental line bundle and Euler characteristic. Under mild conditions on the Mukai vector, the moduli spaces $M_A(v)$ are smooth and projective. Their Albanese varieties are $A\times \hat {A}$ , and we denote a fiber of the Albanese morphism by $K_A(v)$ .
If defined over ${\mathbb C}$ , the variety $K_A(v)$ is a hyperkähler variety of dimension $v^22$ and is deformation equivalent to the generalized Kummer variety $K_n(A)\cong K_A(1,0,n1)$ , where $n:=\frac {v^2}{2}1$ , which is given by the fiber over $0$ of the summation map acting on the Hilbert scheme of length $(n+1)$ points on A. Following Fu and Li [Reference García and Lax17], who study these varieties over other fields, we call the $K_A(v)$ symplectic varieties (see Proposition 2.5). There are four known deformation types of hyperkähler varieties: $\operatorname {\mathrm {K3}}^{[n]}$ type, Kummer type (or $\operatorname {\mathrm {Kum}}_n$ type) and the two sporadic examples of O’Grady [Reference O’Grady49, Reference Ploog50]. The varieties $K_A(v)$ are of Kummer ntype. It has been shown [Reference Mongardi and Wandel41, Prop. 2.4] that under a latticetheoretic condition, if $n+1$ is a prime power, any hyperkähler of Kummer ntype is the fiber of the Albanese map of a moduli space of stable objects on an abelian surface A. So varieties $K_A(v)$ do not exhaust the class but are, at this point, the best understood.
In [Reference Honigs21], Hassett and Tschinkel analyze the cohomology of complex generalized Kummer fourfolds $K_2(A)$ . They show that $H^4(K_2(A), {\mathbb Q})$ is generated by $H^2(K_2(A),{\mathbb Q})$ and an $81$ dimensional vector space spanned by the cycle classes of $81$ distinct K3 surfaces in $K_2(A)$ . These surfaces are each contained in the fixed locus of a symplectic involution of the form $t_x^*\iota ^*$ , where $\iota $ is multiplication by $1$ on A, and $t_x$ is translation by a point of the threetorsion $A[3]$ of A. Hassett and Tschinkel use deformation theory to show that the middle cohomology for any hyperkähler variety X of $\operatorname {\mathrm {Kum}}_2$ type has a similar decomposition. The cohomology of Kummertype hyperkähler varieties is also studied in [Reference Gulbrandsen19].
In this paper, we extend these results by characterizing the Galois action on the $\ell $ adic étale cohomology of fourfolds $K_A(v)$ over nonclosed fields. As one might expect from the results of Hassett–Tschinkel, there is an $81$ dimensional subspace of $H^4_{\mathrm {\acute{e}t}}(K_2(A)_{\bar {k}},{{\mathbb Q}}_{\ell }(2))$ whose Galois action is determined by the structure of $A[3]$ . However, deformationtheoretic tools are too coarse to keep track of how the Galois action changes for other fourfolds $K_A(v)$ , which we find depends on v:
Theorem 1.1 (Theorem 4.4, Proposition 4.6)
Suppose $K_A(v)$ is a smooth, projective variety over an arbitrary field k. Then, there is a subgroup $G_{A_{{\bar {k}}}}(v) \leqslant (A_{{\bar {k}}}\times \hat {A}_{{\bar {k}}})[3]$ and a Galois equivariant isomorphism
where V is the $80$ dimensional subrepresentation of the permutation representation ${\mathbb Q}_{\ell }[G_{A_{\bar {k}}}(v)]$ such that
and the trivial representation ${\mathbb Q}_{\ell }$ is the span of $(0,0)\in G_{A_{\bar {k}}}(v)$ . The Galois action on the group ring ${\mathbb Q}_{\ell }[G_{A_{\bar {k}}}(v)]$ is induced by the action on $G_{A_{\bar {k}}}(v)$ .
By a generalization of the work of Yoshioka [Reference Yoshioka56], this means the Galois action on the middle cohomology is determined by the action on $H^2_{\mathrm {\acute{e}t}}(A_{\bar {k}},{\mathbb Q}_{\ell }(1))$ and the action on the subgroup $G_{A_{\bar {k}}}(v)$ , which is the kernel of the isogeny $\phi \colon A\times \hat {A} \to A\times \hat {A}$ given by $(x,y)\mapsto (\phi _M(y)sx,\phi _L(x)+ry)$ (See Section 3.1). This stands in surprising contrast to the case of moduli spaces of sheaves on K3 surfaces – symplectic varieties of K3 $^{[n]}$ type – where the cohomology representations depend only on that of the K3 surface [Reference Frei, Honigs and Voight15, Thm. 2].
As a consequence, by studying the even cohomology of $K_2(A)$ for A defined over a number field, we are able to show the following result on derived equivalence:
Corollary 1.2 (Corollary 5.8)
Let A be an abelian surface over a number field k for which the permutation representations associated to $A_{\bar {k}}[3]$ and $\hat {A}_{\bar {k}}[3]$ are not isomorphic. Then, $K_2(A)$ and $K_2(\hat {A})$ are not derived equivalent over k.
In forthcoming work [Reference Fu and Li16] on Galois actions on torsion subgroups of abelian surfaces, examples of such abelian surfaces are constructed. Intriguingly, this corollary shows that if $K_2(A)$ and $K_2(\hat {A})$ are derived equivalent after base change to ${\mathbb C}$ , then the kernel of the Fourier–Mukai transform cannot be given by naturally associated bundles that would descend to the field of definition for A. Corollary 1.2 complements the recent work of Magni [Reference Markman37], which provides a sufficient condition for the existence of such equivalences over algebraically closed fields of characteristic zero.
The cohomology group V in Theorem 1.1 is generated by K3 surfaces contained in the fixedpoint loci of symplectic involutions on $K_A(v)$ . We give a casebycase explicit description of $G_A(v)$ , and hence, an explicit description of these symplectic involutions, which dictate the Galois action on V.
By work of Boissière–NieperWisskirchen–Sarti in [Reference Bolognesi and Massarenti8], Hassett–Tschinkel in [Reference Honigs21] and Kapfer–Menet in [Reference Kass31], for any hyperkähler variety X over ${\mathbb C}$ of $\operatorname {\mathrm {Kum}}_{n1}$ type, the kernel
consists of symplectic automorphisms of X; when $\dim X=4$ , the kernel contains all of the symplectic involutions of X. We give an explicit description of this kernel for hyperkähler varieties $K_A(v)$ of any dimension at least $4$ over ${\mathbb C}$ .
Theorem 1.3 (Theorem 3.15)
Suppose $K_A(v)$ is a smooth, projective variety over $k={\mathbb C}$ . Then,
consists of automorphisms of the following two forms:
where $\kappa =\iota ^*$ if $K_A(v)$ is an Albanese fiber over symmetric line bundles, and otherwise $\kappa $ is a composition of $\iota ^*$ with a translation. The $\kappa _{(x,y)}$ are symplectic involutions of $K_A(v)$ , and when $\dim K_A(v)=4$ , these are all of the symplectic involutions.
In the complex case, the group $G_{A}(v)$ also appears in [Reference Matsushita38] as $\Gamma _v$ . Markman defines $\Gamma _v$ as the kernel of the map $\phi $ above as well as in terms of Clifford algebras (§10.1, Remark 4.3 op.cit.). The result [Reference Matsushita38, Lemma 10.1] and its proof show $\Gamma _v$ embeds into the monodromy group of $K_A(v)$ , acts trivially on $H^2(K_A(v),{\mathbb Z})$ and $H^3(K_A(v),{\mathbb Z})$ , and that $M_A(v)$ is isomorphic to a quotient of $A\times \hat {A}\times K_A(v)$ by an action of $\Gamma _v$ . Thus, the fact that the automorphisms $L_y\otimes t_x^*$ are symplectic is not new, but we provide a proof to make our study of this family selfcontained.
Beyond their analysis of the middle cohomology for $K_2(A)$ , Hassett and Tschinkel explicitly describe the fixedpoint loci of the symplectic involutions. They show that the locus fixed by the standard involution contains the Kummer K3 surface
as well as a unique isolated point supported at the identity element $0$ . Tarí in [Reference Tarí54] finishes the description by showing there are 35 more isolated points, which are tuples of twotorsion points of A. The deformation invariance of the symplectic involutions implies that the fixed locus of any $\iota _{(x,y)}$ in $K_A(v)$ also consists of a K3 surface and 36 isolated points [Reference Kass31, Thm. 7.5].
Motivated by these results, we seek a similar description of the fixedpoint loci in fourfolds $K_A(0,l,s)$ , whose general member is a degree $s+3$ line bundle on a genus 4 curve in the linear system $L$ with $c_1(L)=l$ . These moduli spaces admit a Lagrangian fibration, which aids in our study. We give the following description:
Theorem 1.4 (Theorem 7.2)
The K3 surface in the fixedpoint locus of $\iota ^*$ acting on $K_A(0,l,s)$ is elliptically fibered with four singular fibers of type $I_1$ and $10$ singular fibers of type $I_2$ .
The singular fibers in this elliptic fibration agree with a natural elliptic fibration on the Kummer K3 surface $K_1(A)$ when A is $(1,3)$ polarized – a necessary condition for $K_A(0,l,s)$ to be a fourfold. The K3 surface appears to be closely connected to the relative Jacobian of $K_1(A)\to {\mathbb P}^1$ .
We also describe the isolated points in the fixedpoint locus using the Abel map for the curves in $L$ .
Outline
In Section 2, we provide a brief introduction to moduli spaces of sheaves, and Kummertype varieties arising from them, over arbitrary fields. In Section 3, we identify which automorphisms of $M_A(v)$ given by translation and tensoring by a degree 0 line bundle restrict to automorphisms of $K_A(v)$ , and then show how these give rise to the description of the symplectic automorphisms discussed in Theorem 1.3. We also begin the analysis of the fixedpoint loci for the symplectic involutions. In Section 4, we study the middle cohomology of fourfolds $K_A(v)$ , proving Theorem 1.1. In Section 5, we compare our results to questions about derived equivalences between abelian surfaces and their generalized Kummer fourfolds. Namely, we give criteria in Section 5.1 for when a derived equivalence between abelian surfaces A and B induces an isomorphism between $G_A(v)$ and $G_B(v)$ , and we prove Corollary 1.2 in Section 5.2.
