2.1 Lattices and isometries

Let L be a lattice, which is a finitely generated free $\mathbb {Z}$ -module equipped with a nondegenerate, integer valued symmetric bilinear form. For two vectors $u,v\in L$ , we denote by $u.v\in \mathbb {Z}$ the image by the bilinear form, and we let $v^2 := v.v$ . We assume that L is even unless stated otherwise (i.e., $v^2\in 2\mathbb {Z}$ for all $v\in L$ ). Finally, we denote by $O^+(L)\leq O(L)$ the group of orientiation-preserving isometries of L.

In what follows, all $ADE$ root lattices are assumed to be negative definite. The Leech lattice $\mathbb {L}$ is the unique (up to isometry) negative definite even unimodular lattice of rank 24 that contains no $(-2)$ -vectors. We denote by U the hyperbolic plane lattice, which is the unique even unimodular lattice of rank 2, up to isometry.

We denote by $D_L:=L^\vee /L$ the discriminant group of L. For any lattice isometry $f\colon L\xrightarrow {\cong } L'$ , we denote by $D_f\colon D_L\xrightarrow {\cong } D_{L'}$ the induced isometry.

Let $G\leq O(L)$ be a group of isometries. We denote by $L^G:=\{v\in L\mid g(v)=v, \,\, \forall g\in G\}$ and by $L_G:=(L^G)_L^\perp $ the associated invariant and coinvariant sublattices, respectively.

2.2 Primitive extensions

For nondegenerate lattices, all morphisms are injective – we therefore talk about embeddings.

Given an even lattice S, the proof of [Reference Nikulin36, Proposition 1.15.1] describes a procedure to classify, up to isomorphism, primitive sublattices of an even lattice, with given signature and discriminant group, that are isometric to S. The proof makes use of the notion of primitive extensions, which we recall now.

Let S and T be even lattices and let L be an overlattice of $S\oplus T$ . Such an overlattice $S\oplus T\subseteq L$ is called a primitive extension if both composite embeddings $S\hookrightarrow S\oplus T\hookrightarrow L$ and $T\hookrightarrow S\oplus T\hookrightarrow L$ are primitive.

Definition 2.8. We define a glue map between S and T to be an isomorphism of finite abelian groups

$$\begin{align*}D_S\geq H_S\xrightarrow{\gamma} H_T\leq D_T\end{align*}$$

such that $x^2+\gamma (x)^2\in 2\mathbb {Z}$ for all $x\in H_S$ . We moreover call $H_S$ and $H_T$ the glue domains of $\gamma $ .

Let $(S, s)$ and $(T, t)$ be two even lattices with isometry, where $s\in O(S)$ and $t\in O(T)$ . Let $D_S\geq H_S\xrightarrow {\gamma } H_T\leq D_T$ be a glue map. The glue map $\gamma $ is called $(s, t)$ -equivariant if $H_S$ and $H_T$ are respectively $D_s$ -stable and $D_t$ -stable, and if it satisfies the equivariant gluing condition:

(EGC) $$ \begin{align} \gamma\circ (D_{s})_{\mid H_S} = (D_{t})_{\mid H_T}\circ \gamma. \end{align} $$

We call $(L_\gamma , f_\gamma )$ an equivariant primitive extension of $(S, s)$ and $(T, t)$ .

2.3 Borcherds lattice

A common technique to classify finite groups of symplectic birational transformations of IHS manifolds is to relate them to isometry groups of the Leech lattice $\mathbb {L}$ . Unfortunately, this lattice is too small for our purposes, and so we will require some results about isometries of the Borcherds lattice:

$$ \begin{align*}\mathbb{B}:=U\oplus \mathbb{L}.\end{align*} $$

The lattice $\mathbb {B}$ is even unimodular, of signature $(1,25)$ , and unique in its genus. Let us fix a basis $\{e,s\}$ for $U\subset \mathbb {B}$ with $e^2=0, s^2=-2$ and $e. s=1$ .

The group of isometries of $\mathbb {B}$ is known and it has been studied for instance by Conway in [Reference Conway, Sloane, Bannai, Borcherds, Leech, Norton, Odlyzko, Parker, Queen and Venkov11, Chapter 27]. For the reader’s convenience, we recall Conway’s results, following an exposition of Brandhorst and Mezzedimi [Reference Brandhorst and Mezzedimi7].

We denote by $\mathcal {P}$ one of the connected components of

$$\begin{align*}\left\{ x\in \mathbb{B}\otimes \mathbb{R}\,\mid\, x^2> 0\right\};\end{align*}$$

we call it the positive cone of $\mathbb {B}$ . Note that $O(\mathbb {B}) = \{\pm {id}\}\times O^+(\mathbb {B})$ and $- {id}$ does not preserve $\mathcal {P}$ – we can see that the group $O^+(\mathbb {B})$ is the stabiliser of $\mathcal {P}$ in $O(\mathbb {B})$ .

Let us denote by $\Delta := \{r\in \mathbb {B}\;\mid \; r^2=-2\}$ the set of roots of $\mathbb {B}$ and denote by $W(\mathbb {B})\leq O^+(\mathbb {B})$ the subgroup generated by the reflections in the roots $r\in \Delta $ . The group $W(\mathbb {B})$ is the so-called Weyl group of $\mathbb {B}$ , and it acts simply transitively on the set of connected components, or chambers, of

$$\begin{align*}\Gamma := \mathcal{P}\setminus \bigcup_{r\in\Delta}r^\perp.\end{align*}$$

For any chamber $D\subseteq \Gamma $ , we let $P_D := \{v\in \mathbb {B}\;\mid \; v.x>0,\,\forall x\in \overline {D}\}$ , and moreover, we define

$$\begin{align*}\Delta_D:= \{ r\in P_D\cap \Delta\;\mid\; r-r'\notin P_D\cap \Delta,\, \forall r'\in P_D\cap\Delta\}\end{align*}$$

the set of simple roots of D [Reference Brandhorst and Mezzedimi7, §2.6].

For the rest of the paper, we refer to e and $D_0$ as Conway’s vector and Conway’s chamber, respectively. We moreover denote by $\mathcal {D} := \overline {D_0}$ the closure of $D_0$ in $\mathcal {P}$ , and we let $\mathrm {Aut}(\mathcal {D})$ be the subgroup of isometries of $O^+(\mathbb {B})$ preserving $\mathcal {D}$ . Since $\mathcal {D}$ is the closure of a fundamental domain for the action of $W(\mathbb {B})$ on $\mathcal {P}$ , we can identify $\mathrm {Aut}(\mathcal {D})$ with $O^+(\mathbb {B})/W(\mathbb {B})$ [Reference Brandhorst and Mezzedimi7, §2.6].

We would like to study finite subgroups of $\mathrm {Aut}(\mathcal {D})$ in order to determine their coinvariant sublattices. For this, we will need to understand the structure of $\mathrm {Aut}(\mathcal {D})$ .

Definition 2.16 (Eichler–Siegel transformation).

