Automorphisms and Periods of Cubic Fourfolds

We classify the symplectic automorphism groups for cubic fourfolds. The main inputs are the global Torelli theorem for cubic fourfolds and the classification of the fixed-point sublattices of the Leech lattice. Among the highlights of our results, we note that there are 34 possible groups of symplectic automorphisms, with 6 maximal cases. The six maximal cases correspond to 8 non-isomorphic, and isolated in moduli, cubic fourfolds; six of them previously identified by other authors. Finally, the Fermat cubic fourfold has the largest possible order (174,960) for the automorphism group (non-necessarily symplectic) among all smooth cubic fourfolds.


Introduction
Cubic fourfolds are some of the most intensely studied objects in algebraic geometry in connection to rationality questions and to constructing compact hyper-Kähler manifolds. What sets the cubic fourfolds apart is that they are Fano fourfolds whose middle cohomology is of level 2 with h 3,1 = 1 (i.e., up to a Tate twist looks like the cohomology of a K3 surface). Consequently, the moduli space of cubic fourfolds behaves very similarly to the moduli space of polarized K3 surfaces. Specifically, Voisin [Voi86] proved a global Torelli theorem for cubic fourfolds. Later, Hassett [Has00] identified some natural Noether-Lefschetz divisors C d (for d ∈ Z + with d ≡ 0, 2 (mod 6)) in the moduli space of cubic fourfolds, and conjectured that the image of the period map is the complement of C 2 and C 6 . This was subsequently verified by Laza [Laz09,Laz10] and Looijenga [Loo09]. More recently, the second author [Zhe17] proved a stronger version of the Torelli theorem: the automorphisms of cubic fourfolds are detected by (polarized) Hodge isometries.
The purpose of this paper is to use the period map to study and classify the possible symplectic automorphism groups (Definition 2.8) for cubic fourfolds. The model for our study is the well-known case of K3 surfaces. Namely, a consequence of the Torelli theorem for K3 surfaces is that there is a close connection between the automorphism group Aut(Y ) of a K3 surface Y and the Hodge isometries on H 2 (Y, Z). Nikulin [Nik79a] started a systematic investigation of the possible finite automorphism groups for K3 surfaces by means of lattice theory ( [Nik79b]). This study culminated with the celebrated result of Mukai [Muk88] relating the classification of the finite groups of symplectic automorphisms acting on K3 surfaces with certain subgroups of the Mathieu group M 23 . Kondō [Kon98] simplified Mukai's proof by relating this classification problem to the isometries of the Niemeier lattices. Kondō's approach avoids the Leech lattice (the unique Niemeier lattice containing no roots), but it turns out that a related construction that involves only the Leech lattice L behaves more uniformly and adapts to higher dimensions ( [GHV12], [Huy16]). In particular, one sees that all the symplectic automorphism groups G occurring are subgroups of the Conway group Co 0 (= O(L)) satisfying a certain rank condition on the fixed-point sublattice L G .
The higher dimensional analogue of the K3 surfaces are the hyper-Kähler manifolds (simply connected, compact Kähler manifold, carrying a unique holomorphic symplectic 2-form). Due to Verbitsky's Torelli Theorem and recent results on Mori cones of hyper-Kähler manifolds (e.g., [BM14], [BHT15], [HT16]), the approach to automorphisms via lattices that works for K3 surfaces can be extended to the case of hyper-Kähler manifolds of K3 [n] type, leading to a flurry of activity on the subject. In particular, we note the work of Mongardi [Mon13a,Mon16] who started a systematic study of the symplectic automorphisms of hyper-Kähler manifolds of K3 [n] type. Around the same time, Höhn and Mason [HM16] have completed the classification of the fixed-point sublattices L G of L with respect to subgroups G of Co 0 (the case G is cyclic was previously done by Harada-Lang [HL90]). Using this classification, in subsequent work [HM14], Höhn and Mason have completed Mongardi's analysis for hyper-Kähler manifolds of K3 [2] , obtaining an analogue of Mukai's results in the 4-dimensional case. There are 15 maximal groups ([HM14, Table 2]) that are listed in Table 1 in our paper.
Theorem 1.2. Let X be a smooth cubic fourfold with symplectic automorphism group G = Aut s (X). Let S := S G (X) be the covariant lattice (i.e., the orthogonal complement of the invariant sublattice of H 4 (X, Z) under the induced action of G). Then one of the following situations holds: (0) rank(S) = 0, G = 1.
Moreover, all the 34 pairs (G, S) in above cases can arise from smooth cubic fourfold with G the symplectic automorphism group. In fact, the dimension of the moduli space of cubic fourfolds with associated pair (G, S) is 20 − rank(S).
(Here, F i , N i , L i denote cubic, quadric, and linear polynomials respectively. We denote by 1 n (k 1 , . . . , k 6 ) the diagonal matrix (ζ k1 , . . . , ζ k6 ) ∈ SL(6), where ζ is a primitive n-root of unity.) cubic surface as a subgroup of W (E 6 ). For cubic threefolds, we are not aware of a systematic study of their automorphism groups (see [GAL11,GAL19], [Adl78] for some results). Using the period map of Allcock-Carlson-Toledo [ACT11] (see also [LS07]), and ideas from this paper, we are able to relate the classification of the automorphisms groups for cubic threefolds to the Suzuki sporadic group Suz (N.B. an index 6 extension of Suz is isomorphic to the centralizer of an order 3 element in Co 0 ; see [Wil83]). To our knowledge this relationship is new, we plan to return to it in future work.
We note that once a Leech pair (G, S) as in Theorem 1.2 is specified, one obtains a moduli space M (G,S) of dimension 20 − rank(S) parametrizing cubic fourfolds X with G ⊂ Aut s (X) (e.g., see [Mon13a,Ch. 5], [YZ18]). However, it is not necessary that this moduli space is irreducible. This corresponds to S having different primitive embeddings into the primitive lattice Λ 0 for cubic fourfolds (the existence of the embedding S ֒→ Λ 0 is essentially the content of Theorem 1.2). It is thus a natural question to study uniqueness of S ֒→ Λ 0 for the pairs (G, S) occurring in Theorem 1.2. The analogue question for (unpolarized) K3 surfaces was studied by Hashimoto [Has12] (similarly, for polarized symplectic involutions, see [vGS07]). Here, we are restricting ourselves to the maximal cases (i.e., rank(S) = 20), as those are the most interesting cases. For instance, these cases give interesting examples of maximal algebraic cubics (in the sense of maximal possible rank for the group of algebraic cycles H 4 (X, Z) ∩ H 2,2 (X); equivalently the transcendental lattice T is negative definite of rank 2). We obtain a somewhat surprising result: while there are 6 groups that occur (cf. Theorem 1.2(9)), there are 8 cubic fourfolds (automatically isolated in moduli) corresponding to them. Six out of the eight cases are identified in [HM14, Table 2]; we are not able to give equations for the remaining two special cubics (cases X 2 (A 7 ) and X 2 (M 10 ) below). Theorem 1.8. Let (G, S) be a Leech pair such that rank(S) = 20 and there exists a smooth cubic fourfold X with G = Aut s (X) and (G, S) ∼ = (G, S G (X)). We denote by T the orthogonal complement of S in H 4 0 (X, Z). Then we have and only have the following possibilities: (1) G = 3 4 : A 6 , the corresponding cubic fourfold is the Fermat one X(3 4 : A 6 ) = V (x 3 1 + x 3 2 + x 3 3 + x 3 4 + x 3 5 + x 3 6 ) and T = −(6 3 6) = A 2 (−3). Moreover, this is the only smooth cubic fourfold with a symplectic automorphism of order 9. It holds Aut(X)/ Aut s (X) ∼ = Z/6.
Structure of the paper. In section 2, we introduce and briefly review the properties of the period map for cubic fourfolds. Additionally, in §2.3, we review the notion of Borcherds marking for cubic fourfolds. In the following section 3, we review the necessary material on the Leech lattice, Niemeier lattices, and Conway group. These two review sections (specific to our situation) are complemented by two appendix sections, which cover very standard material, but which nonetheless might be helpful to the reader. Specifically, in Appendix A, we collect results in lattice theory (mostly due to Nikulin) which are essential in our arguments.
In Appendix B, we review some basic facts and notations for finite groups.
The main content going into the proof of Theorem 1.2 is discussed in sections 3 and 4. First, following Mongardi's work, we introduce the notion of Leech pair (Definition 3.3), and give a key lemma (Lemma 3.4). We then focus on the polarized case. In particular, we establish a criterion (Theorem 4.5) for a Leech pair (G, S) to arise from a group of symplectic automorphisms for some cubic fourfold X. In §4.5 we prove Theorem 1.8 using methods from Lattice theory.
The remaining two sections are complementing our main classification result. Namely, in section 5, we partially discuss the completely analogous (and somewhat easier) situation for degree 2 and 6 K3 surfaces. Finally, while the focus of this paper is on symplectic automorphisms, we make some comments on the non-symplectic case in section 6. In particular, we determine the full automorphism groups of the 8 maximal cases of Theorem 1.8 (see Proposition 6.12). This allows us to distinguish geometrically the two cases of Theorem 1.8(2) with Aut s (X) ∼ = A 7 (i.e., one has a anti-symplectic involution, while the other does not). As a consequence of this classification, we also obtain that the maximal possible order of automorphism group for a cubic fourfold is 174, 960 which is reached only by the Fermat cubic fourfold (an analogous result for K3 surfaces was obtained by Kondō [Kon99]).
Acknowledgement. Most of the work was done while the second author visited Stony Brook during the Spring 2018 semester. His stay was supported by Tsinghua Scholarship for Overseas Graduate Studies. He thanks Stony Brook for hosting him and he is grateful to his advisor, Eduard Looijenga, for constant support and helpful discussions on related topics. The research of the first author was partially supported by NSF grants DMS-1254812 and DMS-1802128.
After the posting of our manuscript, we have learned of the work of Ouchi [Ouc19], who explores the interplay between automorphisms of cubic fourfolds and the automorphisms of the associated K3 category (the Kuznetsov component). We thank G. Ouchi for sharing an early version of his work, and for some comments on our paper. We are also grateful to S. Mukai for sharing with us some of his partial work on the classification of automorphisms of cubic fourfolds (from late eighties). As a consequence, we have updated some of our notations (and added some remarks) to be aligned with Mukai's work.

Automorphisms and periods
In this section we review some well-known facts, which are the starting point of our classification of the automorphism groups for cubic fourfolds. First, the Global Torelli Theorem (Thm. 2.3 and Prop. 2.4) allows one to reduce the classification of automorphisms for cubic fourfolds to the classification of automorphisms of Hodge structures, which in turn is essentially a lattice theoretic question. Classically, this approach was successfully applied to the case of K3 surfaces (Nikulin, Mukai, Kondō and others). More recently, it was (partially) adapted to the case of hyper-Kähler manifolds of K3 [n] type. The Fano variety F (X) of a cubic fourfold X is a hyper-Kähler of K3 [2] type. Thus, the classification of automorphisms of X is closely related to the classification of automorphisms of F (X). We review this in §2.2 below. Finally, the difference to most of related work that we cite is that we need to keep track of the polarization. It turns out that it is better to keep track of a "Borcherds polarization" instead of the natural polarization of X (or equivalently F (X)). We introduce this notion in §2.3.