The second half of the paper is dedicated to studying the fixedpoint locus of $\iota ^*$ for fourfolds $K_A(0,l,s)$ over ${\mathbb C}$ , including the proof of Theorem 1.4. In Section 6, we study the general geometry of $K_A(0,l,s)$ and the fixedpoint locus, and then focus on the elliptic fibers of the K3 surface in Section 7. In Section 8, we describe the isolated points in the fixedpoint locus.
Notation
We write the standard involution on an abelian surface A, the morphism multiplying by $1$ in the group law of A, as $\iota \colon A\to A$ . We write $K_n(A)$ for the generalized Kummer variety of dimension $2n$ . In particular, we write $K_1(A)$ for the Kummer K3 surface of A.
For a smooth projective variety X over a field k, let $X_{{\bar {k}}}:=X\times _k {\bar {k}}$ . We denote by $\widetilde {H}(X_{{\bar {k}}},{\mathbb Z}_{\ell })$ the $\ell $ adic Mukai lattice of X, which is the direct sum of the even cohomology twisted into weight zero:
This lattice is given the usual Mukai pairing. For $X=A$ an abelian surface, $(\alpha , \beta ) =\alpha _0\beta _4+\alpha _2\beta _2\alpha _4\beta _0$ . We will always assume that our Mukai vectors v satisfy the conditions given in Setting 2.4 unless indicated otherwise.
Throughout, $D(X)$ denotes the bounded derived category of coherent sheaves on X.
2 Moduli spaces over arbitrary fields
Let A be an abelian surface defined over an arbitrary field k.
Definition 2.1. Let $\omega \in H^4_{\mathrm {\acute{e}t}}(A_{{\bar {k}}}, {\mathbb Z}_{\ell }(2))$ be the numerical equivalence class of a point on $A_{{\bar {k}}}$ . A Mukai vector on A is an element of
where $N(A)$ is a subgroup of $\widetilde {H}(A_{{\bar {k}}}, {\mathbb Z}_{\ell })$ under the natural inclusion.
Given a coherent sheaf $\mathcal F$ on A, we assign to it a Mukai vector $v(\mathcal F)\in N(A)$ given by its rank, the NéronSeveri class of its determinantal line bundle and its Euler characteristic. We will write this as $v(\mathcal F)=(r,l,s)$ .
By fixing a Mukai vector v and a polarization H on A, we can construct the moduli space $M_{A,H}(v)$ parametrizing Hsemistable sheaves on A. We use the more compact notation $M_A(v)$ . We ask that the Mukai vector satisfies the following conditions in order to ensure that the moduli space is nicely behaved (i.e., is a nonempty, smooth, projective variety over k).
Definition 2.2.

(a) A Mukai vector $v \in N(A)$ is geometrically primitive if its image under ${N(A) \to N(A_{{\bar {k}}})}$ is primitive in the lattice.

(b) A Mukai vector $(r,l,s)$ is positive if one of the following is satisfied:

(i) $r>0$

(ii) $r=0$ , l is effective and $s\neq 0$

(iii) $r=0$ , $l=0$ and $s<0$ .


(c) A polarization $H\in \mathrm {Pic}(A)$ is vgeneric if every Hsemistable sheaf with Mukai vector v defined over ${\bar {k}}$ is Hstable.
A polarization is often vgeneric if it is not contained in a locally finite union of certain hyperplanes in $\operatorname {\mathrm {NS}}(A_{\bar {k}})_{\mathbb R}$ defined in [Reference Kaledin, Lehn and Sorger28, Def. 4.C.1], but this is not always enough to ensure genericity (see, for example, [Reference Frei, Honigs and Voight15, Ex. 1.7]).
When $v^2=0$ and H is vgeneric, Mukai showed that $M_H(v)$ is an abelian surface [Reference Mumford46, Rmk. 5.13]. We focus on the higherdimensional case.
Proposition 2.3. Let $v\in N(A)$ be a geometrically primitive and positive Mukai vector with $v^2\geq 2$ , and let H be a vgeneric polarization on A. Then, $M_A(v)$ is a nonempty, smooth, projective, geometrically irreducible variety of dimension $v^2+2$ over k.
Proof. The projectivity and smoothness are shown in [Reference García and Lax17, Prop. 6.9], which relies on classic results in [Reference Mukai44] as well as [Reference Li and Zou34] for the construction of moduli spaces of semistable sheaves over arbitrary fields. Geometric irreducibility of $M_A(v)$ follows from [Reference Kamenova, Mongardi and Oblomkov29, Thm. 4.1] (note that the authors work over ${\mathbb C}$ , but their proof holds for any algebraically closed field). Finally, the dimension claim follows from [Reference Mukai44, Cor. 0.2] once we know $M_A(v)$ is nonempty; nonemptiness is a consequence of [Reference Yoshioka56, Thm. 0.1] along with a lifting argument as in [Reference García and Lax17, Prop. 6.9] when the field has positive characteristic.
Let $v:=(r,l,s)$ be a Mukai vector as in Proposition 2.3 and let
denote the Fourier–Mukai transform on A, which has kernel the Poincaré bundle P on $A \times \hat {A}$ . In [Reference Yoshioka56, Thm. 4.1], Yoshioka proves over ${\mathbb C}$ that the Albanese variety of $M_H(v)$ is $A\times \hat {A}$ and fixing any $\mathcal F_0\in M_H(v)$ , we define the Albanese morphism as follows:
This construction also shows that over an arbitrary field k, the following map gives the Albanese torsor of $M_H(v)$ :
where m is the NéronSeveri class in the Mukai vector $\Phi _P(v):=(s,m,r)$ , which is the negative of the Poincaré dual of l by [Reference Mukai45, Prop. 1.17].
Setting 2.4. Let A be an abelian surface defined over a field k. Let $v:=(r,l,s)\in N(A)$ be a geometrically primitive and positive Mukai vector with $v^2\geq 6$ and $\operatorname {\mathrm {char}} k \nmid \frac {v^2}{2}$ . Let H be a vgeneric polarization on A. Fix $(L,M)$ a pair of line bundles in $\operatorname {\mathrm {Pic}}^l(A)\times \operatorname {\mathrm {Pic}}^{m}(\hat {A})$ . Let $K_A(v)$ be the fiber of $\operatorname {\mathrm {alb}}$ over $(L,M)$ .
Over ${\mathbb C}$ , [Reference Yoshioka56, Thm. 0.2] shows that $K_A(v)$ is a hyperhähler variety, and the following result generalizes this to other fields.
Proposition 2.5 [Reference Yoshioka56, Thm. 0.2], [Reference García and Lax17, Prop. 6.9]
Suppose we have data as in Setting 2.4. Then, $K_A(v)$ is a smooth, projective symplectic variety of dimension $v^22$ and is deformation equivalent to the generalized Kummer variety $K_{(v^22)/2}(A)$ .
For $K_A(v)$ over a field of characteristic zero, which we may assume is a subfield of ${\mathbb C}$ , $K_A(v)_{\mathbb C}$ is a hyperkähler variety. In positive characteristic, Fu and Li [Reference García and Lax17, Def. 3.1] define a symplectic variety X to be a smooth connected variety, where $\pi _1^{\mathrm {\acute{e}t}}(X)=0$ and X admits a nowhere degenerate closed algebraic $2$ form.
We are interested in symplectic involutions on $K_A(v)$ . We will show in Theorem 3.15 that these all involve the induced action of the standard involution $\iota $ on A. Pullback $\iota ^*$ sends degree $0$ line bundles on A to their inverses. For any line bundle $\mathcal L\in \operatorname {\mathrm {Pic}}(A)$ , the multiplication by n map has the property that $[n]^*\mathcal L\cong \mathcal L^{n^2}\otimes M$ for some $M\in \operatorname {\mathrm {Pic}}^0(A)$ . Thus, $\mathcal L$ and $\iota ^*\mathcal L$ differ by a degree $0$ line bundle, so are always in the same NéronSeveri class.
In order for $\iota ^*$ to give a welldefined morphism on $K_A(v)$ , $K_A(v)$ must be a fiber of the Albanese morphism over a pair of symmetric line bundles L and M, which we prefer to do when possible for notational simplicity. In the case of generalized Kummer varieties $K_{n1}(A)$ or varieties $K_A(v)$ whose Mukai vector has trivial NéronSeveri class, it is always possible to choose the fiber over the structure sheaves of A and $\hat {A}$ . For other choices of Mukai vector, we show in Lemma 2.6 below that over an algebraically closed field, we may always choose such a pair of symmetric line bundles.
Lemma 2.6. Let A be an abelian variety over an algebraically closed field k. Then, any class in $\operatorname {\mathrm {NS}}(A)$ has a symmetric representative. Moreover, there is a short exact sequence of the following form, where $\operatorname {\mathrm {Pic}}^{\operatorname {\mathrm {sym}}}(A)$ is the subgroup of all symmetric line bundles:
Proof. The action of $\iota ^*$ on the following short exact sequence
gives rise to the long exact sequence
where $\operatorname {\mathrm {NS}}^{\operatorname {\mathrm {sym}}}(A)=\operatorname {\mathrm {NS}}(A)$ since, for any line bundle $\mathcal L$ , $\iota ^*\mathcal L$ is in the same NéronSeveri class as $\mathcal L$ . The group $H^1({\mathbb Z}/2{\mathbb Z}, \operatorname {\mathrm {Pic}}^0(A))$ is trivial since crossed homomorphisms correspond to elements in $\operatorname {\mathrm {Pic}}^0(A)$ and principal crossed homomorphisms correspond to choices of element in $\operatorname {\mathrm {Pic}}^0(A)$ that have a square root, which is all of them, since we are working over an algebraically closed field.
The proof above requires the field k to be algebraically closed, but we will often work over a nonclosed field. In that case, the existence of a symmetric line bundle in a given NéronSeveri class is not guaranteed. Rather than working over a finite extension of the ground field in order to acquire a symmetric bundle, we will simply alter $\iota ^*$ by a correction factor to get an associated involution on $K_A(v)$ (see Construction 3.10).