For any $\lambda \in \mathbb {L}$ , we define

$$\begin{align*}\psi_\lambda\colon \mathbb{B}\to \mathbb{B},\; x\mapsto x+ (x.\lambda)e - (x.e)\lambda -\frac{1}{2}(x.e)\lambda^2e.\end{align*}$$

Note that $\psi _{\lambda }\in \mathrm {Aut}(\mathcal {D})$ by the proofs of [Reference Brandhorst and Mezzedimi7, Proposition 3.2, Theorem 4.7]. Further, the assignment

$$\begin{align*}\psi\colon \mathbb{L}\to \mathrm{Aut}(\mathcal{D}),\; \lambda\mapsto \psi_\lambda\end{align*}$$

is an injective group homomorphism (where we view $\mathbb {L}$ as a torsion free abelian group of finite rank under addition). By the definition of Conway’s vector e, which is isotropic, there is an exact sequence of lattices

$$\begin{align*}0\to \mathbb{Z} e\to e^\perp\to \mathbb{L}\to 0\end{align*}$$

inducing an isometry $\kappa \colon e^\perp /\mathbb {Z} e\xrightarrow {\cong }\mathbb {L}$ .

Proof. Since any isometry in $\mathrm {Aut}(\mathcal {D})$ fixes e (Lemma 2.15), the isometry $\kappa $ defines an orthogonal representation

(2.1) $$ \begin{align} \pi\colon \mathrm{Aut}(\mathcal{D})\to O(\mathbb{L}), \end{align} $$

which admits a section $\phi :O(\mathbb {L})\rightarrow \mathrm {Aut}(\mathcal {D})$ , given by extending an isometry of $\mathbb {L}$ to one of $\mathbb {B}$ acting as the identity on $\mathbb {Z} e+\mathbb {Z} s\cong U.$ According to [Reference Brandhorst and Mezzedimi7, Proposition 3.2], the following sequence is exact:

$$\begin{align*}0\to \mathbb{L}\xrightarrow{\psi}\mathrm{Aut}(\mathcal{D})\xrightarrow{\pi} O(\mathbb{L})\to 1.\end{align*}$$

Since all nontrivial elements of $\mathbb {L}$ have infinite order and $O(\mathbb {L})$ is of finite order, we have that $\mathbb {L}\cap O(\mathbb {L})$ is trivial, as a subgroup of $\mathrm {Aut}(\mathcal {D})$ , giving the claim.

Hence, any element $h\in \mathrm {Aut}(\mathcal {D})$ can be uniquely written in the form $\psi _\lambda \circ \phi (g)$ for some $\lambda \in \mathbb {L}$ and some $g\in O(\mathbb {L})$ : we write

$$\begin{align*}h = (\lambda, g).\end{align*}$$

We record some properties of elements $h\in \mathrm {Aut}(\mathcal {D})$ for future use. The proof follows by direct calculation.

Theorem 2.20. Let $H\leq \mathrm {Aut}(\mathcal {D})$ be a subgroup, and let $G := \pi (H)\leq O(\mathbb {L})$ (Equation (2.1)). Then H is of finite order if and only if there exists $n \in \mathbb {Z}$ positive and $v\in \mathbb {L}$ such that for all $h = (\lambda , g)\in H$ ,

$$\begin{align*}g(v) - v = n\lambda.\end{align*}$$

Moreover, if H is of finite order, then $H\cong G$ , and n can be chosen to be $\#(H\cdot s)$ .

Proof. First remark that if H is of finite order, then $H\cap \mathbb {L}$ is the trivial subgroup of $\mathrm {Aut}(\mathcal {D})$ . In particular, $\pi $ restricts to an isomorphism between H and $G := \pi (H)$ . Note, moreover, that for any $h = H\leq \mathrm {Aut}(\mathcal {D})$ , we have $h(e) = e$ (Lemma 2.15)

Suppose that H is of finite order, and denote by $n:= \#(H\cdot s)$ the length of the orbit of s under H. Since H is finite, we have that $w := \sum _{r\in H\cdot s}r$ is fixed by H, and moreover $e.w = n(e.s) = n$ . This implies that there exists $m\in \mathbb {Z}$ and $v\in \mathbb {L}$ such that

$$\begin{align*}w = me+ns+v.\end{align*}$$

Since H fixes e and w, we have that H fixes $ns+v$ . In particular, for all $h = (\lambda , g)\in H$ , we have

(2.2) $$ \begin{align} 0 = h(ns+v) - (ns+v) = (g(v)-v-n\lambda) + \left(\lambda.g(v)-n\frac{\lambda^2}{2}\right)e \in \mathbb{L}\oplus \mathbb{Z} e. \end{align} $$

This implies that $g(v)-v = n\lambda $ , and since this holds for any $h = (\lambda , g)\in H$ , we can conclude.

Conversely, suppose that such $n>0$ and $v\in \mathbb {L}$ exist, and let $h = (\lambda , g)\in H$ be arbitrary. Note that we have that

$$\begin{align*}n\lambda.(g(v)+v) = (g(v)-v).(g(v)+v) = 0.\end{align*}$$

In particular, since $g(v)-v=n\lambda $ , we observe that

$$\begin{align*}2g(v).\lambda = (g(v)-v+g(v)+v).\lambda = (g(v)-v).\lambda = n\lambda^2.\end{align*}$$

According to Equation (2.2), we obtain that $ns+v\in \mathbb {B}^H$ . Since $e\in \mathbb {B}^H$ too, one can find $m\in \mathbb {Z}_{\geq 0}$ large enough such that $me+ns+v\in \mathbb {B}^H$ has positive norm. Since the lattice $\mathbb {B}$ is hyperbolic, we deduce that $\mathbb {B}_H$ is negative definite and the group H acting faithfully on such a lattice must be finite.

Proposition 2.23. Let $H\leq \mathrm {Aut}(\mathcal {D})$ be a finite subgroup and let $G := \pi (H)\cong H$ . Then $\mathbb {B}_H$ and $\mathbb {L}_G$ are isometric if and only if there exists $v\in \mathbb {L}$ such that $g(v)-v=\lambda $ for all $(\lambda , g)\in H$ . If the previous does not hold, then there exists $n>1$ such that

$$\begin{align*}\det(D_{\mathbb{B}_H}) = n^2\det(D_{\mathbb{L}_G})\end{align*}$$

holds.

Proof. According to the proof of Theorem 2.20, we know that there exists $n>0$ and $v\in \mathbb {L}$ such that

$$\begin{align*}\mathbb{B}^H = (\mathbb{Z} e\oplus \mathbb{L}^G)+\mathbb{Z}(ns+v)\end{align*}$$

and $g(v)-v = n\lambda $ for all $(\lambda , g)\in H$ . We define $H_v := \psi _v^{-1}\phi (G)\psi _v$ : by definition of $\phi \colon O(\mathbb {L})\to \mathrm {Aut}(\mathcal {D})$ and the fact that $g(v)-v=n\lambda $ for all $(\lambda , g)\in H$ , we see that elements of $H_v$ are of the form $(n\lambda , g)$ , where $(\lambda ,g)\in H$ . We therefore already note that if $n=1$ (i.e., $g(v)-v=\lambda $ for all $(\lambda ,g)\in H$ ), then $H = H_v$ is conjugate to $\phi (G)$ in $\mathrm {Aut}(\mathcal {D})$ , and thus,

$$\begin{align*}\mathbb{B}_H = \mathbb{B}_{H_v} \cong \mathbb{B}_{\phi(G)} = \mathbb{L}_G.\end{align*}$$

Furthermore, by direct computations, we infer that 𝔹 ^{H _v} = (ℤe ⊕ 𝕃 ^G ) + ℤ(s + v). It hence follows that

$$\begin{align*}\det(D_{\mathbb{B}^H}) = n^2\det(D_{\mathbb{B}^{H_v}})\end{align*}$$

holds. Since $\mathbb {B}$ is unimodular, and $\mathbb {B}_{H_v} \cong \mathbb {L}_G$ , we also obtain that

$$\begin{align*}\det(D_{\mathbb{B}_H}) = n^2\det(D_{\mathbb{L}_G})\end{align*}$$

holds too. From that, it is clear that if $\mathbb {B}_H\cong \mathbb {L}_G$ , then $n=1$ and $g(v)-v=\lambda $ for all $(\lambda ,g)\in H$ .