2.1. Periods for cubics. Let X be a smooth cubic fourfold. The middle cohomology group H 4 (X, Z), with the natural intersection pairing, is a unimodular odd lattice Λ of signature (21, 2) (uniquely specified by these conditions). Let η X ∈ H 4 (X, Z) be the square of the hyperplane class of X. The primitive cohomology H 4 (X, Z) prim = η X ⊥ carries a polarized Hodge structure of K3 type (i.e., Hodge numbers (0, 1, 20, 1, 0)). As lattice, H 4 (X, Z) prim ∼ = Λ 0 where Λ 0 := (E 8 ) 2 ⊕ U 2 ⊕ A 2 (with A 2 and E 8 the standard root lattices, and U the hyperbolic plane). Similarly to the well-known case of K3 surfaces, the period domain for Hodge structures on H 4 (X, Z) prim is the 20-dimensional Type IV period domain (where the script + indicates a choice of one of the two connected components).
Associated to the lattice Λ 0 , there are several natural groups: which preserves the spinor norm on Λ 0 (or equivalently preserves D); The global monodromy group Γ for cubic fourfold is O * (Λ 0 ) (cf. Beauville [Bea86]). Since Γ = O * (Λ 0 ) is an arithmetic group, Γ acts properly discontinuously on D. The resulting analytic variety D/Γ is in fact a quasi-projective variety; we refer to it as the global period domain for cubic fourfolds.
The set of short roots in Λ 0 determines a Γ-invariant hyperplane arrangement H 6 in D. Let C 6 := H 6 /Γ ⊂ D/Γ be the associated Heegner divisor. (ii) A norm 6 vector v in Λ 0 with divisibility 3 is called a long root. The set of long roots in Λ 0 determines a Γ-invariant hyperplane arrangement H 2 in D. Let C 2 := H 2 /Γ ⊂ D/Γ.
Remark 2.2. It is well known that there exists a single {±1} × Γ-orbit of short and long roots respectively, and thus C 6 and C 2 are irreducible divisors. Furthermore, Γ(= O * (Λ 0 )) is generated by reflections in short roots ( [Bea86]), and Γ has index 2 in O(Λ 0 ) with O(Λ 0 )/Γ generated by the class of a reflection in a long root.
Let M be the moduli space of smooth cubic fourfolds. It is a quasi-projective 20-dimensional variety, which can be constructed by GIT (see [Laz09] for a full GIT analysis). By associating with a cubic fourfold X, the Hodge structure on its middle cohomology, one obtains a period map Voisin [Voi86] proved that the Global Torelli Theorem is valid for cubic fourfolds. It follows that P is an open embedding. For the purpose of this paper, it is important to understand also the image of the period map P(M) ⊂ D/Γ. This type of question was first investigated by Hassett [Has00]. In particular, he defined certain Heegner divisors C d in D/Γ (indexed by d ∈ Z + with d ≡ 0, 2 (mod 6)) corresponding to cubic fourfolds containing additional Hodge classes. The relevant divisors here are C 2 = H 2 /Γ and C 6 = H 6 /Γ as defined above. Geometrically, C 6 corresponds to singular cubic fourfolds, while C 2 correspond to degenerations of cubics to the secant to Veronese surface in P 5 . The image of the period map misses the divisors C 2 and C 6 . Conversely, as shown by Laza [Laz10] and Looijenga [Loo09], any period outside these two divisors is realized for some smooth cubic fourfold.
Theorem 2.3 (Voisin, Hassett, Laza, Looijenga). The period map for cubic fourfolds gives an isomorphism of quasi-projective varieties We note that both sides of (2.1) have natural orbifold structures. For instance, since any smooth cubic fourfold is GIT stable ( [Laz09]), the moduli space of smooth cubic fourfolds is a smooth Deligne-Mumford stack M with quasi-projective coarse moduli space M. A natural question is whether the period map P identifies the two sides of (2.1) as orbifolds. This is equivalent to the Strong Global Torelli Theorem, i.e., any isomorphism between the polarized Hodge structures of two smooth cubic fourfolds is induced by a unique isomorphism between the two cubic fourfolds. Using the fact that automorphisms of cubic fourfolds X are induced by linear transformations of the ambient projective space P 5 , and that Aut(X) acts faithfully on the middle cohomology H 4 (X, Z) (e.g., [JL17, Proposition 2.16]), the second author [Zhe17] has verified the Strong Global Torelli Theorem.
Proposition 2.4 ( [Zhe17]). Let X 1 and X 2 be two smooth cubic fourfolds. Assume that there is an isomorphism of polarized Hodge structures (in particular ϕ(η X2 ) = η X1 ). Then, there exists a unique isomorphism f : X 1 ∼ = X 2 such that ϕ = f * . In particular, for any smooth cubic fourfold X, where Aut HS stands for group of Hodge isometries.
Remark 2.5. We note that while the period map extends to an isomorphism of quasi-projective varieties where M ADE is the moduli space of cubics with ADE singularities (see [Laz09,Laz10]), the orbifold structure along the discriminant divisor is different. Simply, a general cubic fourfold with a node (i.e., A 1 singularity) has no automorphism, while on the periods side, there is a Z/2 stabilizer corresponding to the reflection in a short root. Here, it is more convenient to work with the primitive lattices M and Λ 0 .) 2.2. The hyper-Kähler fourfold associated with a cubic fourfold X. For a smooth cubic fourfold X, the Fano variety F (X) of lines on X is a smooth hyper-Kähler fourfold, deformation equivalent to K3 [2] (cf. [BD85]). There is a natural polarization on F (X) induced from the Plücker embedding F (X) ֒→ Gr(1, P 5 ) ⊂ P(∧ 2 (C 6 )). Since any automorphism of X is linear, there is a natural group homomorphism Aut(X) −→ Aut(F (X)).
An automorphism of a hyper-Kähler manifold sends H 2,0 to H 2,0 , hence induces a scalar action on H 2,0 . If the scalar is the identify, the automorphism is called symplectic. Otherwise, it is called non-symplectic. Adapting this to the case of cubic fourfolds, we make the following definition: Definition 2.8. An automorphism of a smooth cubic fourfold X is called symplectic, iff the induced automorphism on F (X) is symplectic. Equivalently, an automorphism of X is symplectic iff the induced action on H 3,1 (X) is the identity. We denote the group of symplectic automorphisms of X by Aut s (X).
Remark 2.9. In view of Theorem 2.3 and Proposition 2.4, it is clear that essential arithmetic input in the classification of automorphisms of cubic fourfolds is the primitive cohomology lattice Λ 0 = H 4 (X, Z) prim ∼ = A 2 ⊕ (E 8 ) 2 ⊕ U 2 . Let us note that the associated hyper-Kähler F (X) has the same primitive lattice. More precisely, H 2 (F (X), Z) carries a natural quadratic form, the so-called Beauville-Bogomolov quadratic form. With respect to this form, there is a natural lattice isometry H 2 0 (F (X), Z)(−1) ∼ = H 4 0 (X, Z), which is also an isomorphism of Hodge structures (see [BD85,Proposition 6]). In particular, via this isomorphism H 2,0 (F (X)) maps to H 3,1 (X), justifying our definition above. In summary, the discussion of this subsection says that the classification of the automorphisms of cubic fourfolds is essentially equivalent to the classification of automorphisms of degree 6 (the degree of the Plücker polarization) polarized hyper-Kähler manifolds of Remark 2.10. One should note that there is a subtle difference to the case of K3 surfaces. While for K3 surfaces the full cohomology lattice H 2 (S, Z) is even unimodular, the full cohomology lattice for cubic fourfolds H 4 (X, Z) is odd unimodular. If one prefers to work with hyper-Kähler manifolds of K3 [2] type, we note that the full cohomology lattice (w.r.t. the Beauville-Bogomolov form) is even, but not unimodular (it is (up to sign) A 1 ⊕ (E 8 ) 2 ⊕ U 3 ).

Borcherds polarizations.
In view of Nikulin's theory [Nik79b], it is preferable to work with even unimodular lattices (compare Remark 2.10). The smallest (with definite orthogonal complement) even unimodular lattice that contains the primitive cubic lattice Λ 0 is the Borcherds lattice B, i.e., the unique even unimodular lattice II 26,2 ∼ = (E 8 ) 3 ⊕ U 2 of signature (26, 2). (Here, we prefer to denote it B and call it the Borcherds lattice in honor of Borcherds, who studied the automorphic forms on the associated Type IV symmetric domain.) Remark 2.11. Even in the K3 case, the embedding of the primitive cohomology lattice for a polarized K3 surface into the Borcherds lattice B turns out to be a powerful arithmetic trick (the geometric reason why it works is not yet completely understood). As examples of applications of this artifice (that we baptized Kondō-Scattone trick), we mention Scattone's work [Sca87] on the Baily-Borel compactification for polarized K3 surfaces, Kondō's work [Kon98] on symplectic automorphisms, and the Gristsenko-Hulek-Sankaran work [GHS07] on the Kodaira dimensions on the moduli spaces of K3 surfaces.
Remark 2.12. We recall that there exist 24 even unimodular lattices of rank 24, called the Niemeier lattices (see §3.1 below). What is relevant here is to note that these lattices are intricately related to the Borcherds lattice B. Namely, given a Niemeier lattice N , then B ∼ = N ⊕U 2 . Conversely, the classification of the Niemeier lattices follows from the classification of isotropic vectors in the hyperbolic lattice II 25,1 (see [CS99]), or equivalently the Type II boundary components (i.e., rank 2 totally isotropic subspaces in B) of the Baily-Borel compactification for the Borcherds period domain.
Returning to cubic fourfolds, in analogy with the work of M -polarized K3 surfaces of Dolgachev [Dol96] and Remark 2.9, we can view a cubic fourfold as being Borcherds E 6 -polarized (i.e., Λ 0 admits a primitive embedding into B with orthogonal complement E 6 ). More interestingly, the periods missing from the image of the period map for cubic fourfolds (see Theorem 2.3), i.e., the divisors C 2 and C 6 , correspond to E 7 and E 6 +A 1 Borcherds polarizations respectively. This allows a more uniform view on "singular" cubic fourfolds (i.e., singular cubics, or degenerations to the Veronese surface) -simply X is singular if it acquires an additional root (i.e., the existing "algebraic" lattice E 6 is enlarged to either E 7 or E 6 + A 1 by adding a root). This is of course equivalent to the more classical view of Hassett [Has00] where H 4 alg,prim = H 4 (X, Z) prim ∩ H 2,2 acquires a short root (equivalently, in terms of Borcherds polarizations E 6 ⊂ E 6 + A 1 ) or long root (case E 7 ). From either perspective, the transcendental lattices (for R Borcherds polarized objects, the transcendental lattice is R ⊥ B ) for the two cases are Λ 2 := 2 ⊕ (E 8 ) 2 ⊕ U 2 , and for C 2 and C 6 respectively. One recognizes the two lattices (up to a sign) Λ 2 and Λ 6 as the primitive lattices for degree 2 and respectively 6 K3 surfaces. There is indeed a close geometric relationship between degree 6 (and respectively degree 2) K3 surfaces and singular cubic fourfolds (respectively degenerations to the Veronese surface); see [Has00], [Laz10]. From the perspective of this paper, the relevant fact is the following easy proposition (see [Laz10,§6]).
