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Gushel–Mukai varieties with many symmetries and an explicit irrational Gushel–Mukai threefold

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Abstract

We construct an explicit smooth Fano complex threefold with Picard number 1, index 1, and degree 10 (also known as a Gushel–Mukai threefold) and prove that it is not rational by showing that its intermediate Jacobian has a faithful \({{\,\mathrm{PSL}\,}}(2,\mathbf{F}_{11}) \)-action. Along the way, we construct Gushel–Mukai varieties of various dimensions with rather large (finite) automorphism groups. The starting point of all these constructions is an Eisenbud–Popescu–Walter sextic with a faithful \({{\,\mathrm{PSL}\,}}(2,\mathbf{F}_{11}) \)-action discovered by the second author in 2013.

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Notes

  1. By Nikulin’s celebrated result [43, Corollary 1.9.4], this means that they have same ranks, same signatures, and that their discriminant forms coincide.

  2. In the given decompositions of the lattice \({{\,\mathrm{Pic}\,}}(\widetilde{Y}_A)\), the summand (2) is not generated by the polarization H, because \({{\mathsf {S}}}\) contains no \((-2)\)-classes.

  3. The principally polarized abelian fivefold \((E_\lambda ^5,\theta ')\) was studied in [1, 2, 29, 48]: it is the intermediate Jacobian of the Klein cubic threefold with equation \(x_1^2x_2+x_2^2x_3+x_3^2x_4+x_4^2x_5 +x_5^2x_1 =0\) in \(\mathbf{P}^4\).

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Acknowledgements

We would like to thank B. Gross, G. Nebe, D. Prasad, Yu. Prokhorov, and O. Wittenberg for fruitful exchanges. Special thanks go to A. Kuznetsov, whose numerous comments and suggestions helped improve the exposition and the results of this article; in particular, Propositions A.2 and A.6 are his.

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Correspondence to Olivier Debarre.

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To Fabrizio Catanese, on the occasion of his 70+1st birthday

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Project HyperK—grant agreement 854361).

Appendices

Appendix A: Automorphisms of double EPW sextics

1.1 A.1 Double EPW sextics and their automorphisms

As in Sect. 2.1, let \(V_6\) be a 6-dimensional complex vector space and let \(A\subset \textstyle {\bigwedge ^{3}}{V}_6\) be a Lagrangian subspace with no decomposable vectors, with associated EPW sextic \(Y_A\subset \mathbf{P}(V_6)\). There is a canonical double covering

$$\begin{aligned} \pi _A:\widetilde{Y}_A\longrightarrow Y_A \end{aligned}$$
(17)

branched along the integral surface \(Y^{\ge 2}_A\). The fourfold \(\widetilde{Y}_A\) is called a double EPW sextic and its singular locus is the finite set \(\pi _A^{-1}(Y^{\ge 3}_A)\) (see [45, Section 1.2] or [20, Theorem B.7]). It carries the canonical polarization \(H:=\pi _A^*\mathscr {O}_{Y_A}(1)\) and the image of the associated morphism \(\widetilde{Y}_A\rightarrow \mathbf{P}(H^0(\widetilde{Y}_A,H)^\vee )\) is isomorphic to \(Y_A\). When \(Y_A^{\ge 3}=\varnothing \), we say that \(Y_A\) is quasi-smooth and \(\widetilde{Y}_A\) is a smooth hyperkähler variety of K3\(^{[2]}\)-type.

Every automorphism of \(Y_A\) induces an automorphism of \(\widetilde{Y}_A\) (see the proof of [20, Proposition B.8(b)]) that fixes the class H. Conversely, let \({{\,\mathrm{Aut}\,}}_H(\widetilde{Y}_A)\) be the group of automorphisms of \(\widetilde{Y}_A\) that fix the class H. It contains the covering involution \(\iota \) of \(\pi _A\). Any element of \({{\,\mathrm{Aut}\,}}_H(\widetilde{Y}_A)\) induces an automorphism of \(\mathbf{P}(H^0(\widetilde{Y}_A,H)^\vee )\simeq \mathbf{P}( V_6)\) hence descends to an automorphism of \(Y_A\). This gives a central extension

$$\begin{aligned} 0\rightarrow \langle \iota \rangle \rightarrow {{\,\mathrm{Aut}\,}}_H(\widetilde{Y}_A) \rightarrow {{\,\mathrm{Aut}\,}}(Y_A)\rightarrow 1. \end{aligned}$$
(18)

As we will check in (22), the space \(H^2(\widetilde{Y}_A, \mathscr {O}_{\widetilde{Y}_A}) \) has dimension 1. It is acted on by the group of automorphisms of \(\widetilde{Y}_A\) and this defines another extension

$$\begin{aligned} 1\rightarrow {{\,\mathrm{Aut}\,}}_H^s(\widetilde{Y}_A) \rightarrow {{\,\mathrm{Aut}\,}}_H(\widetilde{Y}_A) \rightarrow {\varvec{\mu }}_r\rightarrow 1. \end{aligned}$$
(19)

The image of  \(\iota \) in \({\varvec{\mu }}_r\) is \(-1\) and \({{\,\mathrm{Aut}\,}}_H^s(\widetilde{Y}_A)\) is the subgroup of elements of \({{\,\mathrm{Aut}\,}}_H(\widetilde{Y}_A)\) that act trivially on \(H^2(\widetilde{Y}_A, \mathscr {O}_{\widetilde{Y}_A}) \) (when \(Y_A^{\ge 3}=\varnothing \), these are exactly, by Hodge theory, the symplectic automorphisms—those that leave any symplectic 2-form on \(\widetilde{Y}_A\) invariant).

We will show in the next proposition (which was kindly provided by A. Kuznetsov) that these extensions are both trivial. For that, we construct an extension

$$\begin{aligned} 1 \rightarrow {\varvec{\mu }}_2 \rightarrow {\widetilde{{{\,\mathrm{Aut}\,}}}}(Y_A) \rightarrow {{\,\mathrm{Aut}\,}}(Y_A) \rightarrow 1 \end{aligned}$$
(20)

as follows. Recall from (1) that there is an embedding \({{\,\mathrm{Aut}\,}}(Y_A) \hookrightarrow {{\,\mathrm{PGL}\,}}(V_6)\). Let G be the inverse image of \({{\,\mathrm{Aut}\,}}(Y_A)\) via the canonical map \({{\,\mathrm{SL}\,}}(V_6)\rightarrow {{\,\mathrm{PGL}\,}}(V_6)\). It is an extension of \({{\,\mathrm{Aut}\,}}(Y_A)\) by \({\varvec{\mu }}_6\) and we set \({\widetilde{{{\,\mathrm{Aut}\,}}}}(Y_A):=G/{\varvec{\mu }}_3\).

The action of G on \(V_6\) induces an action on \(\textstyle {\bigwedge ^{3}}{V}_6\) such that \({\varvec{\mu }}_6\) acts through its cube, hence the latter action factors through an action of \( {\widetilde{{{\,\mathrm{Aut}\,}}}}(Y_A) \). The subspace \(A \subset \textstyle {\bigwedge ^{3}}{V}_6\) is preserved by this action, hence we have a morphism of central extensions

(21)

Lemma A.1

The vertical morphisms in (21) are injective.

Proof

Let \(g\in G\subset {{\,\mathrm{SL}\,}}(V_6)\). Assume that g acts trivially on A. Then it also acts trivially on \(A^\vee \). There is a G-equivariant exact sequence \(0 \rightarrow A \rightarrow \textstyle {\bigwedge ^{3}}{V}_6 \rightarrow A^\vee \rightarrow 0\) which splits G-equivariantly because G is finite. It follows that G also acts trivially on \(\textstyle {\bigwedge ^{3}}{V}_6\). The natural morphism \({{\,\mathrm{PGL}\,}}(V_6) \rightarrow {{\,\mathrm{PGL}\,}}(\textstyle {\bigwedge ^{3}}{V}_6)\) being injective, g is in \({\varvec{\mu }}_6\). Finally, \({\varvec{\mu }}_6/{\varvec{\mu }}_3 \) acts nontrivially on A, hence g is in \({\varvec{\mu }}_3\) and its image in \( {\widetilde{{{\,\mathrm{Aut}\,}}}}(Y_A)\) is 1. This proves that the middle vertical map in (21) is injective.