3.1 IHS manifolds of $OG10$ type

An IHS manifold is a simply connected, compact, Kähler manifold X such that $H^0(X,\Omega ^2_X)$ is generated by a nowhere degenerate holomorphic $2$ -form $\sigma _X$ . In this paper, we are focused on IHS manifolds that are deformation equivalent to O’Grady’s 10-dimensional exceptional example [Reference O’Grady37]. Such a manifold X is said to be of $OG10$ type.

It is well known that $H^2(X,\mathbb {Z})$ admits a quadratic form $q_X$ , known as the Beauville–Bogomolov–Fujiki form. The form $q_X$ is integral and nondegenerate, and the isometry class of the lattice $(H^2(X, \mathbb {Z}), q_X)$ is invariant under deformation. By [Reference Rapagnetta41], for an IHS manifold of $OG10$ type, there is an isometry

$$\begin{align*}\eta: (H^2(X,\mathbb{Z}), q_X)\xrightarrow{\cong}\Lambda := U^3\oplus E_8^2\oplus A_2.\end{align*}$$

A choice of such an isometry $\eta $ is called a marking, and we say that $(X, \eta )$ is a marked IHS manifold. Two such marked IHS manifolds $(X, \eta )$ and $(X', \eta ')$ are called equivalent if there exists an isomorphism $f\colon X\xrightarrow {\cong } X'$ such that $\eta ' = \eta \circ f^\ast $ .

Marked IHS manifolds of $OG10$ type are classified up to equivalence in a coarse moduli space which we denote $\mathcal {M}_{OG10}$ . This space is neither Hausdorff nor connected, and two inseparable points in a same connected component define birational IHS manifolds of $OG10$ type [Reference Huybrechts21, Theorem 4.3]. The period map

$$\begin{align*}\begin{array}{ccccc} \mathcal{P}&\colon& \mathcal{M}_{OG10}&\to&\Omega_{OG10} := \left\{\mathbb{C}\omega\in\mathbb{P}(\Lambda\otimes\mathbb{C})\;\mid\;\omega^2 = 0, \; \omega. \overline{\omega}> 0\right\}\\ &&(X, \eta) &\mapsto&[\eta_{\mathbb{C}}(\sigma_X)] \end{array}\end{align*}$$

relates a marked IHS manifold with its Hodge structure, where $\eta _{\mathbb {C}}$ denotes $\eta $ extended over $\mathbb {C}$ . We have the following crucial result:

3.2 Global Torelli

Let X be an IHS manifold of $OG10$ type. We denote by $\mathrm {Aut}(X) \leq \mathrm {Bir}(X)$ the groups of automorphisms and birational transformations of X, respectively. A birational transformation $f\in \mathrm {Bir}(X)$ is well defined in codimension one, and so we obtain a Hodge isometry $f^*:H^2(X,\mathbb {Z})\rightarrow H^2(X,\mathbb {Z})$ .

Let $(X, \eta )$ be a marked IHS manifold of $OG10$ type. The marking $\eta $ gives rise to an orthogonal representation

$$\begin{align*}\eta_{\ast}\colon \text{Bir}(X)\to O(\Lambda), \,\,\,\,\, f\mapsto \eta (f^\ast)^{-1}\eta^{-1}\end{align*}$$

which is faithful [Reference Mongardi and Wandel34, Theorem 3.1]. By [Reference Onorati39, Theorem 5.4], the image of $\eta _{\ast }$ lies in $O^+(\Lambda )$ , the group of orientation-preserving isometries.

The aim of this paper is to classify finite groups of symplectic birational transformations of IHS manifolds of $OG10$ type. Using the surjectivity of the period map $\mathcal {P}$ and the injectivity of $\eta _{\ast }$ for any marked IHS manifolds of $OG10$ type $(X, \eta )$ , our approach is to classify symplectic finite subgroups of $O^+(\Lambda )$ . We explain now how to determine whether a given finite subgroup of $O^+(\Lambda )$ is symplectic.

Let again $(X, \eta )$ be a marked IHS manifold of $OG10$ type. Any birational transformation of X preserves the birational Kähler cone $\mathcal {BK}(X)$ ; the structure of this cone for a manifold of $OG10$ type is well understood [Reference Mongardi and Onorati33]. In particular, the walls of $\overline {\mathcal {BK}(X)}$ are defined by the hyperplanes $D^\perp \subset \mathcal {C}(X)$ , where D is a stably prime exceptional divisor [Reference Markman27, §5], and $\mathcal {C}(X)$ denotes the connected component of the positive cone of X containing a Kähler class. We define the following set of vectors:

$$\begin{align*}\mathcal{W}^{pex}:=\{v\in \Lambda : v^2=-2\}\cup \{v\in \Lambda : v^2=-6,\; \mathrm{div}_{\Lambda}(v)=3\}.\end{align*}$$

It follows that $\mathcal {BK}(X)$ is contained in an exceptional chamber – that is, a component of

$$\begin{align*}\mathcal{C}(X)\setminus \bigcup_{v\in \mathcal{W}^{pex}} v^\perp\end{align*}$$

(see [Reference Mongardi and Onorati33, Theorem 3.2]). Using this description, we can rephrase the Global Torelli theorem (due to Huybrechts, Markman and Verbitsky) in a way that is more suited for the study of symplectic birational transformations of X. This will provide us with criteria for when a finite group $H\leq O^+(\Lambda )$ is symplectic.

3.3 Classification problems

Recall that $\Lambda := U^3\oplus E_8^2\oplus A_2$ is the lattice isometric to the second integral cohomology lattice of any IHS manifold of $OG10$ type. We aim to classify finite groups $H\leq O^+(\Lambda )$ that are symplectic (i.e., induced by a finite group G of symplectic birational transformations for IHS manifolds X of $OG10$ type). By the discussion in Section 3.2, this is equivalent to classifying, up to conjugacy, finite subgroups $H\leq O^+(\Lambda )$ with negative definite coinvariant sublattice, and such that $\Lambda _H\cap \mathcal {W}^{pex} = \varnothing $ .

We use the action of such a group $H\leq O^+(\Lambda )$ on the discriminant group $D_\Lambda $ in order to aid this classification. Note that, as abelian groups, $D_\Lambda \cong \mathbb {Z}/3\mathbb {Z}$ , and in particular, there is an exact sequence

$$\begin{align*}1\to O^{+,\#}(\Lambda)\to O^+(\Lambda)\to O(D_\Lambda)\to 1,\end{align*}$$

where $O(D_\Lambda )$ has order 2 generated by $- {id}$ . Hence, if $H\leq O^+(\Lambda )$ is a finite subgroup, we have an exact sequence

(3.1) $$ \begin{align} 1 \to H^\#\to H\to \mu_2, \end{align} $$

where $H\to \mu _2$ is defined by the discriminant representation of H.