Proposition 2.13. (i) There is a unique primitive embedding of Λ 0 into B, with orthogonal complement E 6 ; in another words, Λ 0 ⊕ E 6 can be saturated as B in a unique way. (ii) There is a unique primitive embedding of Λ 6 into B, with orthogonal complement A 1 ⊕ E 6 ; in another words, Λ 0 ⊕ A 1 ⊕ E 6 can be saturated as B in a unique way. (iii) There is a unique primitive embedding of Λ 2 into B, with orthogonal complement E 7 ; in another words, Λ 0 ⊕ E 7 can be saturated as B in a unique way.

Automorphisms and the Conway group
Via the Global Torelli Theorem, we have reduced the study of automorphisms for cubic fourfolds to the study of automorphisms of Hodge structures. This is in turn a question about the symmetries (satisfying certain properties) of the underlying cohomology lattice L. In the case of a finite group of symplectic automorphisms G acting on the cohomology lattice L, Nikulin made two key observations: i) the covariant lattice S G (L) is a definite lattice (this is equivalent to the symplectic condition), and ii) S G (L) does not contain any effective algebraic cycle (in fact, the symplectic condition implies that the algebraicity is automatic). In particular, for K3 surfaces, by Riemann-Roch, S G (L) (which is negative definite in this case) should not contain any −2 classes (or equivalently roots). The same holds for hyper-Kähler manifolds of K3 [n] type (e.g., by involving Markman's theory of prime exceptional divisors) and for cubic fourfolds (i.e., there is no norm 2 vector in S G (L); e.g., as a consequence of Theorem 2.3). Normally, one would try to classify S G (L) and its embeddings into the cohomology lattice L. However, using Nikulin's theory, Kondō made the observation that (in the geometric situations considered here: K3s, K3 [n] , or cubics) S G (L) embeds into one of the Niemeier lattices N , and furthermore G extends to an isometry of N (thus G ⊂ O(N )). Niemeier lattices N show up here since they are the smallest even unimodular definite lattices N containing S G (L) for any G. The lattice N being definite is important as the associated orthogonal group O(N ) is finite. Kondō [Kon98] successfully applied this approach to the classification of symplectic automorphisms for K3 surfaces. Kondō avoids the Leech lattice L (namely, he noted that A 1 ⊕ S G (L) embeds into N for K3 surfaces, and thus N = L), but in fact, since S G (L) contains no roots, it is possible to embed it into the Leech lattice L (cf. [GHV12], [Huy16]). Considering embeddings into the Leech lattice L leads to a more uniform behavior. Note however that there is a trade-off here: we deal with a single larger group Co 0 := O(L) versus 23 smaller groups O(N ) for N = L. With the advent of more powerful computational tools, and a better understanding of the Leech lattice (esp. relevant here is [HM16]), we can work throughout with the Leech lattice.
In this section, we briefly review the Leech lattice, the Conway group, and introduce the key concept (due to Mongardi, but with origins going back to Nikulin) of Leech pair. We then close with the Höhn-Mason [HM16] classification of the fixed-point lattices for the Leech lattice L. The material here is standard (and applies equally to K3s and K3 [n] s); we will apply it in the following section to the actual classification of the automorphisms of cubic fourfolds.
3.1. The Leech Lattice and the Conway group. We recall the following classification result of Niemeier.
Theorem 3.2 (Niemeier). Up to isometry, there exist 24 even unimodular positive definite lattices N of rank 24. Let R ⊂ N be the sublattice spanned by the roots (i.e., norm 2 vectors) of N . Then R is of one of the following 24 types: A lattice N as in the theorem is called a Niemeier lattice. In all but one of the cases N is spanned (over Q) by roots. The remaining case, i.e., the Niemeier lattice containing no roots, is called the Leech lattice, and we denote it by L. The automorphism of the Leech lattice is the Conway group The center of Co 0 is just µ 2 = {±id}, and the quotient is one of the largest sporadic simple groups. In fact, |Co 0 | = 2 22 · 3 9 · 5 4 · 7 2 · 11 · 13 · 23(∼ 8 · 10 18 ).
As we will see below, a group G of symplectic automorphisms for K3 surfaces, hyper-Kähler manifolds of type K3 [n] , or cubic fourfolds can be realized as a subgroup of the Conway group Co 0 . Thus, only the prime factors 2, 3, 5, 7, 11, 13, and 23 can occur in ord(G). For K3 surfaces, only the primes p ≤ 7 can occur, while for cubics all primes p ≤ 11 occur (compare Theorem 4.15). In particular, the Fano variety F (X) of a cubic fourfold X admiting an order 11 symplectic automorphism will give an example of an exotic automorphism (i.e., not induced from K3 surfaces) on a hyper-Kähler of K3 (i) S is positive definite, (ii) S does not contain any 2-vector, (iii) G fixes no nontrivial vector in S, (iv) the induced action of G on the discriminant group A S is trivial.
The condition (iv) of the Definition 3.3 should be understood as saying that given a primitive embedding S ֒→ L into a unimodular lattice L, the action of G on S extends (acting trivially on S ⊥ L ) to L. The condition (iii) complements this by saying that S is the covariant lattice for the action of G on L. Note then that the smallest unimodular lattice satisfying the first 2 conditions of the definition above is the Leech lattice L. Obviously, any sublattice of the Leech lattice will also satisfy (i) and (ii) of the definition. Thus choosing a subgroup G ⊂ Co 0 (= O(L)), the associated covariant lattice S G (L) in L will give an example of Leech pair (G, S G (L)). The following proposition says the converse: under a mild condition on the rank of S (satisfied in the geometric context relevant to this paper), the Leech pair (S, G) is obtained as a covariant lattice in L. This argument seems to occur first in [GHV12, Appendix B] (see also [Huy16, Prop. 2.2]; related arguments go back to Scattone [Sca87] and Kondō [Kon98]). For completeness, we sketch the proof.
Proposition 3.4. For a Leech pair (G, S) the following two statements are equivalent: (i) rank(S) + l(q S ) ≤ 24, (ii) There exists a primitive embedding of S into the Leech lattice L.
Once these two condition are fulfilled, there is an action of G on L with (G, S) ∼ = (G, S G (L)).
Proof. Assume (ii), and denote by K the orthogonal complement of the given primitive embedding of S into L. Then l(q S ) = l(q K ) ≤ rank(K) = 24 − rank(S). Thus (ii) implies (i). Now assume (i). Since l(q S ) ≤ 24 − rank(S) < rank(L ⊕ U ) − rank(S), by Nikulin's existence Theorem A.8, there exists a primitive embedding S ֒→ L ⊕ U . Denote by N the orthogonal complement of S in L ⊕ U . Then N has signature (25 − rank(S), 1). Thus N R intersects with the positive cone of L⊕ U . Since S contains no 2-vector, N R intersects with one of the chambers of the positive cone of L ⊕ U .
Let w ∈ U be primitive and isotropic. The vector w ∈ L ⊕ U is called a Weyl vector 5 . We call a vector v ∈ L ⊕ U with (v, v) = 2 and (v, w) = −1 a Leech root. By [CS99,Chap. 27], the automorphism group of L ⊕ U is generated by reflections with respect to Leech roots. Therefore, there exists a chamber C 0 given by for any Leech root v}. By adjusting the embedding S ֒→ L ⊕ U via an automorphism of L ⊕ U , we may assume that N R intersects with C 0 , hence G leaves the chamber C 0 stable. By [Bor84], G fixes the Weyl vector w. Equivalently, w ∈ N . Then we have: which gives rise a primitive embedding of S into the Leech lattice L. The group action of G on S extends to an action on L with (G, S) ∼ = (G, S G (L)).
Corrolary 3.5. For a Leech pair (G, S) satisfying the statements in Lemma 3.4, there is an embedding G ֒→ Co 0 , with image avoiding −id unless rank(S) = 24.
3.3. Höhn-Mason classification of saturated Leech pairs. In view of the discussion above, to classify the Leech pairs relevant to the classification of automorphisms, one can proceed by considering subgroups G ⊂ Co 0 and the associated covariant lattice S G (L). The only issue is that there might be several groups G leading to the same covariant lattice. For the classification of automorphism groups, we are interested in the maximal cases (i.e., in G = Aut s (X) and not subgroups G ′ ⊂ G that happens to have the same invariant/covariant lattice). The following two definitions formalize this idea.
Definition 3.6. A Leech pair (G, S) is called saturated, if G is the maximal group acting faithfully on S and trivially on the discriminant group A S . Let G be a finite group acting on the Leech lattice L. One can consider the (point-wise) stabilizer G ′ of L G . Obviously, G ⊆ G ′ , L G = L G ′ , and G ′ is the largest group stabilizing L G . The induced action of G ′ on A SG(L) ∼ = A L G is trivial. Conversely, every automorphism of S G (L) which trivializes A SG(L) can be extended to an automorphism of L which stabilizes L G . Thus G ′ is equal to the automorphism group of S G (L) trivializing the discriminant. The Leech pair (G, S G (L)) is saturated if and only if G = G ′ .
We denote by A the set of conjugacy classes of sub-pairs of (Co 0 , L). There is a natural poset structure on A . Denote by A sat the sub-poset of A consisting of saturated Leech pairs. A fixed-point sublattice of L is the invariant sublattice L G for some G ⊂ Co 0 . It is clear that associating with (G, S) ∈ A the fixed-point sublattice L G gives rise to a one-to-one correspondence between A sat and the set of (Co 0 -)orbits in the set of fixed-point sublattices of the Leech lattice L. The fixed-point sublattices of L were classified by Höhn and Mason [HM16]. This classification will play a key role for us. For further reference, we mention: Remark 3.9. Harada and Lang [HL90] classified all fixed-point sublattices K which are induced by actions of cyclic groups G ∼ = Z/n on the Leech lattice. The information contained in [HL90] is sometimes richer and more handy than that in [HM16].

The case of cubic fourfolds
In this section, we are classifying the symplectic automorphism groups of smooth cubic fourfolds. First, following the standard argument for K3 surfaces and hyper-Kähler manifolds, we establish that a group G acting symplectically on a cubic X, determines a Leech pair (G, S = S G (X)), which further can be embedded into the Leech lattice L (Corollary 4.3). Since S arises from a cubic fourfold X, it is clear that S embeds into the primitive lattice Λ 0 . By Theorem 2.3 (we use the surjectivity part), this is essentially also a sufficient condition. We state this, in terms of the Borcherds polarization (see §2.3) as an iff criterion in Theorem 4.5. Using this criterion, the actual classification ( §4.4) is accomplished by using the Höhn-Masson [HM16] (see also [HL90]) classification of the fixed-point sublattices in the Leech lattice, and Fu's classification ( [Fu16]) of automorphism groups of prime-power orders. The uniqueness of embeddings in the maximal cases (Theorem 1.8) is discussed in §4.5. 4.1. Leech pairs associated to symplectic automorphisms on cubic fourfolds and K3 surfaces. A finite group of symplectic automorphisms on a K3 surface, on a hyper-Kähler manifold of K3 [n] type, or on a cubic fourfold leads to a Leech pair. The argument essentially goes back to Nikulin [Nik79a], and was refined recently in the context of groups of symplectic automorphisms for hyper-Kähler manifolds (see esp. [Huy16] and [Mon13a]). We review the situation for the cases relevant to us: cubic fourfolds and polarized K3 surfaces.