3 Symplectic involutions on $K_A(v)$
In [Reference Bolognesi and Massarenti8, Cor. 5(2)], the authors show that, for $X=K_{n1}(A)$ over ${\mathbb C}$ , the kernel of
is isomorphic to ${\mathbb Z}/2{\mathbb Z} \ltimes ({\mathbb Z}/n{\mathbb Z})^4$ , generated by $\iota $ and translation by elements of $A[n]$ . In fact, this group of automorphisms is isomorphic to ${\mathbb Z}/2{\mathbb Z} \ltimes ({\mathbb Z}/n{\mathbb Z})^4$ for any hyperkähler variety X of $\operatorname {\mathrm {Kum}}_{n1}$ type, since it is a deformation invariant [Reference Honigs21, Thm. 2.1]. Moreover, when $\dim X=4$ , $\ker \nu $ contains all of the symplectic involutions [Reference Kass31, Thm. 7.5(i)]. Markman identifies a subgroup $\Gamma _v\cong ({\mathbb Z}/n{\mathbb Z})^4$ of $\ker \nu $ when $X=K_A(v)$ as coming from the kernel of $\phi $ defined below [Reference Matsushita38, §10.1]. In this section, we give an explicit description of $\ker \nu $ for $K_{A}(v)_{\bar {k}}$ when we are in the more general Setting 2.4 and k is arbitrary; this will allow us to understand the action of the Galois group on the fixedpoint loci of the involutions in $\ker \nu $ .
In Section 3.1, we identify which automorphisms of $M_A(v)$ given by translation and tensoring by a degree 0 line bundle restrict to automorphisms of $K_A(v)$ and show they form a group isomorphic to $({\mathbb Z}/n{\mathbb Z})^4$ . We also identify the group of such automorphisms when v is not primitive. The other automorphism needed to generate $\ker \nu $ is $\iota ^*$ when $K_A(v)$ is the Albanese fiber over symmetric line bundles; in Section 3.2, we produce an involution $\kappa $ to replace $\iota ^*$ in the more general setting. We then study the fixed loci of the compositions of $\kappa $ with the automorphisms produced in Section 3.1. In Section 3.3, we show that these compositions are symplectic and act trivially on $H^2(K_A(v),{\mathbb Z})$ .
3.1 Automorphisms from translation and tensor
In this section, we work with data as in Setting 2.4, with the additional assumption that k is an algebraically closed field, and we define $n:=\frac {v^2}{2}$ . Because $k=\bar {k}$ and $\operatorname {\mathrm {char}} k \nmid n$ , we have $A[n]\cong ({\mathbb Z}/n{\mathbb Z})^{4}$ .
We recall that given a line bundle $\mathcal {L}\in \operatorname {\mathrm {Pic}}(A)$ , $\phi _{\mathcal L}\colon A\to \hat {A}$ is defined by $\phi _{\mathcal L}(x):=t_x^*\mathcal L\otimes \mathcal L^{1}$ , where $t_x\colon A\to A$ is translation by a point $x\in A$ . We denote by $L_y\in \operatorname {\mathrm {Pic}}^0(A)$ the line bundle corresponding to a point $y\in \hat {A}$ . Note that $\phi _{\mathcal L}$ is dependent only on the NéronSeveri class of $\mathcal L$ , so we will use the notation $\phi _{[\mathcal L]}$ .
Pullback by the translation map and tensoring by degree $0$ line bundles give automorphisms of $M_A(v)$ , and we are interested in when these automorphisms respect the Albanese morphism. That is, we identify in Theorem 3.1 below which of the $L_y\otimes t_x^* \in \operatorname {\mathrm {Aut}} M_A(v)$ restrict to automorphisms of $K_A(v)$ .
Theorem 3.1. Let v be a Mukai vector as in Setting 2.4. There are exactly $n^4$ elements $(x,y)\in A\times \hat {A}$ for which the automorphism $L_y\otimes t_x^*$ on $M_A(v)$ restricts to an automorphism on $K_A(v)$ . These elements form a subgroup
whose set of kpoints is isomorphic to $({\mathbb Z}/n{\mathbb Z})^4$ .
The elements of $G_A(v)$ are the solutions to the following equations on $\hat {A}$ and A, where l and m are the NéronSeveri classes of L and M:
Equivalently, $G_A(v)$ is the kernel of the following isogeny:
The proof of Theorem 3.1 requires analysis of $\phi _l$ and $\phi _m$ . We will crucially need the following lemma.
Lemma 3.2 (Yoshioka [Reference Yoshioka56, Lem. 4.2])
where $\chi :=\chi (L)=\chi (M)=\frac {l^2}{2}=n+rs$ .
Additionally, we recall that for any $\mathcal F\in D(A)$ ,
This follows from [Reference Mukai43, (3.1)]. Though the statement is not quite identical to the one we give here, it immediately follows from biduality of the Poincaré bundle [Reference Huybrechts26, 9.12].
Proof of Theorem 3.1
The main issue in this proof is that maps of the form $L_y\otimes t_x^*$ are not, in general, welldefined as automorphisms on $K_A(v)$ . Given $\mathcal F\in K_A(v)$ , $L_y\otimes t_x^*\mathcal F$ has the same Mukai vector as $\mathcal F$ but may not have the same image under the Albanese morphism. For instance, pullback by $t_x^*$ , in general, preserves Néron–Severi classes of line bundles and acts trivially on the structure sheaf, but it does not act trivially on all line bundles.
We therefore seek the $(x,y)\in A\times \hat {A}$ that satisfy the following conditions:
We may rewrite these conditions as the equations (3.1). Equivalently, these $(x,y)$ are the kernel of the map $\phi $ in (3.2).
Precomposing the map $\phi $ with $\psi \colon A\times \hat {A} \to A\times \hat {A}$ , where $\psi (x,y)= (\phi _m(y)rx,\phi _l(x)+sy)$ , and applying Lemma 3.2, we have
Thus, $\phi \circ \psi =[n]$ , so $\phi $ is surjective and is hence an isogeny. Similarly, $\psi \circ \phi =[n]$ and $G_A(v)\leqslant (A\times \hat {A})[n]$ .
We show $G_A(v)\cong ({\mathbb Z}/n{\mathbb Z})^4$ in Lemma 3.6. This will require an understanding of preimages of elements under $\phi _l$ and $\phi _m$ , which we study in Claims 3.4 and 3.5.
Remark 3.3. Since the maps $\phi _l$ and $\phi _m$ are determined by the Néron–Severi classes of L and M, the proof of Theorem 3.1 shows that the automorphisms of $M_A(v)$ given by elements of $G_A(v)$ will restrict to automorphisms of not just one, but any fiber of the Albanese morphism on $M_A(v)$ .
Furthermore, for any $(x,y)\in (A\times \hat {A})[n]$ , the automorphism $L_y\otimes t_x^*$ induces a permutation of the Albanese fibers and if $(x,y)\notin G_A(v)$ , this permutation does not have any fixed fibers.
If we extend the domain of $\det \times \det \Phi _P$ to elements of $D(A)$ with Mukai vector v (by mapping to the Grothendieck group before taking determinants), $L_y\otimes t_x^*$ acts on the fibers of this map as well.
Before proving Lemma 3.6, we need results on the kernels of $\phi _l$ and $\phi _m$ :
Claim 3.4. Let $p\neq \operatorname {\mathrm {char}} k$ be a prime and $\chi \neq 0$ . Suppose $p^q$ is the highest power of p dividing $\chi $ . Then, the group of ppower torsion points in $\ker \phi _l\cong \ker \phi _m$ is
where $0\leq n_1\leq n_2$ and $n_1+n_2=q$ . If $n_1>0$ , then L and M are $p^{n_1}$ st powers of other line bundles.
If L and M are separable, we may define their polarization type to be the termwise product of pairs $(p^{n_1},p^{n_2})$ as p varies over primes dividing $\chi $ (cf. [Reference Birkenhake and Lange6, §2]).
Proof. Since $\ker \phi _l \cong \ker \phi _{l}$ and $\chi (L) \neq 0$ , we may assume that L is ample. The proof of Riemann–Roch for abelian varieties in [Reference Naruki47, §16] implies that the degree of $\phi _l$ is $\chi ^2$ . The structure of $\ker \phi _l \cap A[p^q]$ is then determined by Lemma 3.2 and the fact that the Weil pairing $e^L$ on the ptorsion is skewsymmetric [Reference Naruki47, §20, Thm. 1]. Since $\phi _m$ is the negative of the dual of $\phi _l$ [Reference Birkenhake and Lange6, §2], the group structure of ppower torsion points in $\ker \phi _m$ is isomorphic to that in $\ker \phi _l$ . The last statement is a consequence of [Reference Naruki47, §23, Thm. 3].
The images of any two elements of the same order under the compositions $\phi _l\circ \phi _m$ or $\phi _m\circ \phi _l$ will have the same order. However, $\phi _l$ and $\phi _m$ do not respect orders in this way.
Claim 3.5. Let $p\neq \operatorname {\mathrm {char}} k$ be a prime dividing $\chi $ , and assume that l and m are not pth multiples of other classes, so $n_1=0$ . Suppose $p^d\mid \chi $ for some $d\in \mathbb {N}$ .

(a) Suppose $u\in A[p^d]\cap \ker {\phi _l}$ . Then, the preimage of u in $\hat {A}[p^d]$ under $\phi _m$ is of the form $b+({\mathbb Z}/p^d{\mathbb Z})^2$ for some $b\in \hat {A}[p^d]$ .

(b) Suppose $v\in \hat {A}[p^d]\cap \ker {\phi _m}$ . Then, the preimage of v in $\hat {A}[p^d]$ vnder $\phi _l$ is of the form $a+({\mathbb Z}/p^d{\mathbb Z})^2$ for some $a\in {A}[p^d]$ .
Now, suppose $p^q=\chi $ .

(c) Suppose $u\in A$ and $\phi _l(u)$ has order $p^c$ . Then, the preimage of u in $\hat {A}$ under $\phi _m$ is of the form $b+({\mathbb Z}/p^q{\mathbb Z})^2$ for some $b\in \hat {A}[p^{c+q}]$ .

(d) Suppose $v\in \hat {A}$ and $\phi _m(v)$ has order $p^c$ . Then, the preimage of v in ${A}$ under $\phi _l$ is of the form $a+({\mathbb Z}/p^q{\mathbb Z})^2$ for some $b\in \hat {A}[p^{c+q}]$ .