Note that the classification of symplectic finite groups satisfying case (1) has already been completed by the same authors in [Reference Marquand and Muller29]. We treat the two other cases inductively. In what follows, we start by showing how to construct representatives of $O^+(\Lambda )$ -conjugacy classes of finite saturated subgroups of $O^{+,\#}(\Lambda )$ , covering case (2). Then, we cover the last case (3) by adapting the extension approach of [Reference Brandhorst and Hashimoto4, Reference Brandhorst and Hofmann5] to Equation (3.1).

4.1 Finite stable subgroups of $O^+(\Lambda )$

Let $H\leq O^+(\Lambda )$ be a symplectic finite subgroup such that $H=H^\#$ . In particular, we assume ${H\leq O^{+,\#}(\Lambda )}$ . The following holds.

Proof. Let us embed $\Lambda $ primitively into the even unimodular lattice $M := U^5\oplus E_8^2$ , with orthogonal complement $F\cong A_2(-1)$ . Since H is stable, we can extend it to a group of isometries $\widetilde {H}\leq O(M)$ acting as the identity on F, in such a way that $M_{\widetilde {H}} = \Lambda _H$ . Note, moreover, that $F\oplus \Lambda ^H\subseteq M^{\widetilde {H}}$ is an overlattice. Since $\widetilde {H}$ acts trivially on $M^{\widetilde {H}}$ , we see that $\widetilde {H}\to O(M_{\widetilde {H}})$ is injective. Moreover, since M is unimodular, the equivariant gluing condition Equation (EGC) tells us that $\widetilde {H}$ maps into $O^\#(M_{\widetilde {H}})$ . We therefore get an injective morphism

$$\begin{align*}H\xrightarrow{\cong}\widetilde{H}\hookrightarrow O^\#(M_{\widetilde{H}}) = O^{\#}(\Lambda_H).\end{align*}$$

Let $g\in O^\#(\Lambda _H) = O^\#(M_{\widetilde {H}})$ . We can extend g to an isometry $\tilde {g}\in O(M)$ acting as the identity on $M^{\widetilde {H}}$ . Since $F\oplus \Lambda ^H\subseteq M^{\widetilde {H}}$ , we have that $\widetilde {g}$ restricts to an isometry h of $\Lambda = F^\perp _M$ fixing pointwise $\Lambda ^H$ : h is the extension of g to $O(\Lambda )$ with the identity on $\Lambda ^H$ . Moreover, $h\in O^{+, \#}(\Lambda )$ according to Equation (EGC) and Remark 3.7. In particular, h lies in the saturation of H in $O^{+,\#}(\Lambda )$ .

Conversely, for any $h\in O^{+, \#}(\Lambda )$ fixing pointwise $\Lambda ^H$ , we can extend h to an isometry $\widetilde {h}$ of $O(M)$ by extending with the identity on F. Similarly as before, the restrictions of h and $\widetilde {h}$ to $\Lambda _H = M_{\widetilde {H}}$ coincide, and they lie in $O^\#(\Lambda _H)$ . We therefore conclude that the saturation of H in $O^{+, \#}(\Lambda )$ coincides with the extension of $O^\#(\Lambda _H)$ with the identity on $\Lambda ^H$ .

It follows that saturated symplectic finite subgroups H of $O^{+, \#}(\Lambda )$ are completely determined by some primitive sublattices $C\subseteq \Lambda $ which are even negative definite, and such that $O^\#(C)$ fixes no nontrivial vector in C – here, H being defined as the extension of $O^\#(C)$ with the identity on $C^\perp _\Lambda $ . The fixed point free action of $O^\#(C)$ on the sublattice $C\subseteq \Lambda $ is an additional tool in our classification. We thus make the following definition:

By Lemma 4.3, in order to construct saturated symplectic finite subgroups H of $O^{+,\#}(\Lambda )$ , it suffices to construct primitive sublattices $ C\subseteq \Lambda $ where C is stable symplectic, and such that $C\cap \mathcal {W}^{pex} = \varnothing $ . In order to ensure this last condition, we observe the following.

Let us denote by $\Pi :=U^3\oplus E_8^3$ the unique even unimodular lattice of signature $(3,27)$ .

Proof. Let us consider the succession of inclusions

$$\begin{align*}C\subseteq C\oplus E_6\subseteq \Lambda\oplus E_6\subseteq \Pi.\end{align*}$$

We will show that $C\cap \mathcal {W}^{pex} = \varnothing $ . First note that if C is trivial, then the result necessary holds. In what follows, we assume that C, and therefore $O^\#(C)$ , is nontrivial.

Suppose that C has a vector v of square $-6$ such that $\mathrm {div}_\Lambda (v) = 3$ . Since the lattice C is stable symplectic, there exists an isometry $g\in O^\#(C)$ such that $g(v)\neq v$ holds. Note that $g(v)\neq -v$ also holds: in fact, since g is stable, we have equalities

$$\begin{align*}\frac{v}{3}+C = D_g\left(\frac{v}{3}+C\right) = \frac{g(v)}{3}+C\in D_C.\end{align*}$$

If $g(v) = -v$ were to hold, we would have that $\frac {2v}{3}\in C$ ; however, this vector has square $\frac {-8}{3}\notin \mathbb {Z}$ , contradicting that C is an even lattice. Hence, $v':=g(v)$ is not proportional to v, and it still has divisibility 3 in $\Lambda $ (divisibility is preserved under isometry, and $g\in O^\#(C)$ being stable can be seen as an isometry of $\Lambda $ ). Similarly to the proof of [Reference Laza and Zheng26, Theorem 4.5, $(iii)\,{\Rightarrow}\, (i)$ ], it follows that the primitive closure M of $E_6+\mathbb {Z} v+\mathbb {Z} v'$ in $\Pi $ is isometric to $E_8$ . In particular, since $E_8$ has a unique sublattice isometric to $E_6$ (up to isometry), with orthogonal complement isometric to $A_2$ , one concludes that $A_2\cong (E_6)^\perp _M$ embeds into C: this is a contradiction since C contains no $(-2)$ -vectors. Hence, $C\cap \mathcal {W}^{pex}=\varnothing $ .

By Lemma 4.6, we see that any stable symplectic primitive sublattice $C\subseteq \Lambda $ not containing $(-2)$ -vectors is the coinvariant sublattice of a symplectic finite subgroup $H\leq O^{+, \#}(\Lambda )$ . We have therefore reduced our problem to constructing and classifying stable symplectic primitive sublattices of $\Lambda $ not containing $(-2)$ -vectors.

In the following sections, we explain how to recover the abstract isometry class of such hearts. We then make use of the following theorem to eventually classify saturated symplectic finite subgroup $H\leq O^{+, \#}(\Lambda )$ up to conjugacy in $O^+(\Lambda )$ .

4.2 Isometry classes of hearts of $\Lambda $

Recall that $\mathbb {B} := U\oplus \mathbb {L}$ , where $\mathbb {L}$ is the Leech lattice, with fixed basis $\{e,s\}$ for $U\subseteq \mathbb {B}$ such that $e^2 = 0$ , $s^2 = -2$ and $e.s = 1$ . We denote again by $\mathcal {D}$ the closure of Conway’s chamber $D_0$ .

Proof. Since C is negative definite of rank at most 21, and since

$$\begin{align*}\mathrm{rank}(C) + l(D_C) \leq \mathrm{rank}(\Lambda)+l(D_\Lambda) < 26\end{align*}$$

[Reference Nikulin36, Proposition 1.15.1], we see that C embeds primitively into $\mathbb {B}$ [Reference Nikulin36, Corollary 1.12.3].