Assume now that g acts as \(\lambda {{\,\mathrm{Id}\,}}_A\) on A. Its eigenvalues on \(\textstyle {\bigwedge ^{3}}{V}_6\) are then \(\lambda \) and \(\lambda ^{-1}\), both with multiplicity 10. Let \(\lambda _1,\dots ,\lambda _6\) be its eigenvalues on \(V_6\). For all \(1\le i<j<k\le 6\), one then has \(\lambda _i\lambda _j\lambda _k= \lambda \) or \(\lambda ^{-1}\). It follows that if ijklm are all distinct, \(\lambda _i\lambda _j\lambda _k,\lambda _i\lambda _j\lambda _l,\lambda _i\lambda _j\lambda _m\) can only take 2 values, hence \(\lambda _k,\lambda _l,\lambda _m\) can only take 2 values. So, there are at most 2 distinct eigenvalues and one of the eigenspaces, say \(E_{\lambda _1}\), has dimension at least 3. If \(\lambda \ne \lambda ^{-1}\), the eigenspace in \(\textstyle {\bigwedge ^{3}}{V}_6\) for the eigenvalue \(\lambda _1^3\), which is either A or \(A^\vee \), contains \(\textstyle {\bigwedge ^{3}}{E}_{\lambda _1}\). This contradicts the fact that A and \(A^\vee \) contain no decomposable vectors. Therefore, \(\lambda = \lambda ^{-1}\) and g acts as \(\pm {{\,\mathrm{Id}\,}}_A\), and the first part of the proof implies that the image of \(\pm g\) in \( {\widetilde{{{\,\mathrm{Aut}\,}}}}(Y_A)\) is 1. This proves that the rightmost vertical map in (21) is injective. \(\square \)

The following proposition strengthens some results of Beri [8, Proposition 4.1].

Proposition A.2

(Kuznetsov) Let \(A\subset \textstyle {\bigwedge ^{3}}{V}_6\) be a Lagrangian subspace with no decomposable vectors. The extensions (18) and (19) are trivial and \(r=2\); more precisely, there is an isomorphism

$$\begin{aligned} {{\,\mathrm{Aut}\,}}_H(\widetilde{Y}_A)\simeq {{\,\mathrm{Aut}\,}}(Y_A)\times \langle \iota \rangle \end{aligned}$$

that splits (18) and the factor \({{\,\mathrm{Aut}\,}}(Y_A)\) corresponds to the subgroup \({{\,\mathrm{Aut}\,}}_H^s(\widetilde{Y}_A)\) of \({{\,\mathrm{Aut}\,}}_H(\widetilde{Y}_A)\).

Proof

We briefly recall from [45, Section 1.2] (see also [23]) the construction of the double cover \(\pi _A:\widetilde{Y}_A\rightarrow Y_A\). In the terminology of the latter article, one considers the Lagrangian subbundles \(\mathscr {A}_1:=A \otimes \mathscr {O}_{\mathbf{P}(V_6)}\) and \(\mathscr {A}_2:=\textstyle {\bigwedge ^{2}}{T}_{\mathbf{P}(V_6)}(-3)\) of the trivial vector bundle \(\textstyle {\bigwedge ^{3}}{V}_6 \otimes \mathscr {O}_{\mathbf{P}(V_6)}\), and the first Lagrangian cointersection sheaf \( \mathscr {R}_1 := {{\,\mathrm{Coker}\,}}(\mathscr {A}_2\hookrightarrow \mathscr {A}_1^\vee ) \), a rank-1 sheaf with support \(Y_A\). One sets [23, Theorem 5.2(1)]

$$\begin{aligned} \widetilde{Y}_A = {{\,\mathrm{Spec}\,}}(\mathscr {O}_{Y_A} \oplus \mathscr {R}_1(-3)). \end{aligned}$$

In particular, one has

$$\begin{aligned} H^2(\widetilde{Y}_A, \mathscr {O}_{\widetilde{Y}_A}) \simeq H^2(Y_A, \mathscr {R}_1(-3)) \simeq H^3(\mathbf{P}(V_6), \mathscr {A}_2(-3))= H^3(\mathbf{P}(V_6), \textstyle {\bigwedge ^{2}}{T}_{\mathbf{P}(V_6)}(-6))\simeq \mathbf{C}.\nonumber \\ \end{aligned}$$
(22)

The subbundles \(\mathscr {A}_1\) and \(\mathscr {A}_2\) are invariant for the action of \({\widetilde{{{\,\mathrm{Aut}\,}}}}(Y_A)\) on \(\textstyle {\bigwedge ^{3}}{V}_6\), hence the sheaf \(\mathscr {R}_1 \) is \({\widetilde{{{\,\mathrm{Aut}\,}}}}(Y_A)\)-equivariant. Finally, the line bundle \(\mathscr {O}_{\mathbf{P}(V_6)}(-1) \) has a G-linearization (the subgroup \(G\subset {{\,\mathrm{SL}\,}}(V_6)\) was defined right before Lemma A.1). It follows that \(\mathscr {O}_{\mathbf{P}(V_6)}(-3)\) has an \({\widetilde{{{\,\mathrm{Aut}\,}}}}(Y_A)\)-linearization, hence the same is true for the sheaf \(\mathscr {R}_1(-3)\). Therefore, the group \({\widetilde{{{\,\mathrm{Aut}\,}}}}(Y_A)\) acts on \(\widetilde{Y}_A\) and fixes the polarization H.

Observe now that since the nontrivial element of \( {\varvec{\mu }}_2 \subset {\widetilde{{{\,\mathrm{Aut}\,}}}}(Y_A)\) acts by \(-1\) on A, hence also on \( \mathscr {R}_1\), and since it acts by \(-1\) on \(\mathscr {O}(-1)\), hence also on \(\mathscr {O}(-3)\), the group \( {\varvec{\mu }}_2\) acts trivially on \(\mathscr {R}_1(-3)\), hence also on \(\widetilde{Y}_A\). Therefore, the morphism \({\widetilde{{{\,\mathrm{Aut}\,}}}}(Y_A)\rightarrow {{\,\mathrm{Aut}\,}}_H(\widetilde{Y}_A)\) factors through the quotient \({\widetilde{{{\,\mathrm{Aut}\,}}}}(Y_A)/{\varvec{\mu }}_2 = {{\,\mathrm{Aut}\,}}(Y_A)\). In other words, the surjection \({{\,\mathrm{Aut}\,}}_H(\widetilde{Y}_A) \rightarrow {{\,\mathrm{Aut}\,}}(Y_A)\) in (18) has a section and this central extension is trivial.

The action of the group \({{\,\mathrm{Aut}\,}}(\widetilde{Y}_A)\) on the 1-dimensional vector space \(H^2(\widetilde{Y}_A, \mathscr {O}_{\widetilde{Y}_A})\) defines a morphism \({{\,\mathrm{Aut}\,}}(\widetilde{Y}_A)\rightarrow \mathbf{C}^\times \) that maps \(\iota \) to \(-1\). The lift \({\widetilde{{{\,\mathrm{Aut}\,}}}}(Y_A)\rightarrow {{\,\mathrm{Aut}\,}}(Y_A) \hookrightarrow {{\,\mathrm{Aut}\,}}_H(\widetilde{Y}_A)\) acts trivially on \(H^2(\widetilde{Y}_A, \mathscr {O}_{\widetilde{Y}_A})\) because its action is induced by the action of \({{\,\mathrm{PGL}\,}}(V_6)\), which has no nontrivial characters. This gives a surjection \({{\,\mathrm{Aut}\,}}_H(\widetilde{Y}_A) \rightarrow \langle \iota \rangle \) which is trivial on the image of the section \({{\,\mathrm{Aut}\,}}(Y_A) \hookrightarrow {{\,\mathrm{Aut}\,}}_H(\widetilde{Y}_A)\). This implies that the extension (19) is also trivial and \(r=2\). The theorem is therefore proved. \(\square \)

1.2 A. 2 Moduli space and period map of (double) EPW sextics

Quasi-smooth EPW sextics admit an affine coarse moduli space \({\mathbf {M}^{\mathrm {EPW},0}}\), constructed in [46] as a GIT quotient by \({{\,\mathrm{PGL}\,}}(V_6)\) of an affine open dense subset of the space of Lagrangian subspaces in \(\textstyle {\bigwedge ^{3}}{V}_6\).

Let \(\widetilde{Y}\) be a hyperkähler fourfold of K3\(^{[2]}\)-type (such as a double EPW sextic). The lattice \(H^2(\widetilde{Y},\mathbf{Z})\) (endowed with the Beauville–Bogomolov quadratic form \(q_{BB}\)) is isomorphic to the lattice

$$\begin{aligned} L:=U^{\oplus 3}\oplus E_8(-1)^{\oplus 2}\oplus (-2), \end{aligned}$$
(23)

where U is the hyperbolic plane \(\bigl ( \mathbf{Z}^2, \bigl ({\begin{matrix} 0&{} 1\\ 1 &{} 0 \end{matrix}}\bigr )\bigr )\), \( E_8(-1)\) is the negative definite even unimodular rank-8 lattice, and (m) is the rank-1 lattice with generator of square m.