Notation 4.1. Let X be a smooth cubic fourfold, and G ⊂ Aut s (X). We denote by S G (X) the covariant lattice for the induced action of G on H 4 (X, Z). Similarly, if Y is a smooth algebraic K3 surface, and G ⊂ Aut s (Y ) a finite group, we denote by S G (Y ) the covariant lattice for the induced action of G on H 2 (Y, Z)(−1).
Lemma 4.2. Let X be either a smooth cubic fourfold or an algebraic K3 surface with an action of a finite group G ⊂ Aut s (X). Then (G, S G (X)) is a Leech pair.
Proof. The assumption of symplectic automorphism implies that S G (X) ⊂ H 2,2 (X) ∩ H 4 (X, Z) prim . By Hodge index Theorem, S G (X) is positive definite, and by Theorem 2.3, S G (X) contains no short roots (i.e., the period point avoids C 6 ). Since G acts trivially on the invariant cohomology H 4 (X, Z) G and S G (X) = (H 4 (X, Z) G ) ⊥ , it follows that G acts trivially on A SG(X) . Finally, since Aut(X) acts faithfully on H 4 (X), it is clear that G acts faithfully on S G (X). We conclude that (G, S G (X)) is a Leech pair (cf. Def. 3.3).
The argument for K3 surfaces is similar (and due to Nikulin), except for invoking Riemann-Roch to prove that there is no norm 2 vector (corresponding, via our scaling, to a −2 class) in S G (X).
Corrolary 4.3. Let X be either a smooth cubic fourfold or an algebraic K3 surface with a faithful action of a finite group G ⊂ Aut s (X). There exists a primitive embedding of S G (X) into L, and hence an embedding of G into Co 0 with image avoiding −id.
Proof. By Lemma 4.2, (G, S G (X)) is a Leech pair. Since S G (X) has a primitive embedding into a unimodular lattice of rank 23 (or 22) for cubic fourfolds (or K3 surfaces respectively), the rank condition of Proposition 3.4 is satisfied; the claim follows.
Let us now discuss the role of the polarization. If X is a cubic fourfold, any automorphism f is induced from a linear automorphism of the ambient projective space, and thus ϕ = f * preserves the class η ∈ H 4 (X, Z) (recall η is the square of a hyperplane class). It follows that there is a primitive embedding For K3 surfaces Y , the situation is similar, but there is a subtle difference. Namely, under the assumption that Y is algebraic (i.e., NS(Y ) contains an ample class h), and G is finite, any automorphism ϕ ∈ G will preserve some ample class h ′ (e.g., obtained by "averaging" h). This is the set-up of the classical results of Nikulin and Mukai. However, when taking about polarized K3 surfaces, we will fix an ample class h on Y and insist that the automorphism f preserves h (i.e., f * h = h in cohomology). With this assumption, we 13 have again a primitive embedding is the primitive cohomology (we twist the form by −1 to get consistency with the cubic fourfold case).
Remark 4.4. We are not aware of a systematic study of the symplectic automorphisms in the polarized case for any degree (in section 5 below, we will partially discuss the degree 2 and 6 cases as they are tightly connected to the cubic fourfold case). One situation where the polarized case was studied is the symplectic involutions. We recall that Nikulin proved that there is a single class of symplectic involutions for algebraic K3 surfaces (with notations as above, S G (X) ∼ = E 8 (2)). The polarized symplectic involutions were classified by van Geemen and Sarti [vGS07]; a richer picture emerges (as one needs to keep track of the embedding of E 8 (2) into Λ d , versus the unimodular K3 lattice).
4.1.1. A criterion for Leech pairs to arise from symplectic automorphisms. So far we have discussed how a finite group of symplectic automorphisms G ⊂ Aut s (X) leads to a Leech pair (G, S G (X)), which in turn can be classified by Höhn-Mason [HM16] results. Now we are interested in the converse, given a Leech pair (G, S), when does it come from a symplectic automorphism group G acting on X? By Global Torelli Theorem (and surjectivity of the period map), this becomes a question about embeddings of lattices. For instance, note that (4.1) is a necessary condition if X is a cubic fourfold. In fact, by Theorem 2.3 (and Prop. 2.7), (4.1) is essentially also sufficient, but some care is needed as S needs to avoid both short roots (automatic since (G, S) is a Leech pair) and long roots. To deal with both cases uniformly, it is better to view a smooth cubic fourfold X as being E 6 Borcherds polarized (see §2.3). Based on these considerations, we obtain the following key result which allows us to go back and forth between geometry (automorphisms of X) and arithmetic (fixed-point sublattices of the Leech lattice L).
Theorem 4.5 (Criterion for Leech pairs associated with cubic fourfolds). Let (G, S) be a Leech pair. The following are equivalent: (i) There exists a smooth cubic fourfold X with a faithful and symplectic action of G such that (G, S) ∼ = (G, S G (X)), (ii) There exists a faithful action of G on the Leech lattice L with (G, S) ∼ = (G, S G (L)) and K = L G , such that there exists a primitive embedding of E 6 into K ⊕ U 2 , (iii) There exists an embedding of S ⊕ E 6 into the Borcherds lattice B, such that the image of S is primitive.
Proof. (i) =⇒ (ii): From Corollary 4.3, there exists a primitive embedding S ֒→ L with an extension of the G-action on L such that L G is the orthogonal complement of S in L. We have now two ways to embed S into B, explicitly: Clearly, both embeddings are primitive (e.g., Λ 0 ⊂ B is primitive by Proposition 2.13, and S is primitive in Λ 0 by (4.1)). By Nikulin's results (see Theorem A.9), we know that there is a single conjugacy class of primitive embeddings S ֒→ B. Therefore, we can choose the isomorphism L⊕U 2 ∼ = B, such that the following diagram commutes: Since E 6 does not admit any overlattice, E 6 embeds primitively into K ⊕ U 2 .
(ii) =⇒ (iii): There is the embedding: Notice that S has primitive image in L, hence also has primitive image in B.
(iii) =⇒ (i): The action of G on S induces trivial action on (A S , q S ), hence extend to be an action on B such that its restriction to the orthogonal complement of S trivial. Since S ⊂ B is primitive (by assumption), we get On the other hand, we note that G acts trivially on E 6 ⊂ B (since by construction E 6 ⊂ S ⊥ B ). We view Λ 0 as the orthogonal complement of E 6 in B (cf. Prop. 2.13). Via this identification, the G action on B induces a G action on Λ 0 . By construction S ֒→ Λ 0 (primitive, as S is primitive in B), and clearly (G, S) ∼ = (G, S G (Λ 0 )). We can choose a Hodge structure H on Λ 0 of type (0, 1, 20, 1, 0) (i.e., H is a decomposition of Λ 0,C with the obvious properties) such that H 2,2 ∩ Λ 0 = S (i.e., S is the algebraic lattice). Assuming that S contains no short or long roots, the Global Torelli Theorem (Theorem 2.3) says that there exists a smooth cubic fourfold with H 4 (X, Z) prim ∼ = H (as Hodge structures). Finally, by Proposition 2.4, we conclude that X has a faithful and symplectic action of G such that (G, S G (X)) ∼ = (G, S).
It remains to prove that S ⊂ Λ 0 contains no short or long roots of Λ 0 (see Definition 2.1). By assumption S is a sublattice of the Leech lattice L, so it contains no short roots (i.e., norm 2 vectors). Assume now S contains a long root δ, i.e., (δ, δ) = 6 and div Λ0 (δ) = 3. Since B is obtained by gluing E 6 and Λ 0 , we conclude that δ and E 6 span a E 7 lattice in B. More precisely, there exists ǫ ∈ E 6 (with ǫ 2 = 12 and div E6 (ǫ) = 3) such that (δ + ǫ)/3 ∈ B. Since G acts on S without fixed nonzero vector, there exists g ∈ G such that gδ = δ. We distinguish two cases, either gδ = −δ or not. Assume first gδ = −δ; then g( We conclude v = 2δ/3 ∈ B, but this is a contradiction due to the fact that (δ, δ) = 6 (v will not have integral norm). Thus, we can assume that δ ′ = gδ is a long root non-proportional to δ. Consider the lattice . Then M is a positive definite rank 8 lattice containing two sublattices Sat B ( δ, E 6 ) and Sat B ( δ ′ , E 6 ) of type E 7 . Clearly, M ∼ = E 8 (first, the root sublattice of M is of type E 8 as it is strictly larger than E 7 , then E 8 ⊂ M forces equality for reasons of rank and determinant). It is well known that E 6 admits a unique embedding in E 8 with orthogonal complement A 2 . We get In particular, S contains some short roots, contradicting the fact that (G, S) is a Leech pair. 4.1.2. Moduli of cubics associated with a Leech pair (G, S). We denote by A cub the sub-poset of A consisting of Leech pairs isomorphic to (G, S G (X)) for some smooth cubic fourfold X with G = Aut s (X). It is clear that such a Leech pair (G, S G (X)) is saturated. Therefore we have A cub ⊂ A sat . Our purpose is to determine the poset A cub . We now discuss the geometric loci ("moduli") associated with the elements of this poset. By studying the minimal and maximal loci, in §4.2 and §4.3 respectively, we will be able to complete the proof of our main Theorem 1.2.
Let (G, S) be a Leech pair with G = Aut s (X). As already discussed, it follows that S ⊂ H 2,2 (X) ∩ H 4 (X, Z) prim (i.e., S is a lattice of algebraic cycles on X) and in fact the equality holds generically. Similarly to the well-known situation for K3 surfaces (see Remark 2.9), one can consider the moduli space of S-polarized cubic fourfolds (i.e., cubics with S ֒→ H 4 (X, Z) prim ∼ = Λ 0 primitive), or even (G, S)-polarized (since G acts trivially on A S , the G-action extends to Λ 0 , and thus there is essentially no difference). We obtain a moduli space M (G,S) which is a locally symmetric variety of Type IV (of the same type as the moduli of cubic fourfolds). Some care is needed here. First, M (G,S) can have several irreducible components (corresponding to different primitive embeddings of S into Λ 0 ). Then, since we view M (G,S) as a closed subvariety of M (the moduli of cubic fourfolds), a normalization is needed in order to view it as a locally symmetric variety. Finally, one needs to exclude the restrictions of the hyperplane arrangements H 2 and H 6 (see Theorem 2.3) to the locus of S-polarized cubics (as discussed S does not contain short or long roots, thus this locus is not contained in either H 2 or H 6 ; on the other hand, the restrictions of H 2 and H 6 can lead to multiple irreducible arrangements). We refer to Mongardi [Mon13a] and [YZ18] for further details. To summarize the above discussion, we have: where D is a Type IV domain with a faithful action of an arithmetic group Γ ′ , and H is a Γ ′ -invariant hyperplane arrangement in D. Moreover, dim(F ) = dim(D) = 20 − rank(S).