Let us fix $j\colon C\hookrightarrow \mathbb {B}$ such a primitive embedding. Since $\mathbb {B}$ is unimodular, Equation (EGC) tells us that we can extend $O^\#(C)$ with the identity on $N := j(C)^\perp _{\mathbb {B}}$ to a group $\widetilde {H}\leq O^+(\mathbb {B})$ . In particular, $\mathbb {B}_{\widetilde {H}} = j(C) \cong C$ . Since C does not contain $(-2)$ -vectors, $N\otimes \mathbb {R}$ intersect a chamber D of the positive cone of $\mathbb {B}$ . The group $\widetilde {H}$ acting trivially on N, we have that $\widetilde {H}$ preserves a vector in D, and thus, $\widetilde {H}$ preserves the entire chamber D. Hence, $\widetilde {H}\leq \mathrm {Aut}(\overline {D})$ . Since both $\mathrm {Aut}(\overline {D})$ and $\mathrm {Aut}(\mathcal {D})$ are isomorphic to $O^+(\mathbb {B})/W(\mathbb {B})$ , we obtain that $\widetilde {H}$ is $O^+(\mathbb {B})$ -conjugate to a subgroup $H\leq \mathrm {Aut}(\mathcal {D})$ . In particular, $\mathbb {B}_H\cong \mathbb {B}_{\widetilde {H}}\cong C$ .

Therefore, as abstract lattices, we can view each heart $C\subseteq \Lambda $ as the coinvariant sublattice of a finite subgroup $H\leq \mathrm {Aut}(\mathcal {D})$ . As already noted in Proposition 2.23, in some cases, these are actually primitively embedded into the Leech lattice. Stable symplectic sublattices of the Leech lattice are known and well understood [Reference Höhn and Mason19]. Unfortunately, there exist hearts $C\subset \Lambda $ that embed primitively into $\mathbb {B}$ , but not $\mathbb {L}$ – we call such a hearts exceptional. In the next section, we describe a procedure to recover the abstract isometry class of such exceptional hearts. We then apply Theorem 4.8 to this list of abstract lattices, and to Höhn–Mason list of stable symplectic sublattices of $\mathbb {L}$ , to classify conjugacy classes of hearts in $\Lambda $ .

4.3 Exceptional hearts

In this section, we aim to classify the remaining exceptional hearts: that is, the hearts $C'\subseteq \Lambda $ that embed primitively into $\mathbb {B}$ but not into $\mathbb {L}$ . In order to do so, we will use properties of isometries in $\mathrm {Aut}(\mathcal {D})$ , discussed in Section 2.3. Example 4.12 proves that such exceptional hearts do in fact exist.

Example 4.12. See the Notebook ‘Counterexample’ in the dataset [Reference Marquand and Muller30] for explicit computations and proofs of the following statements. Let C be the negative definite even lattice with Gram matrix

$$\begin{align*} \begin {pmatrix} -4 & 2 & 2 & -2 & -2 & 2 & -2 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & -1 & 1 \\ 2 & -4 & -2 & 0 & 1 & -1 & 2 & -2 & -2 & -1 & -1 & 1 & 1 & -1 & -1 & -2 & -1 & -2 \\ 2 & -2 & -4 & 1 & 0 & -2 & 2 & -2 & -2 & -1 & -1 & -1 & 1 & -1 & 0 & -2 & 0 & -1 \\ -2 & 0 & 1 & -4 & -2 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & -1 & 1 & 0 & 2 & 0 & -1 \\ -2 & 1 & 0 & -2 & -4 & 1 & 0 & 0 & 0 & 1 & 1 & -1 & -1 & 1 & 2 & 2 & -1 & 1 \\ 2 & -1 & -2 & 0 & 1 & -4 & 2 & -2 & -2 & 0 & 0 & 0 & 0 & 0 & -2 & -1 & 1 & -1 \\ -2 & 2 & 2 & 0 & 0 & 2 & -4 & 2 & 2 & -1 & 1 & -1 & -1 & 1 & 0 & 0 & -1 & 2 \\ 2 & -2 & -2 & 0 & 0 & -2 & 2 & -4 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & -1 & 0 & -1 \\ 2 & -2 & -2 & 0 & 0 & -2 & 2 & -1 & -4 & 0 & 1 & -1 & 0 & 0 & -1 & -1 & 0 & -1 \\ 0 & -1 & -1 & 1 & 1 & 0 & -1 & 1 & 0 & -4 & -1 & -1 & 2 & -2 & 0 & -2 & -1 & 0 \\ 0 & -1 & -1 & 1 & 1 & 0 & 1 & -1 & 1 & -1 & -4 & 2 & 2 & -1 & 1 & -1 & 0 & -1 \\ 0 & 1 & -1 & 0 & -1 & 0 & -1 & 1 & -1 & -1 & 2 & -4 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & -1 & -1 & 0 & -1 & -1 & 0 & 2 & 2 & 0 & -4 & 2 & -1 & 1 & 0 & 1 \\ 0 & -1 & -1 & 1 & 1 & 0 & 1 & 1 & 0 & -2 & -1 & 0 & 2 & -4 & 1 & -1 & 0 & -1 \\ 2 & -1 & 0 & 0 & 2 & -2 & 0 & -1 & -1 & 0 & 1 & 0 & -1 & 1 & -4 & -1 & 1 & -1 \\ 2 & -2 & -2 & 2 & 2 & -1 & 0 & -1 & -1 & -2 & -1 & 0 & 1 & -1 & -1 & -4 & 0 & -1 \\ -1 & -1 & 0 & 0 & -1 & 1 & -1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & 0 & -4 & 1 \\ 1 & -2 & -1 & -1 & 1 & -1 & 2 & -1 & -1 & 0 & -1 & 1 & 1 & -1 & -1 & -1 & 1 & -4 \end {pmatrix}. \end{align*}$$

The lattice C lies in the genus $ {II}_{(0,18)}3^{-7}$ , $O^\#(C)\cong C_3\times C_3$ fixes no nontrivial vector in C, and C contains no $(-2)$ -vectors. Using [Reference Nikulin36, Proposition 1.15.1], one can check that the lattice C admits a primitive embedding into $\Lambda $ with orthogonal complement isometric to $U(3)^{ 3}$ .

However, we remark that $\text {rank}(C)+l(D_C) = 25$ : this implies that C does not embed primitively into the Leech lattice, but it does embed primitively into $\mathbb {B}$ .

In order to determine, up to isometry, the stable symplectic sublattices of $\mathbb {B}$ not containing roots that do not embed primitively into the Leech lattice, we know from Proposition 2.23 that we need to look for finite subgroups $H\leq \mathrm {Aut}(\mathcal {D})$ such that there does not exist any $v\in \mathbb {L}$ so that $g(v)-v=\lambda $ for all $(\lambda , g)\in H$ — we refer to such groups as exceptional. Note that since the isometry class of $\mathbb {B}_H$ is preserved under conjugation of H by any element of $O(\mathbb {B})$ , we describe a procedure to recover at least one representative for each $\mathrm {Aut}(\mathcal {D})$ -conjugacy class of finite exceptional subgroups of $\mathrm {Aut}(\mathcal {D})$ .