Fix a class \(h\in L\) with \(h^2=2\). These classes are all in the same O(L)-orbit and

$$\begin{aligned} h^\bot \simeq U^{\oplus 2}\oplus E_8(-1)^{\oplus 2}\oplus (-2)^{\oplus 2}. \end{aligned}$$
(24)

The space

$$\begin{aligned} \begin{aligned} \Omega _{h} :={}&\{ [x]\in \mathbf{P}(L \otimes \mathbf{C})\mid x\cdot h=0,\ x\cdot x=0,\ x\cdot {\bar{x}}>0\}\\ {}={}&\{ [x]\in \mathbf{P}(h^\bot \otimes \mathbf{C})\mid x\cdot x=0,\ x\cdot {\bar{x}}>0\} \end{aligned} \end{aligned}$$

has two connected components, interchanged by complex conjugation, which are Hermitian symmetric domains. It is acted on by the group

$$\begin{aligned} \{g\in O(L)\mid g(h)=h\}, \end{aligned}$$

also with two connected components, which is also the index-2 subgroup \({\widetilde{O}}(h^\bot )\) of \(O(h^\bot )\) that consists of isometries that act trivially on the discriminant group \({{\,\mathrm{Disc}\,}}(h^\bot )\simeq (\mathbf{Z}/2\mathbf{Z})^2\). The quotient is an irreducible quasi-projective variety [3] and the period map

$$\begin{aligned} \wp :{\mathbf {M}^{\mathrm {EPW},0}} \longrightarrow {\widetilde{O}}(h^\bot )\backslash \Omega _{h},\quad [\widetilde{Y}]\longmapsto [H^{2,0}(\widetilde{Y})] \end{aligned}$$
(25)

is algebraic [13]. It is an open embedding by Verbitsky’s Torelli theorem [32, 39, 51].

If \(A\subset \textstyle {\bigwedge ^{3}}{V}_6\) is a Lagrangian such that \(\widetilde{Y}_A\) is smooth with period \([x]\in \mathbf{P}(L \otimes \mathbf{C})\) (well defined only up to the action of \( {\widetilde{O}}(h^\bot )\)), the Picard group \({{\,\mathrm{Pic}\,}}(\widetilde{Y}_A)\) is, by Hodge theory, isomorphic to \(x^\bot \cap L\).

1.3 A.3 The rank lattice \({{\mathsf {S}}}\)

Since it plays a central role in this article, we introduce here the rank-20 lattice \({{\mathsf {S}}}\), first defined in [41, Example 2.9] (see also [42, Example 2.5.9]) where it is denoted by \(S_{11}\). It is defined by the \(20\times 20\) Gram matrix

$$\begin{aligned} { \left( \begin{array}{cccccccccccccccccccc} -4 &{} 1 &{} -2 &{} -2 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} -1 &{} 2 &{} 1 &{} -1 &{} 2 &{} -1 &{} -2 &{} -2 &{} 2 &{} 1 &{} -1 \\ 1 &{} -4 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} 1 &{} -1 &{} 2 &{} -1 &{} -2 &{} 2 &{} 0 &{} -1 &{} 0 &{} 0 &{} -1 &{} -2 &{} 1 \\ -2 &{} -1 &{} -4 &{} -2 &{} -1 &{} -1 &{} 0 &{} 1 &{} 0 &{} -1 &{} 1 &{} 0 &{} -1 &{} 2 &{} -2 &{} -1 &{} -1 &{} 0 &{} 0 &{} 1 \\ -2 &{} -1 &{} -2 &{} -4 &{} 0 &{} 0 &{} -2 &{} 0 &{} -1 &{} 0 &{} 2 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} -1 &{} 1 &{} 0 &{} -1 \\ -1 &{} -1 &{} -1 &{} 0 &{} -4 &{} 1 &{} -1 &{} 2 &{} -2 &{} -1 &{} 1 &{} 0 &{} -1 &{} 0 &{} -2 &{} -2 &{} 0 &{} 1 &{} 1 &{} -1 \\ 1 &{} -1 &{} -1 &{} 0 &{} 1 &{} -4 &{} 0 &{} -1 &{} 0 &{} 1 &{} -2 &{} -1 &{} 0 &{} -1 &{} -1 &{} 0 &{} -1 &{} 0 &{} -1 &{} 1 \\ -1 &{} -1 &{} 0 &{} -2 &{} -1 &{} 0 &{} -4 &{} 1 &{} -2 &{} 1 &{} 1 &{} 1 &{} 0 &{} -1 &{} 0 &{} -1 &{} 0 &{} 2 &{} 0 &{} -2 \\ 1 &{} 1 &{} 1 &{} 0 &{} 2 &{} -1 &{} 1 &{} -4 &{} 0 &{} 0 &{} -1 &{} 1 &{} 1 &{} 0 &{} 2 &{} 1 &{} 0 &{} -1 &{} 1 &{} 0 \\ -1 &{} -1 &{} 0 &{} -1 &{} -2 &{} 0 &{} -2 &{} 0 &{} -4 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} -1 &{} -2 &{} 0 &{} 2 &{} 0 &{} -2 \\ -1 &{} 2 &{} -1 &{} 0 &{} -1 &{} 1 &{} 1 &{} 0 &{} 0 &{} -4 &{} 1 &{} 1 &{} -2 &{} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 0 \\ 2 &{} -1 &{} 1 &{} 2 &{} 1 &{} -2 &{} 1 &{} -1 &{} 0 &{} 1 &{} -4 &{} -2 &{} 2 &{} -1 &{} 0 &{} 0 &{} 0 &{} -1 &{} -2 &{} 1 \\ 1 &{} -2 &{} 0 &{} 1 &{} 0 &{} -1 &{} 1 &{} 1 &{} 1 &{} 1 &{} -2 &{} -4 &{} 1 &{} 0 &{} -1 &{} 0 &{} -1 &{} -1 &{} -2 &{} 2 \\ -1 &{} 2 &{} -1 &{} 0 &{} -1 &{} 0 &{} 0 &{} 1 &{} 1 &{} -2 &{} 2 &{} 1 &{} -4 &{} 0 &{} -1 &{} 0 &{} 0 &{} 1 &{} 2 &{} 0 \\ 2 &{} 0 &{} 2 &{} 1 &{} 0 &{} -1 &{} -1 &{} 0 &{} 0 &{} 1 &{} -1 &{} 0 &{} 0 &{} -4 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} -1 \\ -1 &{} -1 &{} -2 &{} 0 &{} -2 &{} -1 &{} 0 &{} 2 &{} -1 &{} 0 &{} 0 &{} -1 &{} -1 &{} 1 &{} -4 &{} -2 &{} -1 &{} 1 &{} 0 &{} 0 \\ -2 &{} 0 &{} -1 &{} 0 &{} -2 &{} 0 &{} -1 &{} 1 &{} -2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} -2 &{} -4 &{} -2 &{} 2 &{} 0 &{} -1 \\ -2 &{} 0 &{} -1 &{} -1 &{} 0 &{} -1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} -1 &{} 0 &{} 1 &{} -1 &{} -2 &{} -4 &{} 1 &{} 0 &{} 0 \\ 2 &{} -1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 2 &{} -1 &{} 2 &{} 1 &{} -1 &{} -1 &{} 1 &{} 0 &{} 1 &{} 2 &{} 1 &{} -4 &{} 0 &{} 2 \\ 1 &{} -2 &{} 0 &{} 0 &{} 1 &{} -1 &{} 0 &{} 1 &{} 0 &{} 1 &{} -2 &{} -2 &{} 2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -4 &{} 1 \\ -1 &{} 1 &{} 1 &{} -1 &{} -1 &{} 1 &{} -2 &{} 0 &{} -2 &{} 0 &{} 1 &{} 2 &{} 0 &{} -1 &{} 0 &{} -1 &{} 0 &{} 2 &{} 1 &{} -4 \end{array} \right) } \end{aligned}$$

It is negative definite, even, and contains no \((-2)\)-classes. Its discriminant group is \((\mathbf{Z}/11\mathbf{Z})^2\) and its discriminant form is \(\left( {\begin{matrix} -2/11 &{} 0\\ 0 &{} -2/11 \end{matrix}}\right) \). It is in the same genusFootnote 1 as the lattice

$$\begin{aligned} S:= E_8(-1)^{\oplus 2}\oplus \begin{pmatrix} -2 &{}-1 \\ -1 &{} -6 \end{pmatrix}^{\oplus 2}. \end{aligned}$$

However, the lattices \({{\mathsf {S}}}\) and S are not isomorphic (because \({{\mathsf {S}}}\) does not represent \(-2\)) but the indefinite lattices \((2)\oplus {{\mathsf {S}}}\) and \((2)\oplus S\) are by [43, Corollary 1.13.3].

A direct computation shows that the lattice \({{\mathsf {S}}}\) contains the rank-5 lattice with diagonal quadratic form \((-4,-4,-4,-6,-8)\). By [9, Section 6(iii)], the quadratic form on the last four variables represents every even negative integer with the exception of \(-2\), and the first variable can be used to ensure that all these integers can be primitively represented in \({{\mathsf {S}}}\).