Remark 4.7. The definition of M (G,S) makes sense for all Leech pairs (G, S), but in fact it depends only on the saturation, i.e., M (G,S) = M (G ′ ,S) , where G ′ = Aut s (X) for X a general cubic in M (G,S) . In Theorem 1.2, our classification is about saturated pairs, but in the arguments below it is convenient not to require (G, S) to be saturated.
It is clear that the moduli spaces M (G,S) have a natural poset structure that matches with the poset structure on A cub . Theorem 1.2 is organized by the dimensions of M (G,S) ( = ∅) (or equivalently rank(S)).

4.2.
The maximal Leech pairs for cubic fourfolds. We now note that the Leech pairs arising from automorphisms of cubic fourfolds satisfy an easy necessary condition (in terms of the rank of covariant lattice and the rank of the discriminant group). In order to easier relate to the Höhn-Mason classification [HM16], we state the condition in terms of the fixed-point sublattice K in the Leech lattice L.
Proposition 4.9. The equivalent conditions in Theorem 4.5 imply Condition 4.8.
Proof. Since S embeds into Λ 0 which has signature (20, 2), the rank condition is clear. Assume now that (G, S) is a Leech pair with a primitive embedding of S into L, and K is the orthogonal complement of S in L. By Theorem 4.5, there exists a primitive embedding of E 6 into K ⊕ U 2 . Denote by M the orthogonal complement of E 6 in K ⊕ U 2 . We have a saturation E 6 ⊕ M ֒→ K ⊕ U 2 . By Nikulin's glueing theory, there exists an isotropic subspace H of A E6 ⊕ A M , such that Assume first that the glueing group H is trivial, then Remark 4.10. To understand the restriction imposed by Condition 4.8 on Leech pairs, let us consider the case G = Z/2 (i.e., symplectic involutions). According to [HL90] (also [HM16]), there are three nontrivial conjugacy classes of involutions in Co 0 = O(L). The fixed-point sublattices K in the three cases are E 8 (2), D + 12 (2), and BW 16 (the Barnes-Wall lattice), while the covariant lattices S G (L) = K ⊥ L are BW 16 , D + 12 (2), and E 8 (2) respectively. For E 8 (2) and D + 12 (2), it holds rank(K) = l(K) (this holds true whenever K = K ′ (n) for some integral lattice K ′ , n ∈ Z >1 ), while BW 16 obviously satisfies Condition 4.8. We conclude that the only possible Leech pair arising from symplectic involutions on cubic fourfolds is (Z/2, E 8 (2)). To conclude that there is a unique class of symplectic involutions, we would need to prove that there exists a unique primitive embedding of E 8 (2) in Λ 0 . In this particular case, a direct geometric argument (via a diagonalization of the involution) is easier. This concludes item (1) of Theorem 1.2. In §3.3, we have defined a natural poset A on the set of Leech pairs in (Co 0 , L). We are now interested in identifying the maximal Leech pairs (G, S) arising from cubic fourfolds. As noted above, these pairs satisfy Condition 4.8. Focusing on the maximal rank cases, by inspecting [HM16], we note that there are 15 Leech pairs (G, S) ∈ A sat with rank(S) = 20 (or equivalently rank(K) = 4) and satisfying Condition 4.8. In fact, these cases precisely coincide with those of [HM14, Table 9]. For reader's convenience, we list them (sometimes corrected 6 ) in Table ( Remark 4.11. In Table (  We expect that the semi-direct product 3 2 : QD 16 appearing in item 15 is in fact isomorphic to M 2,9 (see §B.2).
It turns out that the 15 groups listed in Table (1) occur as maximal groups of symplectic automorphisms for some hyper-Kähler manifold of K3 [2] type (algebraic, but not polarized). Specifically, it holds:  Table (1). We are interested in the maximal rank cases that can occur for cubic fourfolds, or equivalently the saturated Leech pairs (G, S) for which M (G,S) = ∅ and dim M (G,S) = 0. Höhn and Mason [HM14, Table 11] have identified six cases that do occur for cubic fourfolds, and in fact they gave explicit equations of cubic fourfolds realizing these groups of automorphisms. Using our Criterion 4.5, we prove the converse: these six cases are all the maximal rank possibilities for cubic fourfolds. Note however (see §4.5 below) that in two of the cases, there are two distinct embeddings of S into Λ 0 , leading to two more isolated cubic fourfolds with large symmetry in addition to the six cubics found by Höhn and Mason. 6 There are some typos in the listing of the discriminant forms in [HM16]. For example, the discriminant form corresponds to case of M 10 is listed as 2 +1 5 4 +1 1 3 −1 5 +1 in [HM16], but this is not allowed in the Conway-Sloane [CS99] notation.
Proof. By Theorem 4.5, we need to determine for which (G, S) among the 15 candidates, there exists an embedding of S ⊕ E 6 into the Borcherds lattice B with the image of S primitive. There are two possibilities for such an embedding S ⊕ E 6 ⊂ B. Either S ⊕ E 6 ⊂ B is primitive or not. If S ⊕ E 6 ⊂ B is not primitive, there exists a coindex 3 saturation S of S ⊕ E 6 , in which S is primitive. Then S embeds primitively into B. Since S ⊕ E 6 (or S) has rank 26, and B is the unique even unimodular lattice of signature (26, 2), by Nikulin's theory, we conclude that S ⊕ E 6 (or S respectively) embeds primitively into B iff there exists a negative definite rank 2 even lattice T with discriminant form q T = −q S⊕E6 (or q T = −q S respectively). By Theorem A.8, such a lattice T exists iff four conditions are satisfied. The first condition on the signature is automatically satisfied here. The remaining conditions are on the discriminant form q T (that is determined by S ⊕ E 6 or the index 3 overlatice S of S ⊕ E 6 ). We do a case by case analysis of the 15 possibilities from (1) The discriminant form of S ⊕ E 6 is 3 +2 9 −1 ⊕ 3 +1 . There is a nontrivial saturation S of S ⊕ E 6 with discriminant form 3 −1 9 −1 . There exists a negative rank 2 even lattice T with discriminant form 3 +1 9 +1 . Thus there exists a primitive embedding of S into B with orthogonal complement T .
(2) The discriminant form of S ⊕ E 6 is 2 −2 II 3 +2 7 +1 . There is no nontrivial saturation of S ⊕ E 6 . Since 3 2 × 7 is a square in Z 2 , there does not exists a negative rank 2 even lattice with discriminant form 2 −2 II 3 +2 7 −1 . Thus there does not exist embedding of S ⊕ E 6 into B.

4.3.
Cubics with special groups (cyclic, Klein, and S 3 ) of automorphisms. Theorem 4.14 classifies the 0-dimensional moduli spaces M (G,S) . The top dimensional moduli spaces M (G,S) will correspond to small groups G. In particular, the minimal elements in the poset A cub can be determined by considering G to be a cyclic group of prime order. The cubics with symplectic action of prime order were studied previously, especially by Fu [Fu16] (see also [GAL11]), who classified all the possibilities for prime-power symplectic automorphisms. . Let X = V (F ) ⊂ P 5 be a smooth cubic fourfold with a symplectic action by a prime-power order cyclic group G = g . We can choose coordinates (x 1 , x 2 , · · · , x 6 ) on P 5 , and generator g ∈ G, such that (g, F ) belong to one of the following cases (N.B. the cases are arranged such that the associated moduli space F is irreducible and non-empty): (0) ord(g) = 1, g = id, dim(F ) = 20, and F any smooth cubic.
From the lattice theoretic approach (our main approach in this paper), Fu's classification is closely related to Harada-Lang classification [HL90] of fixed-point sublattices in the Leech lattice with respect to cyclic groups (see Remark 4.10 for the case of involutions). In fact, using the lattice theoretic approach and [HL90], we can improve Fu's result. Specifically, the following holds: Theorem 4.17. Let G be a cyclic group acting symplectically on some smooth cubic fourfold X (i.e., G ⊂ Aut s (X)). Then, the order of |G| is one of the following: |G| ∈ {1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15} Furthermore, the following holds: (1) (Prime-power Cases). For the cases |G| = p k , we have the following correspondences among Fu's classification, Harada-Lang classification and Höhn-Mason classification.
Remark 4.18. The maximal cases in item (3) above are in fact unique. This is proved in §4.5 below. Thus, considering cubic fourfolds with a symplectic action by a cyclic group of order ≥ 9 gives four of the maximal cases listed in Theorem 1.8.
Proof. Harada-Lang [HL90] classified the conjugacy classes of cyclic subgroups in the Conway group Co 0 and their associated fixed lattices K (recall S = K ⊥ L ). The necessary Condition 4.8 says (in particular) rank(K) ≥ 4 and that K is not divisible as a lattice (i.e., K = K ′ (n) for some integral, not necessarily even, lattice K ′ and integer n ≥ 2, because in this situation rank(K) = l(A K )). Inspecting the list of [HL90] in the prime-power order case gives an easy match with the list of Theorem 4.15 (essentially, there is only one possibility for (G, S) once the order of G and the rank of S are specified). The pairs (G, S) are not saturated, but the knowledge of K (essentially, rank and discriminant) suffices to identify the relevant case in Höhn-Mason [HM16] list, and to find the saturated pair (G ′ , S) (with G ⊂ G ′ ).
Assuming that n = |G| has at least 2 prime divisors, and that K is a non-divisible lattice of rank at least 4, leaves only the following cases in [HL90]: −6 D , 6 E , −10 E , −12 H , 14 B and 15 D . As before, for each case we can associate a unique saturated Leech pair from [HM16]. Using Theorem 4.5 (our main criterion), cases −10 E and 14 B can not arise from cubic fourfolds, while the others can occur. Finally, the cases −12 H and 15 D correspond to maximal cases (i.e., rank(K) = 4, or equivalently rank(S) = 20). Considering also the cases of order 9 and 11 identified in Theorem 4.15, we obtain item (3) (compare also with Theorem 4.14).
Remark 4.19. Let us comment on the two apparent repetitions in the matching of the cases in Theorem 4.17. First, the two order 9 cases (case (8) and (9)) correspond to a unique cubic fourfold, in fact the Fermat cubic fourfold X = V (x 3 1 + · · · + x 3 6 ) ⊂ P 5 , which has Aut s (X) = 3 4 : A 6 . The fact that we list two cases of order 9 in Theorem 4.15 corresponds to the existence of two non-conjugate cyclic subgroups of order 9 in 3 4 : A 6 (induced from the two conjugacy classes of order 3 elements in A 6 ). For reference, we note (cf. [HL90, Case 9 C ]) that the fixed-point lattice K is     4 1 1 2 1 4 1 2 1 1 4 −1 2 2 −1 4     which has discriminant form 3 +2 9 +1 . The cases (4) and (5) of order 3 lead to the same Leech pair (G, S) (with K = S ⊥ Λ being the Coxeter-Todd lattice), but in this case the two (8-dimensional) families of cubics are different corresponding to the fact that S has two different primitive embeddings into the lattice Λ 0 (= A 2 ⊕ (E 8 ) 2 ⊕ U 2 ). The other order 3 case (namely (6)) is easily distinguished; it corresponds to K being E * 6 (3) which has discriminant form 3 +5 .
Remark 4.20. Let us also note that the order 6 case −6 D in fact coincides with the case 3 C . This is clear by noticing that they both correspond to case 35 in [HM16] (with saturated group 3 1+4 : 2). This also follows by inspecting [HL90]; in both cases K = E * 6 (3) (N.B. E * 6 is not an integral lattice, thus scaling by 3 does not contradict our non-divisibility assumption on K).