Proof. (a) By Lemma 3.2, the composition $\phi _l\circ \phi _m$ is given by multiplication by $\chi $ . Thus, $\phi _m\circ \phi _l$ acts on $A[p^d]$ as the zero map, and hence,
By Claim 3.4, $A[p^d]\cap \ker \phi _l$ has $p^{2d}$ elements and $\phi _m$ acting on $\hat {A}[p^d]$ is a $p^{2d}$ to $1$ map. It follows by counting that $\operatorname {\mathrm {im}} \phi _m_{\hat {A}[p^d]} = A[p^d]\cap \ker \phi _l$ . By Claim 3.4, the preimage of u is as stated.
Part (b) follows analogously.
(c) By Lemma 3.2, the preimage of $\phi _l(u)$ under $\phi _l\circ \phi _m$ consists of elements of order $p^{c+q}$ . By Lemma 3.4, the result follows.
Part (d) follows analogously.
The following result is proved using a casebycase argument. The explicit argument given has the advantage of aiding in the analysis of examples. See [Reference Matsushita38, Lemma 10.1] for an approach using deformations over ${\mathbb C}$ .
Lemma 3.6. The solutions to the equations (3.1) form a group isomorphic to $({\mathbb Z}/n{\mathbb Z})^4 \leqslant (A\times \hat {A})[n]$ .
Proof. Case 1: $\chi =0$ .
Both L and M must have degree $0$ , so $\phi _l$ and $\phi _m$ are both the $0$ morphism. The equations (3.1) simplify to
Furthermore, $n=rs$ . Since $v=(r,l,s)$ is positive and $v^2\geq 4$ , we must have $r>0$ and $s<0$ . The solutions consist of all products of $s$ torsion points on A and rtorsion points on $\hat {A}$ .
The group of solutions is isomorphic to $({\mathbb Z}/r{\mathbb Z})^4 \times ({\mathbb Z}/s{\mathbb Z})^4$ , hence $({\mathbb Z}/n{\mathbb Z})^4$ , since in this case, primitivity of the Mukai vector implies r and s are relatively prime.
Now, let p be a prime divisor of n and $p^q$ be the highest power of p dividing n. We treat the remaining cases by analyzing solutions in $(A\times \hat {A})[p^q]$ . We may then conclude by using the Sun Zi Remainder Theorem.
Case 2: $\chi \neq 0$ and at least one of r or s is relatively prime with p.
Suppose r is relatively prime with p. Fix an arbitrary $x\in A[p^q]$ . The equation $\phi _l(x)=ry$ then has exactly one solution y because multiplication by $r$ acts bijectively on $\hat {A}[p^q]$ .
Now we check that $(x,y)$ is a solution to (3.1): Applying $\phi _m$ , we have $\phi _m\circ \phi _l(x)=r\phi _m(y)$ . Using Lemma 3.2, we then have $rsx=r\phi _m(y)$ . Since x and y are $p^q$ torsion, multiplication by $r$ acts bijectively, implying $sx=\phi _m(y)$ .
Thus, for each $x\in A[p^q]$ , there is one $y\in \hat {A}[p^q]$ so that $(x,y)$ is a solution to (3.1). The projection map $(x,y)\mapsto x$ gives an isomorphism from solutions to (3.1) to $A[p^q]\cong ({\mathbb Z}/p^q{\mathbb Z})^4$ .
If s is relatively prime with p, an analogous argument shows there is exactly one solution $(x,y)$ to (3.1) for each $y\in \hat {A}[p^q]$ and that, again, the group of all solutions is isomorphic to $({\mathbb Z}/p^q{\mathbb Z})^4$ .
Cases 1 and 2 have covered all cases where r and s are not both divisible by p. Going forward, we assume $p\mid r$ and $p\mid s$ . If $\operatorname {\mathrm {char}}(k)\neq 0$ , our assumption in Setting 2.4 that $\operatorname {\mathrm {char}}(k)\nmid n$ implies in the following cases that $\operatorname {\mathrm {char}}(k)\neq p$ , and so we may apply Claim 3.4. By the primitivity of the Mukai vector, $n_1=0$ and $n_2$ is equal to the highest power of p dividing $\chi $ .
Let j be the highest power of p dividing r and k be the highest power of p dividing s. If r or s is $0$ , we choose the convention that j or k is $\infty $ .
In each of Cases 3, 4, 5, we handle in stages the situations where q becomes higher and higher relative to j and k. From now on, we assume $j\geq k$ . If $k>j$ , the argument is analogous.
Case 3: $\chi \neq 0$ , $0<k\leq j$ and $q\leq j$ . We observe that $p^q$ is the highest power of p that divides $\chi $ .
Solutions $(x,y)\in (A\times \hat {A})[p^q]$ to the first equation in (3.1) are precisely those where $\phi _l(x)=0$ . By Claim 3.4, the group of such x is isomorphic to $({\mathbb Z}/p^q{\mathbb Z})^2$ .
Fix such an x. We observe that $sx\in A[p^q]$ and $\phi _l(sx)=0$ . By Claim 3.5(a), the preimage of $sx$ under $\phi _m$ in $\hat {A}[p^q]$ is of the form $b+({\mathbb Z}/p^q{\mathbb Z})^2$ for some $b\in \hat {A}[p^q]$ . Thus, there are $p^{4q}$ total solutions.
The projection $(x,y)\mapsto x$ gives a surjective group homomorphism $G_A(v)\twoheadrightarrow ({\mathbb Z}/p^q{\mathbb Z})^2$ . The kernel of this map consists of all solutions where $x=0$ , which by Claim 3.4 is isomorphic to $({\mathbb Z}/p^q{\mathbb Z})^2$ . Since $G_A(v)\leqslant (A\times \hat {A})[p^q]\cong ({\mathbb Z}/p^q{\mathbb Z})^8$ , this short exact sequence shows it must be isomorphic to $({\mathbb Z}/p^q{\mathbb Z})^4$ .
In Cases 4 and 5, we make a reduction argument. We observe that for any $(x,y)\in G_A(v)$ , $(sx,sy)\in G_A(v)\cap (A\times \hat {A})[p^{qk}]$ . In each of Cases 4 and 5, we will show that the map
given by multiplication by s is surjective and $p^{4k}$ to $1$ . This argument may be repeated to reduce each case to previous cases.
Case 4: $\chi \neq 0$ , $0<k \leq j< q$ and $q\leq j+k$ .
We note that $p^q\mid \chi $ . Since $qk\leq j$ , the argument in Case 3 shows that
Let $(u,v)\in G_A(v)\cap (A\times \hat {A})[p^{qk}]$ . We seek $(x,y)\in G_A(v)< (A\times \hat {A})[p^{q}]$ , where $(sx,sy)=(u,v)$ . First, we search for elements y, where $\phi _m(y)=u$ and $sy=v$ . Thus, we look at the preimage of u under $\phi _m$ and analyze which of those elements give v when multiplied by s.
Note that $\phi _l(u)=rv=0$ . Since u is $p^{qk}$ torsion, it is also $p^q$ torsion. So by Claim 3.5(a), the preimage of u under $\phi _m$ in $\hat {A}[p^q]$ is of the form $b+({\mathbb Z}/p^q{\mathbb Z})^2$ , where $b\in \hat {A}[p^{q}]$ . Multiplying by s gives a $p^{2k}$ to $1$ map on the following cosets:
We will now show that v is in the image of this map: the preimage of $su$ under $\phi _m$ in $\hat {A}[p^q]$ is of the form $v+({\mathbb Z}/p^q{\mathbb Z})^2$ . The preimage of $su$ under $\phi _m$ that is $p^{qk}$ torsion is thus of the form $v+({\mathbb Z}/p^{qk}{\mathbb Z})^2$ and has exactly $p^{2(qk)}$ elements. Now, the elements of $sb+({\mathbb Z}/p^{qk}{\mathbb Z})^2$ are $p^{qk}$ torsion, there are $p^{2(qk)}$ of them and their image under $\phi _m$ is $su$ . Thus, these sets are equal, implying $v\in sb+({\mathbb Z}/p^{qk}{\mathbb Z})^2$ . Thus, there are $p^{2k}$ elements $y\in \hat {A}[p^q]$ with the desired properties.
Now we search for elements x, where $\phi _l(x)=ry=\frac {r}{s}v$ and $sx=u$ . Note that since $j\geq k$ , $\frac {r}{s}=\frac {cp^e}{d}$ for some c, d relatively prime with p. We may define multiplying by $\frac {1}{d}$ on ppower torsion points by taking the preimage under multiplication by d since it is a bijection on such points. We examine the preimage of $\frac {r}{s}v$ under $\phi _l$ and analyze which of those elements give u when multiplied by s.
Note $\phi _m(\frac {r}{s}v)=ru=0$ . So by Claim 3.5(b), the preimage of $\frac {r}{s}v$ under $\phi _l$ is of the form $a+({\mathbb Z}/p^q{\mathbb Z})^2$ , where $a\in {A}[p^{q}]$ .
Multiplying by s gives a $p^{2k}$ to $1$ map on the following cosets:
We will now show that u is in the image of this map: The preimage of $rv$ under $\phi _l$ in $A[p^q]$ is of the form $u+({\mathbb Z}/p^q{\mathbb Z})^2$ . The preimage of $rv$ under $\phi _l$ that is $p^{qk}$ torsion is thus of the form $u+({\mathbb Z}/p^{qk}{\mathbb Z})^2$ . The elements of $sa+({\mathbb Z}/p^{qk}{\mathbb Z})^2$ are $p^{qk}$ torsion and their image under $\phi _l$ is $rv$ . Thus, these sets are equal, implying $u\in sa+({\mathbb Z}/p^{qk}{\mathbb Z})^2$ . In summary, there are $p^{2k}$ elements $x\in A[p^q]$ , where $\phi _l(x)=\frac {r}{s}v=ry$ and $sx=u$ .
This shows that (3.3) is a surjective $p^{4k}$ to $1$ map. Since multiplication by s decreases the order of the ppower torsion of an element by exactly $p^k$ , by (3.4) we may conclude that $G_A(v)\cong ({\mathbb Z}/p^q{\mathbb Z})^4$ .
Case 5: $\chi \neq 0$ , $0<k\leq j< q$ and $j+k<q$ . In this case, $p^{j+k}$ divides $\chi $ and no higher powers of p may divide $\chi $ .