Recall from the proof of Lemma 2.17 that we have an exact sequence

$$\begin{align*}0\to \mathbb{L}\xrightarrow{\psi}\mathrm{Aut}(\mathcal{D})\xrightarrow{\pi} O(\mathbb{L})\to 1,\end{align*}$$

where $\pi $ , which is induced by the isometry $e^\perp /\mathbb {Z} e\cong \mathbb {L}$ , admits a section $\phi \colon O(\mathbb {L})\to \mathrm {Aut}(\mathcal {D})$ . Moreover, the same result tells us that $\mathrm {Aut}(\mathcal {D}) = \mathbb {L}\rtimes O(\mathbb {L})$ , so any element $h\in \mathrm {Aut}(\mathcal {D})$ of finite order can be written as a pair $(\lambda , g)$ , where $g := \pi (h)$ and $\lambda \in \mathbb {L}_g$ (Proposition 2.18). The section $\phi $ above sends any $g\in O(\mathbb {L})$ to $\phi (g) := (0, g)\in \mathrm {Aut}(\mathcal {D})$ .

Let $G\leq O(\mathbb {L})$ be a subgroup. We define a $\mathbb {Z}$ -linear map

$$\begin{align*}p_G\colon \mathbb{L}\to\prod_{g\in G}\mathbb{L}_g,\; v\mapsto (g(v)-v)_{g\in G},\end{align*}$$

whose kernel is exactly $\mathbb {L}^G$ . Let us denote by m the order of G. We define moreover

$$\begin{align*}\mathbb{L}\to \prod_{g\in G}\mathbb{L}_g/m\mathbb{L}_g,\; v\mapsto (g(v)-v+m\mathbb{L}_g)_{g\in G}\end{align*}$$

whose kernel is denoted by $K(G)$ . We observe that $K(G)$ contains $\mathbb {L}^G+m\mathbb {L}$ . We denote by $A(G) := K(G)/(m\mathbb {L}+\mathbb {L}^G)$ – it is a finite abelian group. We have seen in Theorem 2.20 and Remark 2.21 that for any finite subgroup $H\leq \mathrm {Aut}(\mathcal {D})$ satisfying $\pi (H) = G$ , there exists a vector $v\in \mathbb {L}$ such that every element $h = (\lambda , g)\in H$ satisfies

$$\begin{align*}g(v) - v = m \lambda\end{align*}$$

and $\lambda \in \mathbb {L}_g$ . In particular, $v\in K(G)$ , and it is uniquely determined by H, up to translation by a vector in $\ker (p_G) = \mathbb {L}^G$ .

Proof. Let us suppose that there exists $\mu \in \mathbb {L}$ such that $\psi _\mu H\psi _\mu ^{-1} = H'$ , where $\psi _{\mu }$ is the Eichler–Siegel transformation associated to $\mu $ (see Definition 2.16). Then, for all $g\in G$ , we have

$$\begin{align*}\psi_\mu\left(\frac{g(v)-v}{m}, g\right)\psi_\mu^{-1} = \left(\frac{g(v')-v'}{m}, g\right),\end{align*}$$

which is equivalent to

$$\begin{align*}g(v-v') - (v-v') = m(g(\mu)-\mu).\end{align*}$$

Thus, we conclude that $v-v'-m\mu \in \mathbb {L}^g$ for all $g\in G$ , meaning exactly that $v-v'\in \mathbb {L}^G+m\mathbb {L}$ . The converse holds similarly, by reversing the order of the arguments.

Remark 4.15. Let $H'\leq \mathrm {Aut}(\mathcal {D})$ be finite such that $G' := \pi (H')$ is $O(\mathbb {L})$ -conjugate to G. Let $f\in O(\mathbb {L})$ be such that $fG'f^{-1} = G$ . Then the group

$$\begin{align*}H := \left\{(f(\lambda'),\, fg'f^{-1})\;\mid\; (\lambda', g')\in H'\right\} = (0, f)H'(0, f^{-1})\end{align*}$$

is conjugate to $H'$ and satisfies $\pi (H) = fG'f^{-1} = G$ .

Therefore, in order to construct at least one representative for each $\mathrm {Aut}(\mathcal {D})$ -conjugacy class of finite exceptional subgroups H of $\mathrm {Aut}(\mathcal {D})$ , we proceed as follows:

1. 1. we start by fixing a stable symplectic sublattice C of the Leech lattice $\mathbb {L}$ [Reference Höhn and Mason19];

2. 2. we compute a set $\mathcal {G}$ of representatives of $O(\mathbb {L})$ -conjugacy classes of finite subgroups G of $O^\#(C)$ such that $C^G=\{0\}$ (Remark 4.15);

3. 3. for any $G\in \mathcal {G}$ , we compute $A(G)$ : if this is trivial, we try a new group (Corollary 4.14);

4. 4. for every $[v]\in A(G)$ nontrivial, we define $H:= \left \{\left (\frac {g(v)-v}{\#G}, g\right )\;\mid \; g\in G\right \}\leq \mathrm {Aut}(\mathcal {D})$ .

Note that according to Lemma 4.13, the $\mathrm {Aut}(\mathcal {D})$ -conjugacy class of H in step (4) does not depend on a choice of a representative for the nontrivial class $[v]\in A(G)$ .

For our purposes, we only need those stable symplectic lattices of $\mathbb {B}$ that can embed into $\Lambda $ . Thus, we conclude the following:

Proof. According to the proof of [Reference Nikulin36, Proposition 1.15.1], a stable symplectic lattice C embeds primitively into $\Lambda $ only if for all prime numbers p

$$\begin{align*}\mathrm{rank}(C)+l(D_p) \leq 24+\delta_{3,p},\end{align*}$$

where $l(D_p)$ is the length of the p-Sylow subgroup of $D_C$ , and $\delta _{3,p} = 1$ if $p=3$ and 0 otherwise. In fact, it follows from the fact that $D_{\Lambda }$ has order 3 and any $\mathbb {Z}_3$ -lattice admits an orthogonal basis. We deduce from Table 2 that there are only three possible isometry classes of exceptional heart $C\subseteq \Lambda $ .

One observes that for all sublattices $C\subseteq \mathbb {B}$ presented in Table 2, we have $\text {rank}(C)+l(D_C)>24$ . In particular, the following holds.

Corollary 4.19. Let C be a stable symplectic lattice of rank at most 21 and not containing $(-2)$ -vectors. Then C embeds primitively into the Leech lattice $\mathbb {L}$ so that $\mathbb {L}_{O^\#(C)} \cong C$ if and only if

$$ \begin{align*}{rank}(C)+l(D_C)\leq 24.\end{align*} $$

Proof. Already note that if C embeds primitively into the Leech lattice, then according to [Reference Nikulin36, Corollary 1.12.3], we have that rank(C) + l(D _C ) ≤rank(𝕃) = 24.

Now suppose that $\mathrm {rank}(C)+l(D_C)\leq 24$ . By this assumption, we know that C embeds primitively into $\mathbb {B}$ and this in a unique way up to the action $O(\mathbb {B})$ [Reference Nikulin36, Proposition 1.14.1]. In particular, we can choose a primitive embedding $j\colon C\hookrightarrow \mathbb {B}$ such that the complement $N:= j(C)^\perp _{\mathbb {B}}$ intersects the interior of $\mathcal {D}$ . Therefore, as in the proof of Lemma 4.10, we have that the finite group $H\leq O^+(\mathbb {B})$ obtained by extending $O^\#(C)$ with the identity on N preserves $\mathcal {D}$ . Thus, C is the coinvariant sublattice of a finite subgroup of $\mathrm {Aut}(\mathcal {D})$ . There are two cases: either H is exceptional or C is isometric to $\mathbb {L}_{\pi (H)}$ . However, the former is not possible since Table 2 tells us that the coinvariant sublattices N of exceptional finite subgroups of $\mathrm {Aut}(\mathcal {D})$ satisfy

$$\begin{align*}\mathrm{rank}(N)+l(D_N)\geq 25.\\[-37pt] \end{align*}$$

5.1 Brandhorst–Hashimoto–Hofmann extension approach

In what follows, we adapt the extension approach as described in [Reference Brandhorst and Hashimoto4, Reference Brandhorst and Hofmann5] to our context. We do not expect that the reader is familiar with the content of the previously cited works. Our presentation of what follows is self contained.