Remark 4.21. The order 11 case is very interesting, as 11 can not occur as a prime order for symplectic automorphisms of K3 surfaces (and thus this example can be used to construct exotic automorphisms for hyper-Kähler's of K3 [2] type; e.g. [Mon13a, §4.5]). The equation of the unique cubic with an order 11 symplectic automorphism is well known, namely x 2 ). From our perspective, this corresponds to case (11 A ) in [HL90]. The saturated Leech pair is (PSL(2, F 11 ), S) and the fixed-point lattice K is  which has discriminant form 11 +2 .
In view of Theorem 4.17, we note that the only cyclic case that needs further investigation is G ∼ = Z/6 (the prime-power cases are covered by Theorem 4.15, while the maximal cases are discussed later in §4.5).
According to Theorem 4.17, there are two order 6 cases relevant for us (6 E and −6 D ). However, the case −6 D was already covered by Theorem 4.15 (cf. Rem. 4.20). The last cyclic group case is handled by the following result.
We now consider the symmetric group S 3 , relevant to item (d2) in Theorem 1.2.
Lemma 4.24. Let X = V (F ) ⊂ P(V ) be a smooth cubic fourfold with symplectic action of G ∼ = S 3 . Then the action of G on P 5 can be lifted to a representation of G on V ∼ = C 6 , and one of the following holds: (1) The representation of G on V is the direct sum of two standard representations of S 3 . The dimension of the moduli space of cubic fourfolds F with such an action is 6.
(2) The representation of G on V is the direct sum of a standard representation, an alternating character, and two trivial characters of S 3 . The dimension of the moduli space of cubic fourfolds F with such an action is 4.

Proof.
A projective representation of S 3 can be lifted as a linear representation. Suppose we have an action of S 3 on V with an invariant smooth cubic form F ∈ Sym 3 (V * ), such that the induced action of S 3 on V (F ) is faithful and symplectic. There are three involutions in S 3 , and their actions on V must have dimensional two (−1)-eigenspace.
There are three linear irreducible representations of S 3 , namely, the trivial character, the alternating character, and the standard representation on C 3 . Since the action of an order 3 element in G is acting faithfully on V , the representation of G on V has the standard representation of S 3 as an irreducible component. It is then clear that the two cases mentioned in the lemma is all the possibilities.
Suppose V is a direct sum of two standard representation. We can choose coordinate (x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ) of V * , such that G ∼ = S 3 is acting via permutating (x 1 , x 2 ), (x 3 , x 4 ), (x 5 , x 6 ) simultaneously. A cubic form which is invariant under this action can be written uniquely as a linear combinations of 14 cubic forms which are also invariant. The centralizer group of S 3 in GL(V ) can be written as two by two matrices. This group has dimension 8. Hence the dimension of the moduli of cubic fourfolds with this action is 14 − 8 = 6. Suppose V is a direct sum of a standard representation, an alternating character, and two trivial characters. We can choose coordinate (x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ) of V * , such that G ∼ = S 3 is acting via permutating (x 1 , x 2 , x 3 ), identically on x 5 , x 6 , and alternatively on x 4 . A cubic form which is invariant under this action can be written uniquely as a linear combinations of 15 cubic forms which are also invariant. The centralizer group of S 3 in GL(V ) has dimension 11. Thus, the dimension of the moduli space of cubic fourfolds with this action is 15 − 11 = 4.

4.4.
Proof of Theorem 1.2. At this point, we can complete the proof on our classification theorem (Theorem 1.2). The main ingredients of our proof are the criterion given by Theorem 4.5, the Höhn-Mason classification [HM16] of the fixed-point sublattices in the Leech lattice L, and Fu's classification discussed above (Theorem 4.15). Nikulin's criterion for the existence of even lattices with specified discriminant form (Theorem A.8) is a well-known tool that we use repeatedly.
Höhn and Mason [HM16] list all possibilities (290 in total) for saturated Leech pairs (G, S). Condition 4.8 allows us to rapidly remove a large number of cases (e.g., about half of the cases have rank(S) ≥ 21). We analyze the remaining cases one by one using Theorem 4.5 (our main criterion) and Nikulin's theory. The most delicate case, rank(K) = 4, was analyzed in detail in Theorem 4.14. The cases when rank(K) ≥ 5 are similar and in fact easier. Namely, as K becomes larger, it is easier to embed E 6 into K ⊕ U 2 (in particular, note that except rank(K) = 4, (E 6 ) ⊥ K⊕U 2 is indefinite, i.e., the "easy" case of Nikulin's theory). By a routine inspection (we only need to compare the rank of K and l p (A K )) of the list of Höhn-Mason, we see that there are 43 cases (among them, there are 12, 12, 5, 5, 2, 3, 2, 1, 1 cases with rank(K) = 5, 6, 7, 8, 9, 10, 12, 16, 24 respectively) in Höhn-Mason list with rank(K) ≥ 5 and satisfying Condition 4.8. Out of these 43 potential cases with rank(K) ≥ 5, only 28 of them satisfy the equivalent conditions in our main criterion Theorem 4.5. We omit the details. Including the 6 cases of maximal rank, we obtain the list of 34 possibilities for (G, S) ∈ A cub . We list them in Theorem 1.2 in decreasing order of dimension of moduli M (G,S) (or equivalently by rank(S)). (Note however that M (G,S) is not necessarily irreducible. When possible, we list also the irreducible components of M (G,S) .) The second part of Theorem 1.2 is to give explicit equations for some of the cases. As discussed above, Theorem 4.15, Lemma 4.22, Lemma 4.23 and Lemma 4.24 give normal equations for cubic fourfolds X which admit faithful actions by some special group G (either cyclic of prime-power order, Z/6, Klein group or S 3 respectively). Starting with this classification, we proceed in two ways. First, we have the saturation procedure: given a normal form F stabilized by such a G, we obtain a stratum F ⊂ M which corresponds to some Leech pair (G ′ , S) ∈ A cub (i.e., in the list of the previous paragraph). It holds G ⊂ G ′ = Aut s (X) for some generic X in F . Typically, using the information on order of G (note ord(G ′ ) is a multiple of ord(G)) and dim F (= 20 − rank(S)) suffices to identify the pair (G ′ , S). As an illustration of this saturation procedure see item (5) case D 10 in Theorem 1.2.
A second way to proceed is to start with (G, S) ∈ A cub , and consider elements of prime-power order g ∈ G (say ord(g) = p k ). By Theorem 4.15, we know the possible normal form(s) F of X with an action by g (similar arguments apply to Z/2 × Z/2 ⊂ G or S 3 ⊂ G). We then try to specialize F so that it admits an action by G ⊃ g (e.g., see proof of Lemma 4.24). Again, the knowledge of the dimension of F (from the normal form) and that of M (G,S) ⊂ F (Hodge theoretically, as S is a prescribed lattice of algebraic cycles) proved very handy in practice.
Concretely, for G = 1, 2, 3 or 4, we can directly apply the second method (G = g ) and (0), (1), (2b), (3a) are clear. For (2a), we can apply the second method for G = 2 2 and use Lemma 4.23. For (3b), we can apply the second method for G = S 3 and use Lemma 4.24. For (5a), we can apply the second method for G = D 12 and use Lemma 4.22. Then applying the first method we see that a generic cubic fourfold described in Lemma 4.22 has symplectic automorphism group D 12 . For items (5b), (7b) and (7c) of Theorem 1.2, we apply a combination of the two methods.
The last case left is (7a). From Harada-Lang [HL90] (case (3 C )), there is a Leech pair (G, S) with G = Z/3Z and K = E * 6 (3). From Höhn-Mason classification, the only saturated Leech sub-pair of (Co 0 , L) with discriminant 3 5 is (3 1+4 : 2, S). Thus this is the saturation of (G, S). By Theorem 4.5, there exist cubic fourfolds with certain order 3 automorphism such that the induced Leech pair is (G, S). The moduli of such cubic fourfold has dimension 2. These cubic fourfolds must be given by case (6) in Theorem 4.15. Using the first method described above, any cubic fourfold with such an order 3 automorphism has automatically symplectic automorphism group 3 1+4 : 2. We conclude case (7a).
4.5. Uniqueness in maximal case. As discussed in the previous subsection, we are able to identify explicit equations for a number of cases in Theorem 1.2. The cases that are more difficult are those with large, nonabelian group. One further complication that can arise is the fact that M (G,S) might not be irreducible. We discuss in detail this situation for the maximal rank case, rank(S) = 20 or equivalently dim M (G,S) = 0. In Theorem 4.14 we have identified six cases for such pairs (G, S). On the other hand, Höhn-Mason [HM14, page 48] have listed for each of these cases a cubic fourfold in M (G,S) . It turns out, that in two of the six cases, there is an additional point in M (G,S) . This is the new content of our Theorem 1.8. Our arguments are lattice theoretic; we do not have explicit equations for these cubic fourfolds with large automorphism groups. We start with two lemmas: Lemma 4.25. For Leech pairs (G, S) with numbers 1, 4, 5, 10, 11, or 13 in Table ( Proof. Direct inspection of Table 9 in [HM14]. From the reduction theory of lattices (e.g., see [CS99, Chap. 15, §3.2]), we have: Lemma 4.26. Every positive rank 2 lattice admits a basis, such that the corresponding intersection matrix Proof of Theorem 1.8. The issue that we need to investigate is the uniqueness of the primitive embedding S ֒→ Λ 0 (where Λ 0 ∼ = A 2 ⊕ (E 8 ) 2 ⊕ U 2 is the primitive cohomology of the cubic fourfold). We let T = S ⊥ Λ0 be the transcendental lattice. The maximal rank case is very special, as T is in fact a negative definite lattice of rank 24 (in all other cases, T is indefinite, the easy case of Nikulin's theory). We now analyze case by case, the six cases of the Theorem 1.8, corresponding to items 1, 4, 5, 10, 11, or 13 in Table (1). (1) For 3 4 : A 6 , the lattice T has discriminant form 3 +1 9 +1 . Using Lemma 4.26, we see that the negative rank 2 even lattices with discriminant 27 are −(2 1 14) and −(6 3 6). Only −(6 3 6) has discriminant form 3 +1 9 +1 . Hence T = −(6 3 6) is unique. A saturation S ⊕ T ֒→ Λ 0 is given by an injective morphism −q T ֒→ q S . Every two such morphisms differ by an automorphism of q S , which is induced by an automorphism of S (from Lemma 4.25). Thus all primitive embeddings of S into Λ 0 with orthogonal complement T give the same primitive sublattice (up to automorphisms of Λ 0 ). Therefore, this case recovers a unique smooth cubic fourfold, which must be the Fermat cubic fourfold V (x 3 1 + x 3 2 + x 3 3 + x 3 4 + x 3 5 + x 3 6 ).