1.4 A.4 Automorphisms of prime order

Let \(\widetilde{Y}\) be a hyperkähler fourfold of K3\(^{[2]}\)-type. In the lattice \((H^2(\widetilde{Y},\mathbf{Z}),q_{BB})\) mentioned in Appendix A.2, we consider the transcendental lattice

$$\begin{aligned} {{\,\mathrm{Tr}\,}}(\widetilde{Y}) :={{\,\mathrm{Pic}\,}}(\widetilde{Y})^\bot \subset H^2(\widetilde{Y},\mathbf{Z}) . \end{aligned}$$

The automorphism group \({{\,\mathrm{Aut}\,}}(\widetilde{Y})\) is known to act faithfully by isometries on the lattice \((H^2(\widetilde{Y},\mathbf{Z}),q_{BB})\) and preserves the sublattices \({{\,\mathrm{Pic}\,}}(\widetilde{Y})\) and \({{\,\mathrm{Tr}\,}}(\widetilde{Y})\). If G is a subset of \({{\,\mathrm{Aut}\,}}(\widetilde{Y})\), we denote by \(T_G(\widetilde{Y})\) the invariant lattice (of elements of \(H^2(\widetilde{Y},\mathbf{Z}) \) that are invariant by all elements of G) and by \(S_G(\widetilde{Y}):=T_G(\widetilde{Y})^\bot \) its orthogonal in \(H^2(\widetilde{Y},\mathbf{Z}) \).

Many results are known about automorphisms of prime order p of hyperkähler fourfolds. We restrict ourselves to the case \(p\ge 11\).

Theorem A.3

Let \(\widetilde{Y}\) be a projective hyperkähler fourfold of K3\(^{[2]}\)-type and let g be a symplectic automorphism of \(\widetilde{Y}\) of prime order \(p\ge 11\). Then \(p= 11\) and there are inclusions \({{\,\mathrm{Tr}\,}}(\widetilde{Y})\subset T_g(\widetilde{Y})\) and \(S_g(\widetilde{Y})\subset {{\,\mathrm{Pic}\,}}(\widetilde{Y})\). The lattice \(S_g(\widetilde{Y})\) is isomorphic to the lattice \({{\mathsf {S}}}\) and \(\rho (\widetilde{Y})=21\). The possible lattices \(T_g(\widetilde{Y})\) are

$$\begin{aligned} \begin{pmatrix} 2 &{}1 &{}0 \\ 1 &{}6&{}0\\ 0&{}0&{}22 \end{pmatrix}\quad or \quad \begin{pmatrix} 6&{}2 &{}2 \\ 2 &{}8&{}-3\\ 2&{}-3&{}8 \end{pmatrix}. \end{aligned}$$

Proof

The proof is a compilation of previously known results on symplectic automorphisms. The bound \(p\le 11\) is [41, Corollary 2.13]. The inclusions and the properties of the lattice \(S_g(\widetilde{Y})\) are in [40, Lemma 3.5], the equality \(\rho (\widetilde{Y})=21\) is in [41, Proposition 1.2], the lattice \(S_g(\widetilde{Y})\) is determined in [42, Theorem 7.2.7], and the possible lattices \(T_g(\widetilde{Y})\) in [12, Section 5.5.2]. \(\square \)

This theorem applies in particular to (smooth) double EPW sextics \(\widetilde{Y}_A\). We are interested in automorphisms that preserve the canonical degree-2 polarization H. By Proposition A.2, the group of these automorphisms, modulo the covering involution \(\iota \), is isomorphic to the group of automorphisms of the EPW sextic \(Y_A\).

Corollary A.4

Let \(\widetilde{Y}_A \) be a smooth double EPW sextic and let g be an automorphism of \(\widetilde{Y}_A\) of prime order \(p\ge 11\) that fixes the polarization H. Then \(p=11\) and Footnote 2

$$\begin{aligned} \begin{aligned} S_{g}(X) \simeq {{\mathsf {S}}},\qquad T_{g}(\widetilde{Y}_A)&\simeq \begin{pmatrix} 2 &{}1 \\ 1 &{}6 \end{pmatrix}\oplus (22),\qquad {{\,\mathrm{Tr}\,}}(\widetilde{Y}_A)\simeq (22)^{\oplus 2}, \\ {{\,\mathrm{Pic}\,}}(\widetilde{Y}_A)= \mathbf{Z}H \oplus {{\mathsf {S}}}&\simeq (2)\oplus E_8(-1)^{\oplus 2}\oplus \begin{pmatrix} -2 &{}-1 \\ -1 &{} -6 \end{pmatrix}^{\oplus 2} . \end{aligned} \end{aligned}$$

In particular, the fourfold \(\widetilde{Y}_A\) has maximal Picard number 21.

Proof

By Proposition A.2, the automorphism g is symplectic (all nonsymplectic automorphisms have even order). Since \(H\in T_{g}(\widetilde{Y}_A)\) and \(q_{BB}(H)=2\), and the second lattice in Theorem A.3 contains no classes of square 2, there is only one possibility for \(T_{g}(\widetilde{Y}_A)\) (see also [42, Section 7.4.4]). There are only two (opposite) classes of square 2 in that lattice, so we find \( {{\,\mathrm{Tr}\,}}(\widetilde{Y}_A)\) as their orthogonal.

We know that \({{\,\mathrm{Pic}\,}}(\widetilde{Y}_A)\) is an overlattice of \(\mathbf{Z}H\oplus S_{g} (\widetilde{Y}_A)\). Since the latter has no nontrivial overlattices (its discriminant group has no nontrivial isotropic elements), they are equal. Finally, the last isomorphism in the statement follows from the discussion at the end of Section A.3. \(\square \)

We prove in Theorem 4.2 that the double EPW sextic \(\widetilde{Y}_{\mathbb A}\) is the only smooth double EPW sextic with an automorphism of order 11 that fixes the polarization H.

In Hassett’s terminology (recalled in [26, Section 4]), a (smooth) double EPW sextic \(\widetilde{Y}_A\) is special of discriminant d if there exists a primitive rank-2 lattice \(K\subset {{\,\mathrm{Pic}\,}}(\widetilde{Y}_A)\) containing the polarization H such that \({{\,\mathrm{disc}\,}}(K^\bot )=-d\) (the orthogonal complement is taken in \((H^2(\widetilde{Y}_A,\mathbf{Z}),q_{BB})\)); this may only happen when \(d\equiv 0,2,4\pmod {8}\) and \(d>8\) [26, Proposition 4.1 and Remark 6.3]. The fourfold \(\widetilde{Y}_A\) has an associated K3 surface if moreover the lattice \(K^\bot \) is isomorphic to the opposite of the primitive cohomology lattice of a pseudo-polarized K3 surface (necessarily of degree d); a necessary condition for this to happen is \(d\equiv 2,4\pmod {8}\) (this was proved in [25, Proposition 6.6] for GM fourfolds but the computation is the same).

Proposition A.5

The double EPW sextic \(\widetilde{Y}_{\mathbb A}\) is special of discriminant d if and only if d is a multiple of 8 greater than 8. In particular, it has no associated K3 surfaces.

Proof

Assume that \(\widetilde{Y}_{\mathbb A}\) is special of discriminant d. Since \({{\,\mathrm{Pic}\,}}(\widetilde{Y}_{\mathbb A})\simeq \mathbf{Z}H\oplus {{\mathsf {S}}}\), the required lattice K as above is of the form \(\langle H,\kappa \rangle \), where \(\kappa \in {{\mathsf {S}}}\) is primitive. Since \({{\,\mathrm{Disc}\,}}({{\mathsf {S}}})\simeq (\mathbf{Z}/11\mathbf{Z})^2\), the divisibility \(\mathop \mathrm{div}\nolimits _{{{\mathsf {S}}}}(\kappa )\) divides 11 and, since \({{\,\mathrm{Disc}\,}}(H^\bot )\simeq (\mathbf{Z}/2\mathbf{Z})^2\) (see [26, (1)]), the divisibility \(\mathop \mathrm{div}\nolimits _{H^\bot }(\kappa )\) divides 2, but also divides \(\mathop \mathrm{div}\nolimits _{{{\mathsf {S}}}}(\kappa )\) (because \({{\mathsf {S}}}\subset H^\bot \)). It follows that \(\mathop \mathrm{div}\nolimits _{H^\bot }(\kappa )=1\). The lattice \(\langle H,\kappa \rangle ^\bot \) therefore has discriminant \(4\kappa ^2\) by the formula [26, (4)].

It follows that \(\widetilde{Y}_{\mathbb A}\) is special of discriminant d if and only if \(d\equiv 0 \pmod 8\) and \({{\mathsf {S}}}\) primitively represents \(-d/4\). The proposition then follows from the discussion at the end of Appendix A.3. \(\square \)

1.5 A.5 Double EPW surfaces and their automorphisms

Let \(Y_A\subset \mathbf{P}(V_6)\) be an EPW sextic, where \(A\subset \textstyle {\bigwedge ^{3}}{V}_6\) is a Lagrangian subspace with no decomposable vectors. By [23, Theorem 5.2(2)], there is a canonical connected double covering

$$\begin{aligned} \widetilde{Y}_A^{\ge 2}\longrightarrow Y_A^{\ge 2} \end{aligned}$$
(26)

between integral surfaces, with covering involution \(\tau \), branched over the finite set \(Y_A^{\ge 3}\).