By the argument in Case 4, we have
We will first extend our result for solutions of order up to $p^{j+2k}$ . For convenience, define $t:=\min \{q,j+2k\}$ .
Let
We seek
so that $(sx,sy)=(u,v)$ . First, we search for elements y, where $\phi _m(y)=u$ and $sy=v$ ; thus, we look at the preimage of u under $\phi _m$ and analyze which of those elements give v when multiplied by s.
If $\phi _l(u)=0$ , then the argument from Case 4 shows that there are $p^{2k}$ elements $y\in \hat {A}[p^{t}]$ , where $\phi _m(y)=u$ and $sy=v$ .
If $\phi _l(u)\in \hat {A}[p^k]\setminus \{0\}$ , then by Claim 3.5(c), the preimage of u under $\phi _m$ in $\hat {A}[p^{t}]$ is of the form $b+({\mathbb Z}/p^{j+k}{\mathbb Z})^2$ , where $b\in \hat {A}[p^{t}]\setminus \hat {A}[p^{j+k}]$ .
Note that $\phi _l(\phi _m(b))=rv$ and by Lemma 3.2, $rsb=(n+rs)b=rv$ . Multiplication by s gives a $p^{2k}$ to $1$ map on the following cosets:
We will now show that v is in the image of this map.
The preimage of $su$ under $\phi _m$ in $\hat {A}[p^{j+k}]$ is of the form $v+({\mathbb Z}/p^{j+k}{\mathbb Z})^2$ . The part of $v+({\mathbb Z}/p^{j+k}{\mathbb Z})^2$ whose image under multiplication by $r$ is $rv$ is of the form $v+({\mathbb Z}/p^{j}{\mathbb Z})^2$ . Since $\phi _m$ maps $sb+({\mathbb Z}/p^{j}{\mathbb Z})^2$ to $su$ and multiplying this coset by $r$ gives $rv$ , by counting elements, $sb+({\mathbb Z}/p^{j}{\mathbb Z})^2=v+({\mathbb Z}/p^{j}{\mathbb Z})^2$ . Hence, $v\in sb+({\mathbb Z}/p^{j}{\mathbb Z})^2$ . Thus, there are $p^{2k}$ elements $y\in b+({\mathbb Z}/p^{j+k}{\mathbb Z})^2$ , where $\phi _m(y)=u$ and $sy=v$ .
Now we search for elements x where $\phi _l(x)=ry=\frac {r}{s}v$ and $sx=u$ . We examine the preimage of $\frac {r}{s}v$ under $\phi _l$ and analyze which of those elements give u when multiplied by s.
If $\phi _m(\frac {r}{s}v)=ru=0$ , we may conclude using the arguments in Case 4. Otherwise, by Claim 3.5(d), the preimage of $\frac {r}{s}v$ under $\phi _l$ in $A[p^{t}]$ is of the form $a+({\mathbb Z}/p^{j+k}{\mathbb Z})^2$ where $a\in {A}[p^{t}]\setminus {A}[p^{j+k}]$ . Note that $\phi _m(\phi _l(a))=ru$ and by Lemma 3.2, $rsa=ru$ .
Multiplying by s gives a $p^{2k}$ to $1$ map on cosets:
We will now show that u is in the image of this map. The preimage of $rv$ under $\phi _l$ is of the form $u+({\mathbb Z}/p^{j+k}{\mathbb Z})^2$ . The part of $u+({\mathbb Z}/p^{j+k}{\mathbb Z})^2$ whose image under multiplication by $r$ is $ru$ is of the form $u+({\mathbb Z}/p^{j}{\mathbb Z})^2$ . We have shown that $\phi _l$ maps $sa+({\mathbb Z}/p^{j}{\mathbb Z})^2$ to $rv$ and multiplying this coset by $r$ gives $ru$ . By counting elements, we have the set equality $sa+({\mathbb Z}/p^{j}{\mathbb Z})^2=u+({\mathbb Z}/p^{j}{\mathbb Z})^2$ . Hence, $u\in sa+({\mathbb Z}/p^{j}{\mathbb Z})^2$ . Thus, there are $p^{2k}$ elements $x\in a+({\mathbb Z}/p^{j+k}{\mathbb Z})^2$ , where $\phi _l(x)=\frac {r}{s}v$ and $sx=u$ . Thus, the following map is surjective and $p^{4k}$ to $1$ :
If $q\leq j+2k$ , we may now conclude, in combination with (3.5), that $G_A(v)\cong ({\mathbb Z}/p^q{\mathbb Z})^4$ .
If $q> j+2k$ , (3.6) shows that $G_A(v)\cap (A\times \hat {A})[p^{j+2k}]\cong ({\mathbb Z}/p^{j+2k}{\mathbb Z})^4$ . The above argument may be repeated for solutions of orders up to $p^{j+3k}$ and then upward inductively to conclude that $G_A(v)\cong ({\mathbb Z}/p^{q}{\mathbb Z})^4$ .
Example 3.7. (a) For $K_2(A)\cong K_A(1,0,3)$ , l and m are the trivial NéronSeveri classes (these are treated in general by Case 1 of the proof of Lemma 3.6), so $\phi _l(x)=0$ and $\phi _m(y)=0$ . The equations (3.1) simplify to $0=y$ and $0=3x$ , which recovers the fact that the group of symplectic automorphisms for $K_2(A)$ is generated by $\iota $ and translation by elements of $A[3]$ [Reference Bolognesi and Massarenti8, Cor. 5(2)].
(b) In Sections 6–8, we consider fourfolds $K_A(v)$ , where $v=(0,l,s)$ for l primitive and $\chi =3$ . If $s\equiv s'\ \textrm {mod}\ 3$ , then $G_A(0,l,s)=G_A(0,l,s')$ , leaving only three possible distinct groups of this form, which are described by a combination of Cases 2 and 3 of Lemma 3.6. Case 2 shows that $G_A(0,l,1)$ and $G_A(0,l,2)$ , though in general distinct, each have one element $(x,y)\in (A\times \hat {A})[3]$ for every $y\in \hat {A}[3]$ : for any $y\in \hat {A}[3]$ , there is one $x\in \ker \phi _l$ so that $t_y^*M\simeq L_{x}\otimes M$ . However, we see from Case 3 that $G_A(0,l,0)$ is the product of $\ker \phi _l$ and $\ker \phi _m$ .
The assumption in Theorem 3.1 that v is primitive is necessary for $G_A(v)$ to be isomorphic to $({\mathbb Z}/n{\mathbb Z})^4$ . In the case where $v=2v_0$ for $v_0$ a primitive Mukai vector with $v_0^2=2$ , which is used to construct O’Grady sixfolds, the solutions to the equations (3.1) are precisely of the form $(A\times \hat {A})[2]\cong ({\mathbb Z}/2{\mathbb Z})^8$ , as is shown in [Reference Mukai42, Lem. 5.1]. We generalize this result by extending Theorem 3.1 to find all solutions to (3.1) for any Mukai vector.
In the following result, we alter our hypotheses by naming the primitive vector of Setting 2.4 $v_0$ and considering a multiple of $v_0$ .
Corollary 3.8. Let $v=(r,l,s)$ be a Mukai vector on an abelian surface A so that $v=dv_0$ , where $v_0=(r_0,l_0,s_0)$ is primitive and $n:=\frac {v_0^2}{2}$ .
Then, the group $G_A(v)$ of solutions $(x,y)\in A\times \hat {A}$ to the following equations
is isomorphic to $({\mathbb Z}/dn{\mathbb Z})^4\times ({\mathbb Z}/d{\mathbb Z})^4$ .
Proof. Let m and $m_0$ be the respective NéronSeveri classes determined by $\Phi _P$ . We see from the definition of $\phi _l$ that $\phi _l=d\cdot \phi _{l_0}$ and likewise, $\phi _m=d\cdot \phi _{m_0}$ : if we choose $(L_0,M_0)\in \operatorname {\mathrm {Pic}}^{l_0}(A)\times \operatorname {\mathrm {Pic}}^{m_0}(\hat {A})$ and $L:=L_0^{\otimes d}$ , $M:=M_0^{\otimes d}$ , then for any $x\in A$ , we have
Since $\phi _{l_0}$ and $\phi _{m_0}$ are group homomorphisms, we have, for any $(x,y)\in A \times \hat {A}$ ,
Thus, a pair $(x,y)\in A\times \hat {A}$ is a solution to (3.7) if and only if
or equivalently, $(dx,dy)$ solves the equations (3.1) given by $v_0$ . By Theorem 3.1, the set of solutions to the equations (3.1) given by $v_0$ is isomorphic to $({\mathbb Z}/n{\mathbb Z})^4\cong G_A(v_0)$ . We may conclude by observing that the set of solutions to (3.7) is given by exactly the elements of $A\times \hat {A}$ that are in $G_A(v_0)$ after being multiplied by d.
3.2 Involutions and fixed loci
Let A be an abelian surface over an arbitrary field k. If $K_A(v)$ is a fiber over symmetric line bundles, then $\iota ^*$ gives an involution of $K_A(v)$ . However, if symmetric bundles do not exist in the appropriate NéronSeveri classes over k, we show here how to define an involution $\kappa $ to replace $\iota ^*$ . For the remainder of the section, we fix a set of data as in Setting 2.4 and hence, fix a variety $K_A(v)$ over k.
We first give a lemma that will allow us to construct the involution $\kappa $ .
Lemma 3.9. Suppose we have an additional choice of line bundles $L'\in \operatorname {\mathrm {Pic}}^l(A)$ , $M'\in \operatorname {\mathrm {Pic}}^m(\hat {A})$ over k. Let $K_A(v)':=\operatorname {\mathrm {alb}}^{1}(L',M')$ . Then, there is an element $(a,b)\in (A\times \hat {A})(k)$ so that $L_b\otimes t_a^*\colon K_A(v) \to K_A(v)'$ is an isomorphism over k. It is unique up to composition with elements in $G_A(v)(k)$ .