Symplectic automorphisms for low degree K3 surfaces
In this section we will discuss the case of K3 surfaces. As we have indicated, the classification of symplectic automorphisms for K3 surfaces was first systematically investigated by Nikulin [Nik79a] via lattice theory, and culminated in the celebrated result by Mukai [Muk88] on a characterization of maximal finite symplectic groups of K3 surfaces via Mathieu group M 23 . Kondō [Kon98] simplified Mukai's proof by embedding the covariant lattice S into a Niemeier lattice (an approach closely related to ours). Xiao [Xia96] gave the complete list of finite symplectic automorphism groups of K3 surfaces by analyzing the combinatorial structures of the singularities of the quotient surface. Hashimoto [Has12] extended Kondō's lattice theoretic approach to give the complete list and analyze the possibilities of geometric realizations.
We briefly discuss here the case of symplectic automorphisms for low degree polarized K3 surfaces, along the lines of our analysis for cubic fourfolds. Our method is lattice theoretic and relies on the Höhn-Mason [HM16] classification. On the other hand, low degree K3 surfaces have projective models. For those K3 surfaces, one can study the automorphisms of the projective model via geometric methods; some partial results exist in the literature (e.g. [Har14], [DIK00], [MPK16]). Our discussion here only matches some of the maximal cases. A further analysis of the interplay between geometry and arithmetic would be interesting. 5.1. General discussion. As in the cubic fourfold case, the main point of our analysis is that for a K3 surface Y with a faithful symplectic action of a finite group G, one gets a Leech pair (G, S G (Y )(−1)) (see Lemma 4.2). The task now is to identify those that occur for Y a polarized K3 surface with a given degree. Similarly to our main criterion (Theorem 4.5) for cubic fourfolds, we obtain the following criterion for Leech pairs to arise from low degree K3 surfaces. Our arguments apply essentially verbatim as in the proof of Theorem 4.5 for the cases when there exists a Borcherds polarization on Y (see §2.3) which is a root lattice. As already discussed, this is the case for degree 2 and 6. It is also true for the degree 4 case (e.g., [LO16, Sect. 1]). Finally, it also applies to elliptic K3 surfaces. By abuse of notation, we call an elliptic K3 surface a degree 0 K3 surface, and we insist that the polarized symplectic automorphisms preserve the class of the fiber and of the section (i.e., the natural U polarization for elliptic K3 surfaces is point-wise fixed by the automorphism).
Theorem 5.1. Given a Leech pair (G, S). Let d ∈ {0, 2, 4, 6} and R d be the root lattice E 8 , E 7 , D 7 , or E 6 ⊕ A 1 for d = 0, 2, 4, 6 respectively. The following three statements are equivalent: (i) there exists a smooth degree d K3 surface S with a symplectic action G which preserves the polarization, such that (G, S) ∼ = (G, S G (X)), (ii) there exists an action of G on L with S = S G (L) and K = L G , such that there exists a primitive embedding of R d into K ⊕ U 2 , (iii) there exists an embedding of S ⊕ R d into the Borcherds lattice B, such that S has primitive image.

26
The maximal rank for S G (Y )(−1) in the K3 case is 19 (or equivalently the orthogonal complement K in the Leech lattice L has rank 5). From Höhn-Mason classification, we identify the following 11 maximal cases in Table (2) 7 ; they correspond precisely to the 11 maximal cases of Mukai. It is interesting to note that all 11 cases have projective models of degree at most 8 (see [Muk88,Example 0.4]). 168 L 2 (7) 4 +1 1 7 +2 4 8 120 Notation 5.2. The notation of the finite groups appearing in Table (2) follows Mukai's appendix to [Kon98] (N.B. there are some small typos in loc. cit.: the group A 4,4 has order 288, instead of 384). For reader's convenience, we recall that the group M 20 is isomorphic to 2 4 : A 5 , the group M 9 is isomorphic to 3 2 : Q 8 , the group T 48 is isomorphic to L 2 (3). The operator * is the central product. Concretely, the group Q 8 * Q 8 is the quotient of Q 8 × Q 8 by the diagonal of corresponding to the center of Q 8 , and it is isomorphic to an extraspecial group 2 1+4 .
Below, we discuss the maximal rank cases for degree 2 and 6 K3 surfaces as those are connected to cubic fourfolds (as discussed, they correspond to "fake cubics", i.e., the Hassett divisors C 2 and C 6 ). The cases of degree 4 K3 surfaces and elliptic K3 surfaces are equally interesting, but less relevant to the core analysis in this paper. We point out however the classification on projective automorphisms of quartic K3 surfaces in [MPK16], and the work [Gar13] on automorphisms of elliptic K3 surfaces. 5.2. The degree 2 K3 case. The maximal symplectic cases for degree 2 K3 surfaces (analogue to Thm. 4.14 for cubics) are listed below.
Proof. Given (G, S, K) from Table (2). By Theorem 5.1, we need to check whether there exists embedding of S ⊕ E 7 into B, such that the image of S is primitive. We have that q E7 = −q A1 = 2 +1 7 . For numbers 1, 2, 4, 5, 6, the lattice S ⊕ E 7 has no nontrivial saturation in which S is primitive, and l 2 (S ⊕ E 7 ) ≥ 3. For number 10, we have l 3 (S ⊕ E 7 ) = l 3 (K) = 3. Therefore, in these cases, there are no embedding of S ⊕ E 7 into B such that the image of S is primitive. We next check for the other cases one by one.
(5) For number 11 in Table (2), the discriminant form of S ⊕ E 7 is 2 +1 1 8 −2 II 3 −1 ⊕ 2 +1 7 , and there is uniquely a nontrivial saturation S of S ⊕ E 7 in which S is primitive. The discriminant form of S is 8 −2 II 3 −1 . There exists a negative rank 2 even lattice T with discriminant form 8 −2 II 3 +1 . Thus there exists a primitive embedding of S into B with orthogonal complement T . The claim follows.
We discuss the geometric realizations for those maximal symplectic groups. The double cover of P 2 branched along a sextic curve is a degree 2 K3. If a group acts on a plane sextic curve, it also acts on the corresponding degree 2 K3 surface. A classification of automorphism groups of plane sextic curves can be deduced from [Har14, Thm. 2.1]. It is discovered by Wiman [Wim96] that the sextic curve 3 ) has an action by A 6 . The corresponding degree 2 K3 surface also admits an action by A 6 , which must be symplectic since A 6 is simple. In [DIK00] the uniqueness of such a sextic curve (with action by A 6 ) is proved.
Another smooth plane sextic with large symmetry (see Remark 2.4 in [Har14]) is (5.3) V (x 6 1 + x 6 2 + x 6 3 − 10(x 3 1 x 3 2 + x 3 2 x 3 3 + x 3 3 x 3 1 )) which has automorphism group equal to the Hessian group H 216 of order 216 (this group can be represented as the affine special linear group ASL(2, F 3 ), or as the projective unitary group PU(3, F 2 )). Actually the degree 2 K3 surface corresponding to this sextic curve has symplectic automorphism group equal to M 9 ∼ = PSU(3, Finally, the group T 48 is realized by the double cover of P 2 with branch curve 5.3. The degree 6 K3 case. The maximal cases in the degree 6 case are listed below. Theorem 5.4. For a degree 6 K3 surface Y with a symplectic action of a finite group G. Suppose rank(S G (Y )) = 19, then (G, S G (Y )) is one of numbers 3, 8, 10 in Table (2). In particular, the group G can be A 6 (see (5.5)), S 5 (see (5.6)), or N 72 (see (5.7)).
Proof. Given (G, S, K) in Table (2). By Theorem 5.1, we need to check whether there exists embedding of S ⊕ E 6 ⊕ A 1 into B, such that the image of S is primitive. We have that q E6⊕A1 = −q A2 ⊕ q A1 = 2 +1 1 3 +1 . For cases with numbers 1, 2, 4, 5, 6, 11, we have l 2 ( S) ≥ 3 for any saturation S of S ⊕ E 6 ⊕ A 1 in which S is primitive. For case with number 9, we have l 3 ( S) ≥ 3 for any saturation S of S ⊕ E 6 ⊕ A 1 . Therefore, for 28 those cases we can not embed S ⊕ E 6 ⊕ A 1 into B with image of S primitive. We consider the other cases one by one.

5.4.
Uniqueness for K3 surfaces. While we don't investigate the uniqueness question here (i.e., analogues of Theorem 1.8), we point out that Hashimoto [Has12,Main Theorem] proved that for three of Mukai's maximal cases (specifically (3), (7), and (8), corresponding to groups A 6 , L 2 (7), and S 5 ) there are exactly two primitive sublattices (up to conjugate) of Λ K3 (−1) isomorphic to S (where, as before, S is the covariant lattice). Each of these cases has at least one realization for either a degree 2 or 6 K3 surface (see (5.5), (5.2), and (5.6) below). As Hashimoto works in the unpolarized case, the moduli of K3 surfaces with symplectic automorphism groups in the above three cases has two connected component, both of dimension 1. The group A 6 is of special interest since it occurs for degree 2 and degree 6 cases (see (5.1) and (5.5)). Interestingly, the two cases are in two different components.
Proposition 5.5. The embeddings of S into Λ K3 (−1) given by the two geometric realizations (5.1) and (5.5) (degree 2 and degree 6) have different orthogonal complements. In particular, these two K3 surfaces belong to different connected components of the moduli space of K3 surfaces with symplectic automorphism group A 6 .
Proof. Let Y 1 , Y 2 be the degree 2 and degree 6 K3 surfaces with A 6 symplectic action respectively. Then the orthogonal complement of S ∼ = S A6 (Y 1 ) ֒→ H 2 (Y 1 , Z)(−1) contains a vector with self-intersection −2, while the orthogonal complement of S ∼ = S A6 (Y 2 ) ֒→ H 2 (Y 2 , Z)(−1) contains a vector with self-intersection −6. (Note that in our conventions we are scaling the cohomology by −1, making the polarization a negative vector. Furthermore, in these maximal cases, S ⊥ is negative definite of rank 3.) On the other hand, from Hashimoto [Has12, Table 10.3, item 79], the orthogonal complement S ⊥ of an embedding of S into Λ K3 (−1) can be either  which contains (−2)-vector but does not contain any (−6)-vector, or  which contains (−6)-vector but does not contain any (−2)-vector. The claim follows.

5.5.
A geometric relation to cubic fourfolds. Notice that the maximal symplectic automorphism groups for degree 2 (see Theorem 5.3) and degree 6 (see Theorem 5.4) also appear in case rank(S) = 19 in Theorem 1.2. This is not a coincidence. The following proposition explains the geometry behind this phenomena.
Proposition 5.6. Let (G, S) be a Leech pair satisfying conditions in Theorem 5.1 for degree d = 2 or 6, then (G, S) is one of the 34 Leech pairs we obtain in our main Theorem 1.2. Especially, the corresponding moduli of cubic fourfolds has dimension one more than that of the degree 2 or 6 K3 surfaces.
Proof. Let (G, S) be a Leech pair such that there is an embedding of Notice that in both situations we have a natural embedding E 6 ֒→ R d . Thus we have an embedding S ⊕ E 6 ֒→ S ⊕ R d ֒→ B with the image of S in B primitive. Therefore, the Leech pair (G, S) arises from symplectic actions of G on certain smooth cubic fourfolds. The dimension of the moduli space of such cubic fourfolds is 20 − rank(S), while the dimension of degree d K3 surfaces with the corresponding symplectic action by G is 19 − rank(S).