We compare automorphisms of \(Y_A\) with those of \(\widetilde{Y}_A^{\ge 2}\). Any automorphism of \(Y_A\) induces an automorphism of its singular locus \(Y_A^{\ge 2}\). This defines a morphism \({{\,\mathrm{Aut}\,}}(Y_A)\rightarrow {{\,\mathrm{Aut}\,}}(Y_A^{\ge 2})\). Since \({{\,\mathrm{Aut}\,}}(Y_A)\) is a subgroup of \({{\,\mathrm{PGL}\,}}(V_6)\) and the surface \(Y_A^{\ge 2}\) is not contained in a hyperplane, this morphism is injective.

Proposition A.6

(Kuznetsov) Let \(A\subset \textstyle {\bigwedge ^{3}}{V}_6\) be a Lagrangian subspace with no decomposable vectors. Any element of \({{\,\mathrm{Aut}\,}}(Y_A)\) lifts to an automorphism of \(\widetilde{Y}_A^{\ge 2}\). These lifts form a subgroup of \( {{\,\mathrm{Aut}\,}}(\widetilde{Y}_A^{\ge 2})\) which is isomorphic to the group \({\widetilde{{{\,\mathrm{Aut}\,}}}}(Y_A)\) in the extension (20) via an isomorphism that takes \(\langle \tau \rangle \) to \({\varvec{\mu }}_2\).

Proof

The proof follows the exact same steps as the proof of Proposition A.2, whose notation we keep. By [23, Theorem 5.2(2)], the surface \(\widetilde{Y}_A^{\ge 2}\) is defined as

$$\begin{aligned} \widetilde{Y}^{\ge 2}_A = {{\,\mathrm{Spec}\,}}(\mathscr {O}_{Y^{\ge 2}_A} \oplus \mathscr {R}_2(-3)), \end{aligned}$$
(27)

where \( \mathscr {R}_2 = (\textstyle {\bigwedge ^{2}}{\mathscr {R}}_1\vert _{Y^{\ge 2}_A})^{\vee \vee }\). As in the proof of Proposition A.2, the group \({\widetilde{{{\,\mathrm{Aut}\,}}}}(Y_A) \) acts on \(\widetilde{Y}^{\ge 2}_A\) and the nontrivial element of \( {\varvec{\mu }}_2 \) acts by \(-1\) on both \(\mathscr {R}_1\) and \(\mathscr {O}(-3)\). It follows that it acts by 1 on \( \mathscr {R}_2\) and by \(-1\) on \( \mathscr {R}_2(-3)\), hence as the involution \(\tau \) on \(\widetilde{Y}^{\ge 2}_A\). This proves the proposition. \(\square \)

It is possible to deform the double cover (26) to the canonical double étale covering associated with the (smooth) variety of lines on a quartic double solid (see the proof of [24, Proposition 2.5]), so we can use Welters’ calculations in [52, Theorem (3.57) and Proposition (3.60)]. In particular, the abelian group \(H_1(\widetilde{Y}_A^{\ge 2},\mathbf{Z})\) is free of rank 20 (and \(\tau \) acts as \(-{{\,\mathrm{Id}\,}}\)) and there are canonical isomorphisms [24, Proposition 2.5]

$$\begin{aligned} \begin{aligned} T_{{{\,\mathrm{Alb}\,}}(\widetilde{Y}_A^{\ge 2}),0}&\simeq H^1(\widetilde{Y}_A^{\ge 2},\mathscr {O}_{\widetilde{Y}_A^{\ge 2}})\simeq A, \\ H^2(Y_A^{\ge 2},\mathbf{C})&\simeq \textstyle {\bigwedge ^{2}}{H}^1(\widetilde{Y}_A^{\ge 2},\mathbf{C})\simeq \textstyle {\bigwedge ^{2}}{(}A\oplus {\bar{A}}). \end{aligned} \end{aligned}$$
(28)

The Albanese variety \({{\,\mathrm{Alb}\,}}(\widetilde{Y}_A^{\ge 2})\) is thus an abelian variety of dimension 10 and one can consider the analytic representation (see Section C.1)

$$\begin{aligned} \rho _a:{{\,\mathrm{Aut}\,}}(\widetilde{Y}_A^{\ge 2})\longrightarrow {{\,\mathrm{GL}\,}}(T_{{{\,\mathrm{Alb}\,}}(\widetilde{Y}_A^{\ge 2}),0})\simeq {{\,\mathrm{GL}\,}}(A). \end{aligned}$$

Recall from Proposition A.6 that there is an injective morphism \({\widetilde{{{\,\mathrm{Aut}\,}}}}(Y_A)\hookrightarrow {{\,\mathrm{Aut}\,}}(\widetilde{Y}_A^{\ge 2})\).

Proposition A.7

Let \(Y_A\) be a quasi-smooth EPW sextic. The restriction of the analytic representation \(\rho _a\) to the subgroup \({\widetilde{{{\,\mathrm{Aut}\,}}}}(Y_A)\) of \( {{\,\mathrm{Aut}\,}}(\widetilde{Y}_A^{\ge 2})\) is the injective middle vertical map in the diagram (21).

Proof

The morphism \(\rho _a\) is the representation of the group \({{\,\mathrm{Aut}\,}}(\widetilde{Y}_A^{\ge 2})\) on the vector space

$$\begin{aligned} T_{{{\,\mathrm{Alb}\,}}(\widetilde{Y}_A^{\ge 2}),0}\simeq H^1(\widetilde{Y}_A^{\ge 2},\mathscr {O}_{\widetilde{Y}_A^{\ge 2}}). \end{aligned}$$

As in the proof of [24, Proposition 2.5]), there are canonical isomorphisms

$$\begin{aligned} H^1(\widetilde{Y}_A^{\ge 2},\mathscr {O}_{\widetilde{Y}_A^{\ge 2}})\simeq H^1(Y_A^{\ge 2},\mathscr {R}_2(-3)) \simeq H^1(Y_A^{\ge 2},\mathscr {O}_{Y_A^{\ge 2}}(3))^\vee , \end{aligned}$$

where the first isomorphism comes from (27) and the second one from Serre duality (because \( \mathscr {R}_2 \) is the canonical sheaf of \(Y_A^{\ge 2} \)).

As in the proof of Proposition A.6, the sheaf \( \mathscr {O}_{Y_A^{\ge 2}}(3)\) has an \({\widetilde{{{\,\mathrm{Aut}\,}}}}(Y_A) \)-linearization, where \({{\,\mathrm{Aut}\,}}(Y_A)\) acts on \(Y^{\ge 2}_A\) by restriction and the nontrivial element of \( {\varvec{\mu }}_2 \) acts by \(-1\) on \( \mathscr {O}_{Y_A^{\ge 2}}(3)\).

By construction, the resolution

$$\begin{aligned} 0\rightarrow (\textstyle {\bigwedge ^{2}}{\mathscr {A}}_2)(-6)\rightarrow (\mathscr {A}_1^\vee \otimes \mathscr {A}_2)(-6)\rightarrow ({{\,\mathrm{Sym}\,}}^2\!\mathscr {A}_1)(-6)\oplus \mathscr {O}_{\mathbf{P}(V_6)}(-6)\rightarrow \mathscr {O}_{\mathbf{P}(V_6)}\rightarrow \mathscr {O}_{Y_A^{\ge 2}}\rightarrow 0 \end{aligned}$$

given in [21, (33)] is \({\widetilde{{{\,\mathrm{Aut}\,}}}}(Y_A) \)-equivariant, hence induces an \({\widetilde{{{\,\mathrm{Aut}\,}}}}(Y_A)\)-equivariant isomorphism

$$\begin{aligned} H^1(Y_A^{\ge 2},\mathscr {O}_{Y_A^{\ge 2}}(3))\simeq H^3(\mathbf{P}(V_6),(\mathscr {A}_1^\vee \otimes \mathscr {A}_2)(-3)) =A^\vee \otimes H^3(\mathbf{P}(V_6), \mathscr {A}_2(-3)). \end{aligned}$$

As already noted during the proof of Proposition A.2, \({\widetilde{{{\,\mathrm{Aut}\,}}}}(Y_A)\) acts trivially on the 1-dimensional vector space \(H^3(\mathbf{P}(V_6), \mathscr {A}_2(-3))=H^3(\mathbf{P}(V_6), \textstyle {\bigwedge ^{2}}{T}_{\mathbf{P}(V_6)}(-6))\). All this proves that the action of \({{\,\mathrm{Aut}\,}}(\widetilde{Y}_A^{\ge 2})\) on \(T_{{{\,\mathrm{Alb}\,}}(\widetilde{Y}_A^{\ge 2}),0}\) is indeed given by the desired morphism. \(\square \)

1.6 A.6 Automorphisms of GM varieties

Let as before \(V_6\) be a 6-dimensional vector space and let \(A\subset \textstyle {\bigwedge ^{3}}{V}_6\) be a Lagrangian subspace with no decomposable vectors. Let \(V_5\subset V_6\) be a hyperplane and let X be the associated (smooth ordinary) GM variety (Sect. 2.2). One has (see (3))

$$\begin{aligned} {{\,\mathrm{Aut}\,}}(X)\simeq \{ g\in {{\,\mathrm{PGL}\,}}(V_6)\mid \textstyle {\bigwedge ^{3}}{g}(A)=A,\ g(V_5)=V_5\}. \end{aligned}$$

Since the extension (20) splits (Proposition A.2), there is a lift

$$\begin{aligned} {{\,\mathrm{Aut}\,}}(X)\longrightarrow {{\,\mathrm{GL}\,}}(A) \end{aligned}$$
(29)

(see (21)) which is injective by Lemma A.1.