Proof. Recall that for any $(x,y)\in A\times \hat {A}$ , applying $L_y\otimes t_x^*$ to an element $\mathcal F\in K_A(v)$ , we have
We also recall that the morphism $\phi \colon A\times \hat {A} \to A \times \hat {A}$ from (3.2) is an isogeny defined over k and sends sends $(x,y)$ to
The element $(a,b)$ desired is precisely a preimage of $(L'\otimes L^{1},M'\otimes M^{1})\in (A\times \hat {A})(k)$ under $\phi $ . Finally, $L_b\otimes t_a^*: K_A(v) \to K_A(v)'$ is an isomorphism since it has an inverse $L_{b}\otimes t_{a}^*$ .
Construction 3.10. Applying $\iota ^*$ gives an isomorphism from $K_A(v)$ to $\operatorname {\mathrm {alb}}^{1}(\iota ^*L,\iota ^*M)$ . By Lemma 3.9, there is an $(a,b)\in (A\times \hat {A})(k)$ such that $L_b\otimes t_a^*$ maps isomorphically from $\operatorname {\mathrm {alb}}^{1}(\iota ^*L,\iota ^*M)$ back to $K_A(v)$ . So we have the following automorphism defined over k:
Remark 3.11. We note that $\kappa $ is an involution. More generally, for any $(c,d)\in A\times \hat {A}$ , the morphism $L_d\otimes t_c^*\iota ^*$ (which in general is an automorphism of $M(v)$ but perhaps not of $K(v)$ ) is an involution on $M_A(v)$ . Indeed, $(\iota \circ t_c)^2=\operatorname {\mathrm {id}}$ on A and $L_d$ is degree $0$ and hence, fixed under pullback by translation; thus, for any $\mathcal F\in M_A(v)$ , we have:
The following are thus involutions of $K_A(v)$ , where $(x,y)\in G_A(v)(k)$ :
Under the simplifying assumption that L and M are symmetric, we may instead denote these involutions as
Lemma 3.12. Let $n:=\frac {v^2}{2}$ be odd. Assume $k=\bar {k}$ and that L and M are symmetric, so $\iota ^*$ is an involution on $K_A(v)$ . Then, $\operatorname {\mathrm {Fix}}(\iota _{(x,y)})$ is a translation of $\operatorname {\mathrm {Fix}}(\iota _{(0,0)})$ ; that is, there exists $(u,w)\in G_A(v)$ so that
More generally, without the assumption that L and M are symmetric, there exists $(u,w)\in G_A(v)$ so that
Proof. Let $\mathcal F\in K_A(v)$ be in the fixed locus of $\iota ^*$ . Pick $(u,w)\in G_A(v)$ so that $2w=y$ and $2u=x$ , which is possible since n is odd. For instance, when $\frac {v^2}{2}=3$ , $K_A(v)$ is a fourfold and the elements of $G_A(v)$ are all threetorsion, so we may choose $(x,y)$ .
Then, $L_{w}\otimes t_{u}^*\mathcal F$ must be fixed by the involution
The other direction of the containment is similar, as is the case with $\iota ^*$ replaced by $\kappa $ .
Proposition 3.13. Let $n:=\frac {v^2}{2}$ be odd, assume $k=\bar {k}$ and let $L'\in \operatorname {\mathrm {Pic}}^l(A)$ and $M'\in \operatorname {\mathrm {Pic}}^m(\hat {A})$ be symmetric line bundles (cf. Lemma 2.6). Fix an involution $\kappa $ as in Construction 3.10 on $K_A(v)$ . Then, the fixed locus of $\kappa $ in $K_A(v)$ is isomorphic to the fixed locus of $\iota ^*=\iota _{(0,0)}$ in $K_A(v)'$ .
Proof. By Lemma 3.9, there is an $(x,y)\in A\times \hat {A}$ so that $L_y\otimes t_x^*$ gives an isomorphism from $K_A(v)$ to $K_A(v)'$ . The composition
may be rearranged to $L_{2y}\otimes t_{2x}^*\iota ^*$ , where $L_{2y}\otimes t_{2x}^*$ gives an isomorphism from $\operatorname {\mathrm {alb}}^{1}(\iota ^*L,\iota ^*M)$ to $K_A(v)$ . Thus, by the uniqueness statement of Lemma 3.9, there is an element $(u,w)\in G_A(v)$ for which $L_w\otimes t_u^*\circ L_{2y}\otimes t_{2x}^*$ is equal to the map $L_{b}\otimes t_{a}^*$ in the definition of $\kappa $ . Then, $L_{y}\otimes t_{x}^* \operatorname {\mathrm {Fix}}(\iota ^*)$ is equal to $\operatorname {\mathrm {Fix}}(\kappa _{(u,w)})$ , which is isomorphic to $\operatorname {\mathrm {Fix}}(\kappa )$ by Lemma 3.12.
Finally, we give the following general result on the action of the Galois group on the geometric fixed loci.
Proposition 3.14. Let k be an arbitrary field. For $(x,y)\in G_{A_{{\bar {k}}}}(v)$ , the action of $\sigma \in \operatorname {\mathrm {Gal}}(k/k)$ sends the fixed locus of $\kappa _{(x,y)}$ in $K_A(v)_{\bar {k}}$ to the fixed locus of $\kappa _{(\sigma ^{1} x,\sigma ^{1} y)}$ .
Proof. Suppose $\mathcal F$ is fixed by $\kappa _{(x,y)}$ . We use the equality $t_x\circ \sigma =\sigma \circ t_{\sigma ^{1}x}$ and the observation that $\sigma $ commutes with $\iota $ and, moreover, $\kappa $ , since $\kappa $ is defined over the ground field k, to simplify the following equation:
Then, we have $\sigma ^*L_y\simeq L_{\sigma ^{1}y}$ , which we may verify using $\Phi _P\colon D(A)\to D(\hat {A})$ :
3.3 Symplectic automorphisms and involutions
Let A be an abelian surface over ${\mathbb C}$ . In the following lemma, we give a generalization of [Reference Bolognesi and Massarenti8, Cor. 5(2)] to hyperkähler varieties $K_A(v)$ over ${\mathbb C}$ .
Theorem 3.15. Suppose we are in Setting 2.4 and we fix an involution $\kappa $ as in Construction 3.10. Then, the kernel of
consists of automorphisms of the form $L_y\otimes t_x^*$ and of the form $\kappa _{(x,y)}:=L_y\otimes t_x^*\kappa $ for $(x,y)\in G_A(v)$ . Thus, for any $(x,y)\in G_A(v)$ , the automorphism $L_y\otimes t_x^*$ is symplectic. The $\kappa _{(x,y)}$ are symplectic involutions of $K_A(v)$ , and when $\dim K_A(v)=4$ , these are all of the symplectic involutions.
Remark 3.16. While $\kappa $ is not unique, by Lemma 3.9, the collection of elements in $\ker \nu $ is independent of the choice made in Construction 3.10.
Proof. Elements of $(x,y)\in A\times \hat {A}$ act on $M_A(v)$ via $L_{y}\otimes t_{x}^*$ . Abelian varieties are pathconnected, so the action of any element in $A\times \hat {A}$ is homotopic to the identity, which implies the induced action on $H^2(M_A(v),{\mathbb Z})$ is trivial. If $(x,y)\in G_A(v)$ , then Theorem 3.1 shows that the action of $L_{y}\otimes t_{x}^*$ restricts to $K_A(v)$ . By [Reference Yoshioka56, Thm. 0.2(2)], the restriction map $H^2(M_A(v),{\mathbb Z})\to H^2(K_A(v),{\mathbb Z})$ is a surjection. Therefore, $L_{y}\otimes t_{x}^*$ acts trivially on $H^2(K_A(v),{\mathbb Z})$ as well.
By [Reference Yoshioka56, Thm. 0.2(2)], there is an isomorphism
where $v^{\perp } \subset H^{even}(A,{\mathbb Z})$ is the orthogonal complement to v under the Mukai pairing. Since $\iota ^*$ acts by $1$ on $H^1(A,{\mathbb Z})$ , it acts trivially on $H^{even}(A,{\mathbb Z})$ .
If we assume L and M are symmetric, $\iota ^*$ is an automorphism of $K_A(v)$ and therefore must act trivially on $H^2(K_A(v),{\mathbb Z})$ . If L and M are not both symmetric, since we are working over an algebraically closed field, we observe that $\kappa $ is a composition of translation to an Albanese fiber over symmetric bundles, application of $\iota ^*$ on that fiber and translation back (cf. proof of Proposition 3.13). Thus, $\kappa $ will act trivially on $H^2(K_A(v),{\mathbb Z})$ as well.
By the discussion above, $\ker \nu $ contains $2n^4$ elements, so by Theorem 3.1, we have identified all of them. The automorphisms in this kernel are clearly symplectic as the symplectic form generates part of $H^2(K_A(v),{\mathbb C})$ .
For any nontrivial choice of $(x,y)\in G_A(v)$ , $L_y\otimes t_x^*$ is not an involution, but by Section 3.2, $\kappa _{(x,y)}$ is an involution on $K_A(v)$ .
Finally, suppose $\dim K_A(v)=4$ . By [Reference Kass31, Thm. 7.5(i)], all of the symplectic involutions of $K_A(v)$ act trivially on $H^2(K_A(v),{\mathbb Z})$ .
4 The middle cohomology of fourfolds $K_A(v)$
In this section, we work with data as in Setting 2.4 with the additional assumption that $v^2=6$ , so $K_A(v)$ is a fourfold. We will prove results characterizing the middle cohomology of $K_A(v)$ when k has characteristic $0$ in Section 4.1. We use these results to characterize the cohomology similarly when k has positive characteristic in Section 4.2 via a brief lifting argument.
4.1 Results in characteristic zero
Assume $K_A(v)$ is defined over an arbitrary field k of characteristic zero, so we may assume without loss of generality that $\bar {k} \hookrightarrow {\mathbb C}$ . In this case, we can identify the Galois representations which make up the middle cohomology of $K_A(v)$ . This will depend on understanding the fixed loci of $\kappa _{(x,y)}$ for $(x,y)\in G_{A_{\bar {k}}}(v)$ .
Proposition 4.1. Suppose $k={\bar {k}}$ . The fixed locus of any involution $\kappa _{(x,y)}$ for $(x,y)\in G_A(v)$ on a fourfold $K_A(v)$ consists of a K3 surface and 36 isolated points.