Let again $\Lambda := U^3\oplus E_8^2\oplus A_2$ . Let $H\leq O^+(\Lambda )$ be finite such that $H^\#$ is nontrivial and $H\neq H^\#$ . Note that $\Lambda $ is unique in its genus, and for any lattice $\Lambda '\cong \Lambda $ , we denote by $\mathcal {W}^{pex}(\Lambda ')$ to be the set $\varphi (\mathcal {W}^{pex})$ for any isometry $\varphi \colon \Lambda \xrightarrow {\cong }\Lambda '$ . We have seen that there is an exact sequence (Equation (3.1))

$$\begin{align*}1\to H^\#\to H\to \mu_2\to 1,\end{align*}$$

where the character $H\to \mu _2$ is induced by the discriminant representation $H\to O(D_\Lambda )$ . In particular, $[H:H^\#] = 2$ and there exists a nonstable isometry $h\in H\setminus H^\#$ such that $H/H^\#$ is generated by $hH^\#$ . The restriction a of h to $\Lambda ^{H^\#}$ has order $2$ , except when $H^\#$ is not saturated in $O^+(\Lambda )$ and $\Lambda ^{H^\#} = \Lambda ^{H}$ , in which case $a = {id}$ (see Proposition 4.23). In both cases, we observe that the definition of $a\in O(\Lambda ^{H^\#})$ does not depend on a choice of a representative of $hH^\#$ . However, for any $h'\in hH^\#$ , the restriction b of $h'$ to $\Lambda _{H^\#}$ does depend on the choice of $h'$ , and it determines an equivariant primitive extension $(\Lambda ^{H^\#}, a)\oplus (\Lambda _{H^\#}, b)\subseteq (\Lambda , h')$ with associated glue map $\gamma $ .

Remark 5.2. Even though the equivariant primitive extension

$$\begin{align*}(\Lambda^{H^\#}, a)\oplus (\Lambda_{H^\#}, b)\subseteq (\Lambda, h')\end{align*}$$

depends on the choice of $h'$ , the associated glue map $\gamma $ does not depend on any $h'\in hH^\#$ or even on a. Indeed, $\gamma $ is the glue map of the primitive extension $\Lambda ^{H^\#}\oplus \Lambda _{H^\#}\subseteq \Lambda $ .

From Proposition 5.5, one sees that two conjugate symplectic finite subgroups of $O^+(\Lambda )$ share the same heart and head. We would like to measure to which extent the converse holds and describe an algorithm to compute representatives of conjugacy classes of such groups with given heart and head.

Now, for each head $(F,a)$ of a given heart C, we would like to classify isomorphism classes of equivariant primitive extensions

(5.1) $$ \begin{align} (F,a)\oplus (C,b)\subseteq (\Lambda_\gamma, h), \text{ where } \Lambda_\gamma\cong \Lambda \text{ and } D_h \neq {id}, \end{align} $$

such that $b\notin O^\#(C)$ if $a= {id}_F$ . Indeed, we know that the extension of $O^\#(C)$ with the identity on F is stable.

Definition 5.7. Let C be a heart and let $(F, a)$ be a head of C of order $n\in \{1,2\}$ . A spine between C and $(F,a)$ is a glue map

$$\begin{align*}D_F\geq H_F \xrightarrow{\gamma} H_C\leq D_C\end{align*}$$

such that

1. 1. $D_aH_F \leq H_F$ ;

2. 2. there exists $b\in O(C)$ such that $b\notin O^\#(C)$ if $a = {id}_F$ , $D_bH_C\leq H_C$ and $\gamma $ is $(a, b)$ -equivariant;

3. 3. the equivariant primitive extension $(F, a)\oplus (C, b)\subseteq (\Lambda _\gamma , h)$ is such that

   1. (a) $\Lambda _\gamma \cong \Lambda $ ;

   2. (b) $h\in O^+(\Lambda _\gamma )\setminus O^\#(\Lambda _\gamma )$ ;

   3. (c) $(\Lambda _\gamma )_{H_\gamma }\cap \mathcal {W}^{pex}(\Lambda _\gamma ) = \varnothing $ where $H_\gamma := \langle O^\#(C),\, h\rangle $ .

If it exists, we call the isometry b a companion of a along $\gamma $ .

It is not hard to see that the previous definitions are such that the heart $C := \Lambda _{H^\#}$ of any symplectic finite subgroup $H\leq O^+(\Lambda )$ is a heart and its head $(\Lambda ^{H^\#}, a)$ is a head of C.

For two lattices $L_1, L_2$ and two groups of isometries $G_i\leq O(L_i)$ ( $i=1,2$ ), we call the pairs $(L_1, G_1)$ and $(L_2, G_2)$ conjugate if there exists an isometry $\psi \colon L_1\to L_2$ such that $G_2 = \psi G_1\psi ^{-1}$ . The following holds.

Proof. We let $h := a\oplus b$ , where a is an isometry of F of order at most 2 and $b \notin O^\#(C)$ if $a = {id}_F$ . Since $O(D_{\Lambda _\gamma })$ has order 2, we have that $D_{h}^2$ is trivial. Thus, $h^2 = {id}_F\oplus b^2$ must lie in $O^\#(C)$ because $O^\#(C)$ is saturated in $O^{+,\#}(\Lambda _\gamma )$ . By definition of a, we moreover know that h has negative definite coinvariant sublattice, which implies that $h\in O^+(\Lambda _\gamma )$ by Remark 3.7. It follows that $H_\gamma := \langle h,\, O^\#(C)\rangle \leq O^+(\Lambda _\gamma )$ also has negative definite coinvariant sublattice, and by definition of $\gamma $ , we obtain that $H_\gamma $ is symplectic (Lemma 3.6). Moreover, $O^\#(C) = H_\gamma ^\#$ is normal in $H_\gamma $ , so we have that $H_\gamma /O^\#(C)=\langle hO^\#(C)\rangle $ has order 2.

Now, if one had chosen $b\neq b'\in O(C)$ such that $H_C$ is $D_{b'}$ -stable and $\gamma \circ (D_a)_{\mid H_F} = (D_{b'})_{\mid H_C}\circ \gamma $ , giving rise to an isometry $h'\in O(\Lambda _\gamma )$ , then $h^{-1}h' = {id}_F\oplus b^{-1}b'$ would act trivially on F with $D_{h^{-1}h'} = D_{h^{-1}}D_{h'}$ trivial. Therefore, $b' \in bO^\#(C)$ since $O^\#(C)$ is saturated in $O^{+, \#}(\Lambda _\gamma )$ , and thus, $\langle O^\#(C),\, h\rangle = \langle O^\#(C),\, h'\rangle $ .