When the dimension of X is either 3 or 5, its intermediate Jacobian \({{\,\mathrm{Jac}\,}}(X)\) is a 10-dimensional abelian variety. By [24, Theorem 1.1], it is canonically isomorphic to \( {{\,\mathrm{Alb}\,}}(\widetilde{Y}_A^{\ge 2}) \) (see (15)). Therefore, there is an isomorphism

$$\begin{aligned} T_{{{\,\mathrm{Jac}\,}}(X),0}{{\,\mathrm{{\mathop {\longrightarrow }\limits ^{{}_{\scriptstyle \sim }}}}\,}}T_{{{\,\mathrm{Alb}\,}}(\widetilde{Y}_A^{\ge 2}),0}. \end{aligned}$$

Together with the isomorphism (28), this gives an analytic representation

$$\begin{aligned} \rho _{a,X}:{{\,\mathrm{Aut}\,}}(X)\longrightarrow {{\,\mathrm{GL}\,}}(T_{{{\,\mathrm{Jac}\,}}(X),0}){{\,\mathrm{{\mathop {\longrightarrow }\limits ^{{}_{\scriptstyle \sim }}}}\,}}{{\,\mathrm{GL}\,}}(A). \end{aligned}$$

Proposition A.8

The analytic representation \(\rho _{a,X}\) coincides with the injective morphism (29). Equivalently, the isomorphism  (15) is \({{\,\mathrm{Aut}\,}}(X)\)-equivariant.

Proof

Assume \(\dim (X)=3\) and choose a line \(L_0\subset X\). The isomorphism \( {{\,\mathrm{Alb}\,}}(\widetilde{Y}_A^{\ge 2}){{\,\mathrm{{\mathop {\rightarrow }\limits ^{{}_{\scriptstyle \sim }}}}\,}}{{\,\mathrm{Jac}\,}}(X) \) was then constructed in [24, Theorem 4.4] from the Abel–Jacobi map

$$\begin{aligned} {{\,\mathrm{\mathsf {A\!J}}\,}}_{Z_{L_0}}:H_1( \widetilde{Y}_A^{\ge 2},\mathbf{Z})\longrightarrow H_3(X,\mathbf{Z}) \end{aligned}$$

associated with a family \(Z_{L_0}\subset X\times \widetilde{Y}_A^{\ge 2}\) of curves on X parametrized by \(\widetilde{Y}_A^{\ge 2}\). Although the family \(Z_{L_0}\) does depend on the choice of \(L_0\), the map \({{\,\mathrm{\mathsf {A\!J}}\,}}_{Z_{L_0}}\) does not.

Let \(g\in {{\,\mathrm{Aut}\,}}(X)\) (also considered as an automorphism of \(\widetilde{Y}_A^{\ge 2}\)). By the functoriality properties of the Abel–Jacobi map [24, Lemma 3.1], we obtain

$$\begin{aligned} {{\,\mathrm{\mathsf {A\!J}}\,}}_{Z_{L_0}}\circ g_*={{\,\mathrm{\mathsf {A\!J}}\,}}_{({{\,\mathrm{Id}\,}}_{X}\times g)^*(Z_{L_0})}= {{\,\mathrm{\mathsf {A\!J}}\,}}_{( g\times {{\,\mathrm{Id}\,}}_{\widetilde{Y}_A^{\ge 2}})_*(Z_{g^{-1}(L_0)})}=g_*\circ {{\,\mathrm{\mathsf {A\!J}}\,}}_{Z_{g^{-1}(L_0)}}, \end{aligned}$$

which proves the proposition. When \(\dim (X)=5\), the proof is similar, except that \(Z_{\Pi _0}\) is now a family of surfaces in X that depends on a plane \(\Pi _0\subset X\). \(\square \)

Appendix B: Representations of the group \(\mathbb {G}\)

The group \(\mathbb {G}:={{\,\mathrm{PSL}\,}}(2,\mathbf{F}_{11})\) is the only simple group of order \(660=2^2\cdot 3\cdot 5\cdot 11\). It is generated by the classes

$$\begin{aligned} a=\begin{pmatrix} 5 &{}0 \\ 0 &{} 9 \end{pmatrix},\quad b=\begin{pmatrix} 3 &{}5 \\ -5 &{} 3 \end{pmatrix},\quad c=\begin{pmatrix} 1 &{} 1 \\ 0 &{} 1 \end{pmatrix}, \end{aligned}$$

of respective orders 5, 6, and 11. We let \( I_2\) be the identity matrix.

The group \(\mathbb {G}\) has 8 irreducible \(\mathbf{C}\)-representations, of dimensions 1, 5, 5, 10, 10, 11, 12, and 12. We give in Table 3 the characters for 4 of these irreducible representations.

Table 3 Partial character table for \(\mathbb {G}\)

As before, we have set (where \(\zeta _{11}=e^{\frac{2i\pi }{11}}\))

$$\begin{aligned} \lambda :=\zeta _{11}^{1^2}+\zeta _{11}^{2^2}+\zeta _{11}^{3^2}+\zeta _{11}^{4^2}+\zeta _{11}^{5^2}=\zeta _{11}+\zeta _{11}^3+\zeta _{11}^4+\zeta _{11}^5+\zeta _{11}^9=\tfrac{1}{2}(-1+\sqrt{-11}). \end{aligned}$$

The representation \(\xi \) (which appears in Sect. 3.1) has a realization in the matrix ring \(\mathscr {M}_5(\mathbf{C})\) for which

$$\begin{aligned} \xi (a)= \begin{pmatrix} 0&{}0&{}0&{}0&{}1\\ 1&{}0&{}0&{}0&{}0\\ 0&{}1&{}0&{}0&{}0\\ 0&{}0&{}1&{}0&{}0\\ 0&{}0&{}0&{}1&{}0 \end{pmatrix},\quad \xi (c)= \begin{pmatrix} \zeta _{11}&{}0&{}0&{}0&{}0\\ 0&{}\zeta _{11}^4&{}0&{}0&{}0\\ 0&{}0&{}\zeta _{11}^5&{}0&{}0\\ 0&{}0&{}0&{}\zeta _{11}^9&{}0\\ 0&{}0&{}0&{}0&{}\zeta _{11}^3 \end{pmatrix} . \end{aligned}$$
(30)

Every irreducible character of \(\mathbb {G}\) has Schur index 1 (see [50, § 12.2], [28, Theorem 6.1]). In particular, the representation \(\textstyle {\bigwedge ^{2}}{\xi }\), having an integral character, can be defined over \(\mathbf{Q}\) and even, by a theorem of Burnside [15], over \(\mathbf{Z}\), that is, by a morphism \(\mathbb {G}\rightarrow {{\,\mathrm{GL}\,}}(10,\mathbf{Z})\). The representation \(\textstyle {\bigwedge ^{2}}{\xi }\) is self-dual, so there is a \(\mathbb {G}\)-equivariant isomorphism

$$\begin{aligned} w:\textstyle {\bigwedge ^{2}}{V}_\xi {{\,\mathrm{{\mathop {\longrightarrow }\limits ^{{}_{\scriptstyle \sim }}}}\,}}\textstyle {\bigwedge ^{2}}{V}_\xi ^\vee , \end{aligned}$$
(31)

unique up to multiplication by a nonzero scalar, and it is symmetric [50, prop. 38].

Appendix C: Decomposition of abelian varieties with automorphisms

We gather here a few very standard notation and facts about abelian varieties. Let X be a complex abelian variety. We denote by \({{\,\mathrm{Pic}\,}}(X)\) the group of isomorphism classes of line bundles on X, by \({{\,\mathrm{Pic}\,}}^0(X)\subset {{\,\mathrm{Pic}\,}}(X)\) the subgroup of classes of line bundles that are algebraically equivalent to 0, and by \({{\,\mathrm{NS}\,}}(X)\) the Néron–Severi group \({{\,\mathrm{Pic}\,}}(X)/{{\,\mathrm{Pic}\,}}^0(X)\), a free abelian group of finite rank. The group \({{\,\mathrm{Pic}\,}}^0(X)\) has a canonical structure of an abelian variety; it is called the dual abelian variety. Any endomorphism u of X induces an endomorphism \({\widehat{u}}\) of \({{\,\mathrm{Pic}\,}}^0(X)\).