Proof. First, suppose $k={\mathbb C}$ . Work of Hassett and Tschinkel [Reference Honigs21] and Tarí [Reference Tarí54] shows that the statement is true for $K_2(A)$ . A discussion of the isolated fixed points in this case is given in Section 6.1.
Every hyperkähler fourfold $K_A(v)$ is deformation equivalent to $K_2(A)$ and by [Reference Honigs21, Thm. 2.1], its group of symplectic involutions is also a deformation invariant. Thus, as in Kapfer and Menet [Reference Kass31, Thm. 7.5], the fixed loci are related by deformation as well, so the statement holds for $K_A(v)$ .
Now let k be any algebraically closed field of characteristic zero. Since A is defined over k, we can assume without loss of generality that $k \hookrightarrow {\mathbb C}$ . Let ${K}_A(v)_{{\mathbb C}} := K_A(v) \times _{k} {\mathbb C}$ and consider the Cartesian square
where $\widetilde {\kappa }_{(x,y)}$ is formed by replacing $\kappa $ with its extension to ${\mathbb C}$ , which we call $\tilde {\kappa }$ . By Theorem 3.15, $\widetilde {\kappa }_{(x,y)}$ is a symplectic involution, and by the argument above, $\operatorname {\mathrm {Fix}}(\widetilde {\kappa }_{(x,y)})$ is a K3 surface $Z:=Z_{(x,y)}\subset {K}_A(v)_{{\mathbb C}}$ plus 36 isolated points.
By [Reference Frei14, Rmk. 3 following Thm 2.3],
This descent of the fixedpoint locus means that $\operatorname {\mathrm {Fix}}(\kappa _{(x,y)})$ consists of a surface $S:=S_{(x,y)}\subset K_A(v)$ along with 36 kpoints. We claim that S is a K3 surface. Indeed, we see via the valuative criterion of properness, using the fact that $\operatorname {\mathrm {Fix}}(\kappa _{(x,y)})$ is a closed subscheme of $K_A(v)$ , that $S \to \operatorname {\mathrm {Spec}} k$ is proper. By flat base change, we have that $H^1(S,\mathcal {O}_S)\otimes {\mathbb C} \cong H^1(Z,\mathcal {O}_{Z})=0$ , and $H^0(S,\omega _S)\otimes {\mathbb C} \cong H^0(Z,\omega _{Z})={\mathbb C}$ , so $\omega _S$ has a nonvanishing global section and hence, is trivial. Finally, S is smooth by [Reference Esteves11, Lem. 4.1], which completes the proof.
See [Reference Kapfer and Menet30] for further discussion of these fixedpoint loci in hyperkählers of Kummer type.
Let k now be arbitrary. Let $S_{(x,y)}\subset K_A(v)_{\bar {k}}$ be the K3 surface in $\operatorname {\mathrm {Fix}}(\kappa _{(x,y)})$ and $s_{(x,y)}\in H^4_{\mathrm {\acute{e}t}}(K_A(v)_{\bar {k}},{\mathbb Q}_{\ell }(2))$ the image of $[S_{(x,y)}]\in \operatorname {\mathrm {CH}}^2 K_A(v)_{\bar {k}}$ under the cycle class map $\operatorname {\mathrm {CH}}^2 K_A(v)_{\bar {k}} \to H^4_{\mathrm {\acute{e}t}}(K_A(v)_{\bar {k}},{\mathbb Q}_{\ell }(2))$ .
Lemma 4.2. For $\sigma \in \operatorname {\mathrm {Gal}}(\bar {k}/k)$ , the induced action on the cycle classes $s_{(x,y)}$ for $(x,y)\in G_{A_{\bar {k}}}(v)$ is given by
Proof. By [Reference Mongardi and Wandel40, Prop. 9.2], the cycle class map is Galois equivariant, so $\sigma ^*s_{(x,y)}$ is the cycle class of $[\sigma ^* S_{(x,y)}]\in \operatorname {\mathrm {CH}}^2 K_A(v)_{\bar {k}}$ . As in the proof of [Reference Mongardi and Wandel40, Prop. 9.2], we have that $[\sigma ^* S_{(x,y)}]$ is the preimage of $S_{(x,y)}$ under $\sigma ^*\colon K_A(v)_{\bar {k}} \to K_A(v)_{\bar {k}}$ . By Proposition 3.14, $(\sigma ^*)^{1}(S_{(x,y)})=S_{(\sigma x, \sigma y)}$ . Thus, we conclude that $\sigma ^*s_{(x,y)}=s_{(\sigma x, \sigma y)}$ , as desired.
Definition 4.3. For a finite Galois module G, let ${\mathbb Q}_{\ell }[G]$ be the ${\mathbb Q}_{\ell }$ vector space with basis given by G, where the action of the Galois group on ${\mathbb Q}_{\ell }[G]$ is determined by the action on G: for $\sigma \in \operatorname {\mathrm {Gal}}({\bar {k}}/k)$ and $\sum _{g_i \in G} a_i g_i \in {\mathbb Q}_{\ell }[G]$ ,
We call ${\mathbb Q}_{\ell }[G]$ the permutation representation.
Recall that when k is not algebraically closed, the group $G_{A_{{\bar {k}}}}(v)$ naturally has the structure of a finite $\operatorname {\mathrm {Gal}}({\bar {k}}/k)$ module.
Theorem 4.4. There is an isomorphism of Galois representations
where V is the $80$ dimensional subrepresentation of ${\mathbb Q}_{\ell }[G_{A_{\bar {k}}}(v)]$ such that
and the trivial representation ${\mathbb Q}_{\ell }$ is the span of $(0,0)\in G_{A_{\bar {k}}}(v)$ .
Remark 4.5. As will be shown in Lemma 5.7, the action of the Galois group on $H^2_{\mathrm {\acute{e}t}}(K_A(v)_{\bar {k}},{\mathbb Q}_{\ell }(1))$ , and hence $\operatorname {\mathrm {Sym}}^2H^2_{\mathrm {\acute{e}t}}(K_A(v)_{\bar {k}},{\mathbb Q}_{\ell }(1))$ , is determined by the action on $H^2_{\mathrm {\acute{e}t}}(A_{\bar {k}},{\mathbb Q}_{\ell }(1))$ .
Proof. By Theorem 3.1, we have $3^4=81$ involutions
where $(x,y)\in G_{A_{\bar {k}}}(v)$ .
As in the proof of Proposition 4.1, let ${K}_A(v)_{{\mathbb C}} := K_A(v)\times _k {\mathbb C}$ and $\widetilde {\kappa }_{(x,y)}\colon {K}_A(v)_{{\mathbb C}} \to {K}_A(v)_{{\mathbb C}}$ the base change of $\kappa _{(x,y)}$ . By Proposition 4.1, $\operatorname {\mathrm {Fix}}(\widetilde {\kappa }_{(x,y)})$ contains a K3 surface $Z_{(x,y)}\subset {K}_A(v)_{{\mathbb C}}$ . This gives 81 distinct K3 surfaces in ${K}_A(v)_{{\mathbb C}}$ , where the distinctness follows from [Reference Honigs21, Thm. 2.1]. Via the cycle class map, these 81 surfaces give corresponding classes $z_{(x,y)}\in H^4({K}_A(v)_{{\mathbb C}}, {\mathbb Q})$ .
Similarly, there are K3 surfaces $S_{(x,y)}\subset K_A(v)_{\bar {k}}$ and corresponding cohomology classes $s_{(x,y)}\in H^4_{\mathrm {\acute{e}t}}({K}_A(v)_{\bar {k}}, {\mathbb Q}_{\ell }(2))$ such that $S_{(x,y)}\times _{\bar {k}} {\mathbb C}= Z_{(x,y)}\subset {K}_A(v)_{{\mathbb C}}$ . Under the comparison and smooth base change isomorphisms
the classes $z_{(x,y)}$ correspond to the classes $s_{(x,y)}$ .
By [Reference Kass31, Thm. 7.5(ii)], the pair $({K}_A(v)_{{\mathbb C}}, \widetilde {\kappa }_{(x,y)})$ is deformation equivalent to the pair $(K_2(A_{{\mathbb C}}), t_{\tau }\circ [\operatorname {\mathrm {Id}}]^{[[3]]})$ for some $\tau \in A_{{\mathbb C}}[3]$ . In particular, these complex manifolds are diffeomorphic and so they have isomorphic cohomology rings. By [Reference Honigs21, Prop. 4.3] (see also the discussion in [Reference Kass31, §6.4]), the ${\mathbb Q}_{\ell }$ span of $\{z_{(x,y)}z_{(0,0)}\}_{(x,y) \in G_{A_{\mathbb C}}(v)}$ is an 80dimensional vector space of $H^4(K_A(v)_{{\mathbb C}},{\mathbb Q}_{\ell }(2))$ which is a direct sum complement to the subspace $\operatorname {\mathrm {Sym}}^2H^2(K_A(v)_{{\mathbb C}},{\mathbb Q}_{\ell }(1))$ .
Since the $s_{(x,y)}$ in $H^4(K_A(v)_{\bar {k}},{\mathbb Q}_{\ell }(2))$ correspond to the $z_{(x,y)}$ , it follows that
is an $80$ dimensional subspace of $H^4_{\mathrm {\acute{e}t}}(K_A(v)_{\bar {k}},{\mathbb Q}_{\ell }(2))$ which is a direct sum complement to $\operatorname {\mathrm {Sym}}^2H^2_{\mathrm {\acute{e}t}}(K_A(v)_{\bar {k}},{\mathbb Q}_{\ell }(1))$ .
By Lemma 4.2, we know that for $\sigma \in \operatorname {\mathrm {Gal}}(\bar {k}/k)$ ,
Thus, V is a Galoisinvariant subspace of $H^4_{\mathrm {\acute{e}t}}(K_A(v)_{\bar {k}},{\mathbb Q}_{\ell }(2))$ . Noting that ${\mathbb Q}_{\ell }[G_{A_{\bar {k}}}(v)]$ is semisimple by Maschke’s Theorem – the Galois action factors through a finite group representation determined by the finite extension of k over which $G_{A_{\bar {k}}}(v)$ is defined – and that $\sigma ^*(s_{(0,0)})=s_{(0,0)}$ , this shows that V is the $80$ dimensional subrepresentation of ${\mathbb Q}_{\ell }[G_{A_{\bar {k}}}(v)]$ such that
where the trivial representation corresponds to $(0,0)\in G_{A_{\bar {k}}}(v)$ . Hence, $H^4_{\mathrm {\acute{e}t}}(K_A(v)_{\bar {k}},{\mathbb Q}_{\ell }(2))$ has the decomposition as stated.