Finally, let $\psi \in O(F, a)$ and let $c\in O(C)$ . Then, for any spine

$$\begin{align*}D_{F}\geq H_{F} \xrightarrow{\gamma_1} H_C\leq D_C\end{align*}$$

between C and $(F, a)$ with companion b of a, the glue map

$$\begin{align*}D_{F}\geq D_{\psi}^{-1}H_F \xrightarrow{\gamma_2 := D_c\circ\gamma_1\circ D_{\psi}} D_cH_C\leq D_C\end{align*}$$

is a spine between C and $(F, a)$ , and $cbc^{-1}$ is a companion of a along $\gamma _2$ . By [Reference Nikulin36, Corollary 1.5.2], $\gamma _1$ and $\gamma _2$ indeed defines isomorphic equivariant primitive extensions, and since $O^\#(C)$ is normal in $O(C)$ , the corresponding resulting isometry $\psi ^{-1}\oplus c\colon L_{\gamma _1}\xrightarrow {\cong }L_{\gamma _2}$ conjugates $\langle O^\#(C), a\oplus b\rangle \leq O^+(\Lambda _{\gamma _1})$ and $\langle O^\#(C), a\oplus cbc^{-1} \rangle \leq O^+(\Lambda _{\gamma _2})$ . The rest of the proof follows from the Definition 5.7.

Hence, the definitions of hearts, heads and spines are consistent with Definition 5.3. The following Theorem gives a converse to Proposition 5.5, and it is a stated in such a way that we can describe an effective algorithm out of it.

Theorem 5.9. Let C be a heart, let $(F,a)$ be a head of C of order $n\in \{1,2\}$ and let S be the set of spines between C and $(F, a)$ . Then the double cosets in

are in bijection with the conjugacy classes of pairs $(\Lambda _\gamma , H_\gamma )$ , where $\Lambda _\gamma \cong \Lambda $ and $H_\gamma \leq O^+(\Lambda _\gamma )$ is a symplectic finite subgroup such that $H_\gamma ^\#$ is saturated in $O^{+, \#}(\Lambda _\gamma )$ , the subgroup $\overline {H_\gamma }\leq O(D_{\Lambda _\gamma })$ has order 2, the heart of $H_\gamma $ is isometric to C and the head of $H_\gamma $ is isomorphic to $(F, a)$ .

Based on Theorem 5.9, the strategy behind the extension approach can be summarised in the following way.

1. 1. Start with a given heart $C\subseteq \Lambda $ , as classified in Proposition 4.20.

2. 2. Determine a set $\mathcal {F}_C(n)$ of representatives of isomorphism classes of heads $(F, a)$ of C of given order $n=1,2$ . Such pairs $(F, a)$ must satisfy that $F\cong C^\perp _\Lambda $ , $a\in O(F)$ has order n and $F_a$ is negative definite or trivial.

3. 3. For each heart C and candidate head $(F, a)\in \mathcal {F}_C(n)$ of order $n=1,2$ , compute and classify equivariant primitive extensions $(F,a)\oplus (C, b)\subseteq (\Lambda ', h)$ , where $\Lambda '\cong \Lambda $ , $D_h\neq {id}$ and where $b\notin O^\#(C)$ if $a= {id}_F$ .

Step (3) is done by constructing spines representing each of the double cosets described in Theorem 5.9.

6.1 LSV manifolds

Let $V\subset \mathbb {P}^5$ be a smooth cubic fourfold. By the construction of [Reference Laza, Saccà and Voisin25, Reference Saccà42], there exists an IHS manifold $X_V$ and a Lagrangian fibration $\pi :X_V\rightarrow (\mathbb {P}^5)^{\vee }$ that compactifies the intermediate Jacobian fibration associated to V. Further, the manifold $X_V$ is an IHS manifold of $OG10$ type. We call such a manifold an LSV manifold associated to V.

We prove Theorem 6.1 in a series of propositions. First, we establish the existence of cubic fourfolds with specified group actions.

Proof. This follows a similar strategy as [Reference Marquand and Muller29, Theorems 4.1 and 5.1]. In what follows, we identify G with a subgroup of $O(\Lambda )$ by fixing a marking of X.

Let $U_1:=U$ be such that $\Lambda ^G =U_1\oplus \Gamma \hookrightarrow \Lambda $ , and denote by $L:=(U_1)^\perp _{\Lambda (-1)}$ . Then L is an even lattice of signature $(20, 2)$ and (recalling that $ADE$ lattices are assumed negative definite),

$$ \begin{align*}L\cong U^2\oplus E_8^2(-1)\oplus A_2(-1).\end{align*} $$

The group G restricts to $G\leq O(L)$ with

$$ \begin{align*}L^G\cong \Gamma(-1) \text{ and } L_G\cong \Lambda_G(-1).\end{align*} $$

We choose a weight 4 pure Hodge structure H of $K3$ type on L, such that $H^{2,2}\cap L = \Lambda _G(-1)$ . In particular, $H^{3,1}\subset L^G$ , and since $\Lambda _G\cap \mathcal {W}^{pex}=\varnothing $ , we can apply the Global Torelli Theorem for cubic fourfolds ([Reference Voisin44], [49, Prop 1.3]). We obtain a smooth cubic fourfold V with $H^4(V,\mathbb {Z})_{prim}\cong H$ as Hodge structures.

First, we assume that G acts trivially on $D_\Lambda $ , and hence on $D_L.$ In order to conclude that G embeds in $\mathrm {Aut}_s(V)$ , we need to extend G to a group of isometries of $H^4(V,\mathbb {Z})$ fixing the square of the hyperplane class $h^2\in H^4(V,\mathbb {Z})$ . We have that $L\oplus \langle h^2\rangle \subset H^4(V,\mathbb {Z})$ . Since G, seen as a subgroup of $O(L)$ , is stable, we can extend G with $ {id}_{\langle h^2\rangle }$ to an isometry group of $H^4(V,\mathbb {Z})$ ; it follows by the Torelli theorem for cubic fourfolds that G embeds into $\mathrm {Aut}(V)$ . Further, in this case, it follows that G acts faithfully symplectically on V.

Next, we assume that G does not act trivially on $D_\Lambda $ . In other words, there exists a short exact sequence as in Equation (3.1):

$$ \begin{align*}1\rightarrow G^\#\rightarrow G\rightarrow \mu_2\rightarrow 1.\end{align*} $$

We can conclude that $G=\langle G^\#, a\rangle $ , where $a\in O(\Lambda )$ acts nontrivially on $D_\Lambda $ . Here, a has even order and $\Lambda ^a$ splits $U_1$ as well, since $\Lambda _a$ is negative definite. We can first embed $G^\#$ into $\mathrm {Aut}_s(V)$ as above. It remains to extend the isometry a.

We restrict a to L to obtain an isometry of L, but acting nontrivially on the discriminant group $D_L$ . As in the proof of [Reference Marquand and Muller29, Theorem 4.3], we can instead extend the isometry $-a\in O(L)$ to an isometry b of $H^4(V,\mathbb {Z})$ , acting trivially on $\langle h^2\rangle ,$ and by the Torelli theorem, we conclude that b is induced by an automorphism of V, which we still denote by $b\in \mathrm {Aut}(V)$ . However, this automorphism b is now antisymplectic: the isometry a acted trivially on $H^{3,1};$ hence, b acts by $- {id}|_{H^{3,1}}$ . The group $\langle G_s, b\rangle \cong G$ is thus a subgroup of $\mathrm {Aut}(V)$ , satisfying the statement of the Proposition.

Finally, since $G_s$ is a group of symplectic automorphisms of a cubic fourfold, the pair $(G_s,\Lambda _{G_s})$ is a Leech pair and occurs in the classification of [Reference Laza and Zheng26].