Given the class \(\theta \in {{\,\mathrm{NS}\,}}(X)\) of a line bundle L on X, we let \(\varphi _{\theta }\) be the morphism

$$\begin{aligned} \begin{aligned} X&\longrightarrow {{\,\mathrm{Pic}\,}}^0(X) \\ x&\longmapsto \tau _x^*L\otimes L^{-1} \end{aligned} \end{aligned}$$

of abelian varieties, where \(\tau _x\) is the translation by x (the morphism \(\varphi _\theta \) is independent of the choice of the representative L of \(\theta \)). When \(\theta \) is a polarization, that is, when L is ample, \(\varphi _{\theta }\) is an isogeny.

We say that a polarization \(\theta \) is principal when \(\varphi _\theta \) is an isomorphism. If \(n:=\dim (X)\), this is equivalent to saying that the self-intersection number \(\theta ^n\) is n!. The associated Rosati involution on \({{\,\mathrm{End}\,}}(X)\) is then defined by \(u\mapsto u':=\varphi _{\theta }^{-1}\circ {\widehat{u}} \circ \varphi _{\theta }\). The map

is an injective morphism of free abelian groups whose image is the group \({{\,\mathrm{End}\,}}^s(X)\) of symmetric elements for the Rosati involution [10, Theorem 5.2.4]. If \(u\in {{\,\mathrm{End}\,}}(X)\), one has \(\varphi _{u^*\theta '}={\widehat{u}}\circ \varphi _{\theta '}\circ u\) hence

$$\begin{aligned} \iota _{\theta }(u^*\theta ')=\varphi _{\theta }^{-1}\circ \varphi _{u^*\theta '} =\varphi _{\theta }^{-1}\circ {\widehat{u}}\circ \varphi _{\theta '}\circ u=u'\circ \varphi _{\theta }^{-1} \circ \varphi _{\theta '}\circ u= u'\circ \iota _{\theta }(\theta ')\circ u. \end{aligned}$$
(32)

Set \({{\,\mathrm{NS}\,}}_\mathbf{Q}(X)={{\,\mathrm{NS}\,}}(X)\otimes \mathbf{Q}\) and \({{\,\mathrm{End}\,}}_\mathbf{Q}(X) ={{\,\mathrm{End}\,}}(X)\otimes \mathbf{Q}\) (both are finite-dimensional \(\mathbf{Q}\)-vector spaces). If the polarization \(\theta \) is no longer principal, or if \(\theta \in {{\,\mathrm{NS}\,}}_\mathbf{Q}(X)\) is only a \(\mathbf{Q}\)-polarization, the Rosati involution is still defined on \({{\,\mathrm{End}\,}}_\mathbf{Q}(X)\) by the same formula and we may view \(\iota _{\theta }\) as an injective morphism

with image \({{\,\mathrm{End}\,}}^s_\mathbf{Q}(X)\) [10, Remark 5.2.5]. Formula (32) remains valid for \(u\in {{\,\mathrm{End}\,}}(X)\) and \(\theta '\in {{\,\mathrm{NS}\,}}_\mathbf{Q}(X)\).

We will also need the so-called analytic representation

It sends an endomorphism of X to its tangent map at 0.

1.1 C.1 \(\mathbf{Q}\)-actions on abelian varieties

Let X be an abelian variety and let G be a finite group. A \(\mathbf{Q}\)-action of G on X is a morphism \(\rho :\mathbf{Q}[G]\rightarrow {{\,\mathrm{End}\,}}_\mathbf{Q}(X)\) of \(\mathbf{Q}\)-algebras. The composition

$$\begin{aligned} G\xrightarrow {\ \rho \ } {{\,\mathrm{End}\,}}_\mathbf{Q}(X) \xrightarrow {\ \rho _a\ } {{\,\mathrm{End}\,}}_\mathbf{C}(T_{X,0}) \end{aligned}$$

is called the analytic representation of G.

Proposition C.1

Let X be an abelian variety of dimension n with a \(\mathbf{Q}\)-action of a finite group G. Assume that the analytic representation of G is irreducible and defined over \(\mathbf{Q}\). Then X is isogeneous to the product of n copies of an elliptic curve.

Proof

This follows from [27, (3.1)–(3.4)] (see also [34, Section 1] and [10, Proposition 13.6.2]). This reference assumes that we have a faithful action \(G\rightarrow {{\,\mathrm{Aut}\,}}(X)\) but only uses the induced morphism \(\mathbf{Q}[G]\rightarrow {{\,\mathrm{End}\,}}_\mathbf{Q}(X)\) of \(\mathbf{Q}\)-algebras. \(\square \)

In the situation of Proposition C.1, we prove that any G-invariant \(\mathbf{Q}\)-polarization is essentially unique.

Lemma C.2

Let X be an abelian variety with a \(\mathbf{Q}\)-action of a finite group G and let \(\theta \) be a G-invariant polarization on X. If the analytic representation of G is irreducible, any G-invariant \(\mathbf{Q}\)-polarization on X is a rational multiple of \(\theta \).

Proof

Let \(g\in G\), which we view as an invertible element of \({{\,\mathrm{End}\,}}_\mathbf{Q}(X) \). Since \(\theta \) is g-invariant, identity (32) (applied with \(\theta '=\theta \) and \(u=g\)) implies \(g'\circ g={{\,\mathrm{Id}\,}}_X\). Let \(\theta '\in {{\,\mathrm{NS}\,}}_\mathbf{Q}(X)\). Applying (32) again, we get

$$\begin{aligned} \iota _{\theta }(g^*\theta ')=g'\circ \iota _{\theta }(\theta ')\circ g=g^{-1}\circ \iota _{\theta }(\theta ')\circ g. \end{aligned}$$

If \(\theta '\) is G-invariant, we obtain \(\iota _{\theta }(\theta ')=g^{-1}\circ \iota _{\theta }(\theta ')\circ g\) for all \(g\in G\). If the analytic representation of G is irreducible, \(\rho _a(\iota _{\theta }(\theta '))\) must, by Schur’s lemma, be a multiple of the identity, hence \(\theta '\) must be a multiple of \(\theta \). \(\square \)

1.2 C.2 Polarizations on self-products of elliptic curves

Let E be an elliptic curve, so that \({\mathfrak {o}}_E:={{\,\mathrm{End}\,}}(E)\) is either \(\mathbf{Z}\) or an order in an imaginary quadratic extension of \(\mathbf{Q}\). We have

$$\begin{aligned} {{\,\mathrm{End}\,}}(E^n)\simeq \mathscr {M}_{n}({\mathfrak {o}}_E)\quad \text {and}\quad {{\,\mathrm{End}\,}}_\mathbf{Q}(E^n)\simeq \mathscr {M}_{n}({\mathfrak {o}}_E\otimes \mathbf{Q}), \end{aligned}$$

and \(\rho _a\) is the embedding of these matrix rings into the ring \(\mathscr {M}_n(\mathbf{C})\) induced by the analytic representation \({\mathfrak {o}}_E\hookrightarrow \mathbf{C}\) of E.

Polarizations on \(E^n\) were studied in particular by Lange in [37]. We denote by \(\theta _0\) the product principal polarization on \(E^n\).

Proposition C.1

Let E be an elliptic curve.

  1. 1.

    The Rosati involution defined by \(\theta _0\) on \({{\,\mathrm{End}\,}}(E^n)\) corresponds to the involution \(M\mapsto {\overline{M}}^T\) on \(\mathscr {M}_{n}({\mathfrak {o}}_E)\).

  2. 2.

    Via the embedding \(\iota _{\theta _0}\), polarizations \(\theta \) on \(E^n\) correspond to positive definite Hermitian matrices \(M_\theta \in \mathscr {M}_{n}({\mathfrak {o}}_E)\). The degree of the polarization \(\theta \) is \( \det (M_\theta )\) and the group of automorphisms \({{\,\mathrm{Aut}\,}}(E^n,\theta )\) is the unitary group

    $$\begin{aligned} \mathbf{U}(n,M_\theta ):= \{M\in \mathscr {M}_{n}({\mathfrak {o}}_E)\mid {\overline{M}}^T M_\theta \, M= M_\theta \}. \end{aligned}$$

Proof

If we write \(E=\mathbf{C}/(\mathbf{Z}\oplus \tau \mathbf{Z})\), the period matrix for \(E^n\) is \(\begin{pmatrix}I_n&\tau I_n\end{pmatrix}\). The first item then follows from [37, Lemma 2.3] and elements of \( {{\,\mathrm{NS}\,}}(E^n)\) correspond to Hermitian matrices. By [10, Theorem 5.2.4], polarizations correspond to positive definite Hermitian matrices and the degree of the polarization is the determinant of the matrix. More precisely, one has [10, Proposition 5.2.3]

$$\begin{aligned} \det (T I_n-M_\theta )=\sum _{j=0}^n (-1)^{n-j}\frac{\theta _0^j\cdot \theta ^{n-j}}{j!(n-j)!}\,T^j . \end{aligned}$$

The last item follows from (32). \(\square \)

Remark C.2

Let G be a finite group with a \(\mathbf{Q}\)-representation \(\rho :\mathbf{Q}[G]\rightarrow \mathscr {M}_{n}(\mathbf{Q})\). For any elliptic curve E, this defines a \(\mathbf{Q}\)-action of G on \(E^n\). It follows from the proposition that any positive definite symmetric matrix \(M_\theta \in \mathscr {M}_{n}(\mathbf{Q})\) such that, for all \(g\in G\),

$$\begin{aligned} \rho (g)^T M_\theta \, \rho (g)= M_\theta \end{aligned}$$

defines a G-invariant \(\mathbf{Q}\)-polarization on \(E^n\). Such a matrix always exists: take for example \(M_\theta :=\sum _{g\in G} \rho (g)^T \rho (g)\) (it corresponds to the \(\mathbf{Q}\)-polarization \( \sum _{g\in G} g^*\theta _0\)).