4.2 Results in positive characteristic via lifting
In this section we observe that, because Kummer varieties $K_A(v)$ defined over a field of positive characteristic lift to characteristic $0$ [Reference García and Lax17], we may use Theorem 4.4 to give a similar description of the middle cohomology.
Proposition 4.6. Suppose we have data as in Setting 2.4, where the base field k has characteristic $p>0$ . Then,
where $V'$ is the $80$ dimensional subrepresentation of ${\mathbb Q}_{\ell }[G_{A_{\bar {k}}}(v)]$ such that
and the trivial representation ${\mathbb Q}_{\ell }$ is the span of $(0,0)\in G_{A_{\bar {k}}}(v)$ .
Proof. As explained in the proof of [Reference García and Lax17, Prop. 6.9], it is possible to lift $K_A(v)$ to characteristic $0$ by lifting its defining data. That is, the data $(A,v,H,L,M)$ defined over k has a lift $(\mathcal {A}, v_W, \mathcal {H}, \mathcal {L}, \mathcal {M})$ to a complete discrete valuation ring W of characteristic zero with residue field k and field of fractions $F:=\operatorname {\mathrm {Frac}} W$ . Note that all of this lifting data can be recovered from a lift of $(A,H,L)$ . Indeed, lifting A automatically gives us a lift of $\hat {A}$ , and lifting line bundles on $\hat {A}$ amounts to lifting their Néron–Severi class; a lift of the Néron–Severi class of M is given by the Néron–Severi class of $\det (\Phi _{\mathcal P}(\mathcal {L}))$ . Call the specialization of $v_W$ to the generic fiber $v_F$ .
There is a surjection of Galois groups
which is given by restricting automorphisms to the ring of integers of $\bar {F}$ and then passing to the quotient $\bar {k}$ . By the smooth base change theorem [Reference Altman and Kleiman1, Exp. XVI, Corollaire 2.2], for $\ell \neq p$ , there are isomorphisms
which are equivariant with respect to the action of $\operatorname {\mathrm {Gal}}(\bar {F}/F)$ on the left and $\operatorname {\mathrm {Gal}}(\bar {k}/k)$ on the right, compatible with (4.1).
The isomorphisms of (4.2) are compatible with the ring structure on cohomology, so the isomorphism $H^4_{\mathrm {\acute{e}t}}(K_{A_{\bar {F}}}(v_F), {\mathbb Q}_{\ell }(2)) \cong H^4_{\mathrm {\acute{e}t}}(K_{A}(v)_{\bar {k}}, {\mathbb Q}_{\ell }(2))$ restricts to an isomorphism
which is again compatible with the respective Galois group actions.
Let the following be the decomposition given by Theorem 4.4:
and let $V' \subset H^4_{\mathrm {\acute{e}t}}(K_{A}(v)_{\bar {k}}, {\mathbb Q}_{\ell }(2))$ be the vector space complement to $\operatorname {\mathrm {Sym}}^2 H^2_{\mathrm {\acute{e}t}}(K_{A}(v)_{\bar {k}}, {\mathbb Q}_{\ell }(1))$ . Using the fact that V is a $\operatorname {\mathrm {Gal}}(\bar {F}/F)$ subrepresentation of $H^4_{\mathrm {\acute{e}t}}(K_{A_{\bar {F}}}(v_F), {\mathbb Q}_{\ell }(2))$ , we conclude that
as $\operatorname {\mathrm {Gal}}(\bar {k}/k)$ representations. In particular, there is an isomorphism $V\cong V'$ which is equivariant with respect to the action of $\operatorname {\mathrm {Gal}}(\bar {F}/F)$ on the left and $\operatorname {\mathrm {Gal}}(\bar {k}/k)$ on the right, again compatible with (4.1).
The subgroup $G_{A_{\bar {F}}}(v_F)\leqslant (A_{\bar {F}}\times \hat {A}_{\bar {F}})[3]$ is given by equations (3.1) determined by $v_F$ , which is part of our lifted data. Thus, since the action of $\operatorname {\mathrm {Gal}}(\bar {F}/F)$ on V is given by $G_{A_{\bar {F}}}(v)$ , the action of $\operatorname {\mathrm {Gal}}(\bar {k}/k)$ on $V'$ must be the one determined analogously by $G_{A_{\bar {k}}}(v)$ .
5 Relation to derived equivalences
There are a number of results related to derived equivalences of smooth, projective symplectic varieties. For example, if X and Y are derived equivalent smooth complex projective surfaces, then $D(\operatorname {\mathrm {Hilb}}^nX)\cong D(\operatorname {\mathrm {Hilb}}^nY)$ [Reference Polishchuk51, Prop. 8]. If X and Y are K3 surfaces, then the converse holds, and if two moduli spaces of stable sheaves $M_X(v)$ and $M_Y(v')$ are derived equivalent, then X and Y are also derived equivalent [Reference Birkenhake and Lange5, Cor. 9.7]. If X and Y are derived equivalent K3 surfaces over any field k, then the $\ell $ adic étale cohomologies of any moduli $M_X(v)$ and $M_Y(v')$ of equal dimension are isomorphic as $\operatorname {\mathrm {Gal}}(\bar {k}/k)$ representations [Reference Frei, Honigs and Voight15, Thm. 2]. However, it is still an open question when such moduli are derived equivalent.
In the direction of symplectic varieties of Kummer type, complex abelian surfaces A and B are derived equivalent if and only if there is an isomorphism $K_1(A)\cong K_1(B)$ between their associated Kummer K3 surfaces [Reference Hudson and Barth24, Reference Stellari53]. This result has also been proved for abelian surfaces over fields of odd characteristic [Reference Magni36]; the relation between Kummer surfaces and twisted derived equivalence of abelian surfaces has been examined in [Reference Li and Zou35, Thm. 6.5.2]. While A and $\hat {A}$ are always derived equivalent over their field of definition, it is not known exactly when there is a derived equivalence between the generalized Kummer fourfolds $K_2(A)$ and $K_2(\hat {A})$ . Recently, it was shown that, over an algebraically closed field of characteristic zero, they are derived equivalent when A has a polarization of exponent coprime to $3$ [Reference Markman37, Theorem 1].
Given these results, we ask the following two questions, which we examine in Sections 5.1 and 5.2, respectively.
Question 1. Suppose we have a derived equivalence of abelian surfaces $D^b(A)\cong D^b(B)$ . How do the groups $G_A(v)$ introduced in Section 3 interact with the Rouquier isomorphism $A\times \hat {A}\simeq B\times \hat {B}$ ?
Question 2. Under what conditions are irreducible symplectic fourfolds of Kummer type derived equivalent?
Throughout this section, we will assume we are working with data as in Setting 2.4 and that all varieties $K_A(v)$ are an Albanese fiber over symmetric line bundles.
5.1 Compatibility with the Rouquier isomorphism
Proposition 5.1 (Rouquier, cf. [Reference Huybrechts26, Prop. 9.45])
Let A and B be abelian varieties and $F\colon D(A) \to D(B)$ a derived equivalence. There is an isomorphism $f \colon A\times \hat {A} \to B \times \hat {B}$ , called the Rouquier isomorphism, which maps $(a,\alpha )\in A\times \hat {A}$ to the unique element $(b,\beta )\in B\times \hat {B}$ so that the following diagram commutes:
The following proposition gives some results addressing Question 1.
Proposition 5.2. Let A and B be abelian surfaces over a field k, and let $v=(r,l,s)\in N(A)$ and $v'=(r',l',s')\in N(B)$ .
Let $F\colon D(A)\to D(B)$ be a derived equivalence such that $F(v)=v'$ . Then, the base change of the Rouquier isomorphism to the algebraic closure $\bar {k}$ restricts to a group scheme isomorphism
under any of the following conditions:

(a) For any elements $\mathcal F,\mathcal G\in M_A(v)$ such that $\operatorname {\mathrm {alb}}(\mathcal F)=\operatorname {\mathrm {alb}}(\mathcal G)$ , we have $\det (F(\mathcal F))=\det (F(\mathcal G))$ and $\det (\Phi _P\circ F(\mathcal F))=\det (\Phi _P\circ F(\mathcal G))$ ;

(b) F is a stabilitypreserving Fourier–Mukai transform; that is, if $E\in M_A(v)$ , then $F(E)$ is in $M_B(v')$ ; or

(c) $k={\mathbb C}$ and $\frac {v^2}{2}= 3$ (i.e., $K_A(v)$ is a fourfold).
We note that the isomorphism (5.2) implies that the actions of $\operatorname {\mathrm {Gal}}(\bar {k}/k)$ on $G_{A_{\bar {k}}}(v)$ and $G_{B_{\bar {k}}}(v')$ are isomorphic.
Proof. Let $(a,\alpha )\in G_{A_{\bar {k}}}(v)$ . By Remark 3.3, to prove that $(b,\beta ):=f_{\bar {k}}(a,\alpha )\in G_{B_{\bar {k}}}(v')$ , it suffices to produce an element $\mathcal H\in D(B)$ , where $v(\mathcal H)=v'$ , $\det (\mathcal H)=\det (L_{\beta }\otimes t_b^*\mathcal H)$ , and $\det (\Phi _P(\mathcal H))=\det (\Phi _P(L_{\beta }\otimes t_b^*\mathcal H))$ .
Under condition (a), for any $\mathcal F\in M_A(v)$ , we may take $\mathcal H:=F(\mathcal F)$ . In this case, we have $L_{\beta }\otimes t_b^*\mathcal H=F(L_{\alpha }\otimes t_a^*\mathcal F)$ . Since
condition (a) allows us to conclude that $\mathcal H$ has the needed property.
Under condition (b), F restricts to an isomorphism $M_A(v)\to M_B(v')$ and by the universal property of the Albanese morphism, there is a commutative diagram as follows:
Thus, F satisfies condition (a).
By [Reference Honigs21, Prop. 4.3] if $K_A(v)$ is a fourfold, the intersection of the fixed loci of $\kappa $ and $(L_{\alpha }\otimes t_a^*)\kappa $ acting on