The analytic representation is \(\rho _\mathbf{C}:\mathbf{C}[G]\rightarrow \mathscr {M}_{n}(\mathbf{C})\). If it is irreducible, every G-invariant \(\mathbf{Q}\)-polarization on \(E^{ n}\) is, by Lemma C.2, a rational multiple of \(\theta \).

We end this section with the construction of an explicit abelian variety of dimension 10 with a \(\mathbb {G}\)-action, such that the associated analytic representation is the irreducible representation \(\textstyle {\bigwedge ^{2}}{\xi }\), together with a \(\mathbb {G}\)-invariant principal polarization. Set \( \lambda :=\tfrac{1}{2}(-1+\sqrt{-11})\) and consider the elliptic curve \(E_\lambda :=\mathbf{C}/\mathbf{Z}[\lambda ]\), which has complex multiplication by \(\mathbf{Z}[\lambda ]\).

Proposition C.3

There exists a principal polarization \(\theta \) on the abelian variety \(E_\lambda ^{10}\) and a faithful action \(\mathbb {G}\hookrightarrow {{\,\mathrm{Aut}\,}}(E_\lambda ^{10},\theta )\) such that the associated analytic representation is the irreducible representation \(\textstyle {\bigwedge ^{2}}{\xi }\) of \(\mathbb {G}\).

Proof

By [49, Table 1], there is a positive definite unimodular \(\mathbf{Z}[\lambda ]\)-sesquilinear Hermitian form \(H'\) on \(\mathbf{Z}[\lambda ]^5\) with an automorphism of order 11. Its Gram matrix in the canonical \(\mathbf{Z}[\lambda ]\)-basis \((e_1,\dots ,e_5)\) of \(\mathbf{Z}[\lambda ]^5\) is

$$\begin{aligned} \begin{pmatrix} 3 &{} 1-{\bar{\lambda }} &{}-\lambda &{}1&{}-{\bar{\lambda }} \\ 1-\lambda &{} 3 &{} -1&{} - \lambda &{}1 \\ -{\bar{\lambda }} &{} -1 &{} 3&{}\lambda &{}-1+\lambda \\ 1 &{} -{\bar{\lambda }}&{} {\bar{\lambda }}&{}3&{}1-{\bar{\lambda }} \\ - \lambda &{} 1 &{} -1+{\bar{\lambda }}&{} 1-\lambda &{}3 \end{pmatrix} \end{aligned}$$

and its unitary group has order \(2^{3}\cdot 3\cdot 5\cdot 11=1\,320 \) [49].

By Proposition C.1, this form defines a principal polarization \(\theta '\) on the abelian variety \(E_\lambda ^5\) and the group \({{\,\mathrm{Aut}\,}}(E_\lambda ^5,\theta ')\) has order \(1\,320 \); in particular, it contains an element of order 11. It follows from [7] that the group \({{\,\mathrm{Aut}\,}}(E_\lambda ^5,\theta ')\) is isomorphic to \(\mathbb {G}\times \{\pm 1\} \) and the faithful representation \(\mathbb {G}\hookrightarrow {{\,\mathrm{Aut}\,}}(E_\lambda ^5,\theta ')\hookrightarrow \mathbf{U}(5,H')\) given by Proposition C.1 is \(\xi \).Footnote 3

The Hermitian form \(H'\) on \(\mathbf{Z}[\lambda ]^5\) induces a positive definite unimodular Hermitian form H on \(\textstyle {\bigwedge ^{2}}{\mathbf{Z}}[\lambda ]^5=\mathbf{Z}[\lambda ]^{10}\) by the formula

$$\begin{aligned} H(x_1\wedge x_2,x_3\wedge x_4):=H'(x_1,x_3)H'(x_2,x_4)-H'(x_1,x_4)H'(x_2,x_3). \end{aligned}$$

The matrix of H (in the basis \((e_{12},e_{13},e_{14},e_{15},e_{23},e_{24},e_{25},e_{34},e_{35},e_{45})\)) is

$$\begin{aligned} \left( {\begin{matrix} 4&{} 2\lambda &{}-1 -2\lambda &{}-1-\lambda &{}-2+2\lambda &{}-\lambda &{}-1-2 \lambda &{}-2-\lambda &{}1&{} -2 \\ 2{\bar{\lambda }}&{}6 &{}-1+2\lambda &{}-1+2\lambda &{}6+2\lambda &{}-2+\lambda &{}-4+\lambda &{}\lambda &{}-\lambda &{}2+\lambda \\ -1-2{\bar{\lambda }}&{}-1+2{\bar{\lambda }} &{} 8&{}5+2\lambda &{}-2-2\lambda &{} 5+2\lambda &{}3+2\lambda &{}1-2\lambda &{}1&{}-1-2\lambda \\ -1-{\bar{\lambda }}&{}-1+2{\bar{\lambda }} &{} 5+2{\bar{\lambda }}&{} 6&{}-1-2 \lambda &{}4&{}5+2\lambda &{}-1-\lambda &{}-1-\lambda &{}-1-\lambda \\ -2+2{\bar{\lambda }}&{} 6+2{\bar{\lambda }}&{}-2-2{\bar{\lambda }} &{}-1-2{\bar{\lambda }} &{} 8&{}2\lambda &{} -2+3\lambda &{} 2\lambda &{} -2-\lambda &{}3+\lambda \\ -{\bar{\lambda }} &{} -2+{\bar{\lambda }}&{} 5+2{\bar{\lambda }} &{}4 &{} 2{\bar{\lambda }}&{} 6&{} 5+2\lambda &{}0 &{}-1 &{}-\lambda \\ -1-2 {\bar{\lambda }}&{}-4+{\bar{\lambda }} &{}3+2{\bar{\lambda }} &{} 5+2{\bar{\lambda }}&{}-2+3{\bar{\lambda }} &{}5+2{\bar{\lambda }} &{} 8&{}2 &{}-1+\lambda &{}-1 -2\lambda \\ -2-{\bar{\lambda }} &{}{\bar{\lambda }} &{}1-2{\bar{\lambda }} &{}-1-{\bar{\lambda }} &{} 2{\bar{\lambda }} &{}0 &{}2 &{} 6&{}2+2 \lambda &{} -2\lambda \\ 1&{}-{\bar{\lambda }} &{}1 &{}-1-{\bar{\lambda }} &{} -2-{\bar{\lambda }} &{} -1&{}-1+{\bar{\lambda }} &{}2+2{\bar{\lambda }} &{}4 &{} -2 \\ -2&{}2+{\bar{\lambda }}&{}-1-2{\bar{\lambda }}&{}-1-{\bar{\lambda }}&{}3+{\bar{\lambda }} &{}-{\bar{\lambda }} &{}-1 -2{\bar{\lambda }} &{} -2{\bar{\lambda }} &{}-2 &{} 4 \end{matrix}}\right) . \end{aligned}$$
(33)

By Proposition C.1 again, the form H defines a principal polarization \(\theta \) on the abelian variety \(E_\lambda ^{10}\), the group \({{\,\mathrm{Aut}\,}}(E_\lambda ^{10},\theta )\) contains \(\mathbb {G}\), and the corresponding analytic representation is \(\textstyle {\bigwedge ^{2}}{\xi }\). \(\square \)

The \(\mathbb {G}\)-action on \(E_\lambda ^{10}\) in the proposition is not the \(\mathbb {G}\)-action described in Remark C.2 (otherwise, since \(\mathbb {G}\)-invariant polarizations are proportional, the matrix (33) would, by Lemma C.2, have rational coefficients): these actions are only conjugate by a \(\mathbf{Q}\)-automorphism of \(E_\lambda ^{10}\).

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Debarre, O., Mongardi, G. Gushel–Mukai varieties with many symmetries and an explicit irrational Gushel–Mukai threefold. Boll Unione Mat Ital 15, 133–161 (2022). https://doi.org/10.1007/s40574-021-00293-6

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