Remark 5.7. The above proposition tells that if we have a family of fake cubic fourfolds with symplectic action by a finite group G, then we can smooth the fake cubic fourfolds to smooth ones, preserving the action of G. What we obtain is a family (of one more dimension) of cubic fourfolds with symplectic action of G such that the generic fibers are smooth.
Let us briefly discuss the geometry behind Proposition 5.6 (and Remark 5.7). For simplicity, we restrict to the case of nodal cubic fourfolds (parametrized by the Hassett divisor C 6 ). A singular cubic fourfold can be written as (5.8) X 0 = V (f 2 (x 1 , . . . , x 5 )x 6 + f 3 (x 1 , . . . , x 5 )) ⊂ P 5 for some homogeneous polynomials f 2 , f 3 of degree 2 and 3 respectively. Note that the equation above singles out the singular point p = (0, . . . , 1) ∈ X. The linear projection from p π : X 0 P 4 is a birational equivalence. The inverse map π −1 : P 4 X 0 has indeterminacy locus the degree 6 K3 surface More precisely, assuming Y is smooth, X 0 has a unique singular point p which is either of type is singular), and it holds This establishes a Hodge correspondence (essentially an identification) between the Hodge structure on H 4 (X 0 ) (still pure) and H 2 (Y )(−1). In terms of automorphism, note that since the polarized automorphisms of Y are induced from projective transformations, i.e., G = Aut(Y ) pol ⊂ PGL(5), G acts by automorphisms on X 0 . The group G preserves the quadric V (f 2 ) ⊂ P 5 and its strict transform E in X 0 = Bl Y P 4 . But then E is precisely the exceptional divisor of X 0 = Bl p X 0 → X 0 . We conclude that G acts on X 0 by automorphisms preserving the singular point p.
6. Some remarks on the full automorphism groups for smooth cubic fourfolds In this section we discuss about automorphisms and automorphism groups of smooth cubic fourfolds in general (i.e. without the symplectic assumption). We first discuss some general structure results in §6.1 (the same arguments apply to K3 surfaces or hyper-Kähler manifolds). In §6.2, we obtain some estimate on "how non-symplectic" the automorphism group of a cubic fourfold can be. Finally, in §6.3, we give some arithmetic conditions for smooth cubic fourfolds to admit non-symplectic automorphisms of order 2, 3 or 4, and then use this to find the full automorphism groups for smooth cubic fourfolds with rank(S) = 20.
6.1. Basic structures of the full automorphism groups. Let X be a smooth cubic fourfold, and G = Aut(X) the automorphism group. The induced action of G on H 3,1 (X) gives a character χ : G −→ C × , with kernel the symplectic automorphism group G s = Ker(χ). The image of χ is a cyclic group which we denoted by G. We have the following short exact sequence of finite groups: As before, the symplectic part G s ⊂ Aut(X) induces a Leech pair (G s , S). Denote by T (X) ⊂ H 4 (X, Z) the transcendental lattice of X. Note T (X) ⊂ H 4 (X, Z) Gs prim = S ⊥ Λ0 . The induced action of the full automorphism group G on H 4 (X, Z) (or H 4 (X, Z) prim ) preserves the algebraic and transcendental lattices. Since G s acts trivially on T (X), the action of G on T (X) factors through an action of G on T (X). Clearly, the action of G preserves the Hodge structure on T (X), and in particular it preserves the subspace H 3,1 ∼ = C ⊂ T (X). Choosing σ a generator of H 3,1 (i.e., σ is the class of a (3, 1) form on X), we see that G acts on σ by roots of unity, i.e., if ξ ∈ G is a generator then ξ.σ = ζσ for some root of unity ζ( = 1) ∈ U (1) ⊂ C * . We then note: Lemma 6.1. The induced action of G on T (X) is faithful and has no non-zero fixed vectors.
Proof. Suppose not faithful, then there exists g ∈ G \ G s such that the induced action of g on H 4 (X, Z) leaves T (X) invariant. But this implies that g fixes H 3,1 (X), which is a contradiction to the assumption g / ∈ G s . Suppose there is a non-zero vector v ∈ T (X) fixed by G. Then, denoting as above by ξ a generator of G, and σ a generator of H 3,1 , we have which forces σ, v = 0. Thus, v ∈ H 2,2 ∩ H 4 (X, Z), a contradiction. (Alternatively, the Hodge structure on T (X) is irreducible. The fixed locus of G is a sub-Hodge structure, and thus can only be trivial.) Denote by n the order of G (i.e., G ∼ = Z/n). Standard algebra leads to the following: Corrolary 6.2. We have ϕ(n) rank(T (X)). Here ϕ is the Euler function.
Proof. Let ξ be a generator of G, and ζ a primitive n-root of unity such that ξ.σ = ζσ for σ ∈ H 3,1 (X). The arguments of the previous lemma, easily give that all the eigenvalues of ξ on T (X) are primitive n-roots of unity. The characteristic polynomial p ξ of ξ as an isomorphism of T (X) is rational. It follows that p ξ is a power of the cyclotomic polynomial. The claim follows.
6.2. Order of the non-symplectic part. The list of smooth cubic fourfolds with prime order automorphism is known. Specifically, according to [GAL11, Theorem 3.8] there are 13 irreducible families 8 of cubics with a prime order automorphism. In particular, Proposition 6.3. A prime factor of the order of the automorphism group of a smooth cubic fourfold can only be 2, 3, 5, 7, or 11. A non-symplectic prime-order automorphism of a smooth cubic fourfold can have order 2 or 3.
Proof. The list of prime orders is a consequence of [GAL11, Theorem 3.8]. The second part follows by noticing that 7 of the 13 cases were already identified in Theorem 4.15 as the symplectic cases (see also Remark 4.16). The symplectic cases cover all the cases involving the primes 5, 7, and 11. The claim follows. By Proposition 6.3, the order of G has only prime factors 2 or 3. Thus, we can write n(= |G|) = 2 k 3 l . From Corollary 6.2 and the fact T (X) ⊂ S ⊥ Λ0 we get: (6.1) ϕ(n) = ϕ(2 k 3 l ) ≤ 22 − rank(S).
As mentioned the induced action of G on H 4 (X, Z) preserves the algebraic and transcendental lattices. In fact G preserves also the covariant lattice S(= S Gs (X)).
Lemma 6.4. The induced action of G on H 4 (X, Z) leaves S stable.
Proof. The subgroup G s is normal in G = Aut(X). Thus for any g ∈ G, gG s g −1 = G s . By definition, S is the orthogonal complement of the invariant lattice Λ Gs . Clearly G s = gG s g −1 leaves every vector in gΛ Gs invariant. It follows that gΛ Gs = Λ Gs . By taking orthogonal complements, we get that g leaves S stable.
The action of G on S induces a homomorphism π : G −→ Aut(q S ). Since G s acts trivially on q S , the homomorphism π descends to a morphism π : G −→ Aut(q S ).
Proof. Suppose g ∈ G \ G s acts trivially on q S . Thus, the action of g on S is by isometries preserving the discriminant. As previously discussed, any such isometry of S can be lifted to a symplectic automorphism of X. Thus, there exists h ∈ G s , such that the restrictions of g and h to S are the same. Replacing g by gh −1 , we can assume (wlog) that g acts trivially on S.
Replacing g by a power g k , we can further assume that g has prime order. By Proposition 6.3, we can assume that g is either of order 2 or 3.
By the classification in [GAL11] and the discussion in [YZ18,§6], there are two conjugacy classes of nonsymplectic involutions, with corresponding moduli spaces arithmetic quotient of type IV domains having dimensions 10 and 14 (N.B. the 14 dimensional case is discussed in detail in [LPZ18]). In particular, the invariant sublattice of Λ 0 (which contains S) is of rank 12 or 8 respectively, contradicting rank(S) ≥ 13. The order 3 case is similar. Namely, there are 4 conjugacy classes of of non-symplectic order three automorphisms, with corresponding moduli spaces arithmetic ball quotients of dimensions 4, 6, 7 and 10 (N.B. the 10dimensional case is [ACT11]). Again, the automorphism g can not leave a sublattice of rank at least 13 of Λ 0 invariant, a contradiction. 8 The case F 2 5 in [GAL11, Theorem 3.8] should be excluded, as the corresponding family contains only singular cubic fourfolds. This was pointed out in [BCS16].
The proposition above is very useful in the cases where S is of large rank, or equivalently G s is relatively large; this is the case of interest in this paper. In fact, note that most of the cases in Theorem 1.2 satisfy rank(S) ≥ 13. It would be interesting to classify the possible orders n = 2 k 3 l of non-symplectic automorphisms on a cubic fourfold, especially we do not know what is the largest possible such n (compare (6.1)). These cases will have essentially trivial symplectic automorphism group, thus they should be handled by different methods.
Remark 6.6. A major difference between the lattice theoretic methods in the symplectic and anti-symplectic cases is that the covariant lattice N for an anti-symplectic automorphism contains the transcendental lattice T (X), and thus (except the case rank(T (X)) = 2) N is indefinite (in particular, O(N ) is typically infinite).
6.3. Maximal cases. We conclude our discussion of the automorphism groups of cubic fourfolds, with a discussion of the full automorphism group for the 8 maximal cases (with respect to symplectic automorphisms) identified in Theorem 1.8. These are the most interesting cases from the perspective of this paper, and they are particularly suitable to classification (compare Prop. 6.4 and Rem. 6.6, and note rank(S) = 20, rank(T ) = 2).
Since we assume rank(S) = 20, the transcendental lattice T (X) is the orthogonal complement of S(−1) in H 4 (X, Z) prim and has rank 2. From Equation (6.1) we get that the possible orders for the non-symplectic part G are n = 2, 3, 4, or 6. We discuss first the case of anti-symplectic involutions.
We have the following necessary condition for a smooth cubic fourfold with maximal symplectic symmetry to admit an anti-symplectic involution.
Proposition 6.8. Let X be a smooth cubic fourfold with rank(S) = 20. Suppose there exists an antisymplectic involution on X, then the composition of S ⊕ E 6 ֒→ H 4 0 (X, Z) ⊕ E 6 ֒→ B is not primitive.
Proof. By Lemma 6.4, the induced involution ι * on H 4 0 (X, Z) preserves S = S Gs (X). Since ι * equals to −id on the orthogonal complement of S in H 4 (X, Z), the invariant sublattice M = H 4 0 (X, Z) ι * of H 4 0 (X, Z) is contained in S. Suppose j : S ⊕ E 6 ֒→ B is primitive, then the inclusion j : M ⊕ E 6 ֒→ B is also primitive.
On the other hand, the involution ι * on H 4 0 (X, Z) extends to an involution on B, with restriction to E 6 trivial. The invariant sublattice of B under the action of ι * is M ⊕ E 6 . This is a contradiction, because the invariant sublattice (in a unimodular lattice) of an involution has 2-group as its discriminant group, while |A E6 | = 3.
In particular, this allows us to distinguish the two cases of Theorem 1.8(2) with symplectic automorphism group A 7 . Namely, we note that cubic fourfold with A 7 automorphisms identified by Höhn-Mason has an extra symplectic involution, while the other can not have.
Similarly, we get: Corrolary 6.10. The cubic fourfold X 2 (A 7 ), and those with symplectic automorphism groups G s = L 2 (11) and M 10 , have no anti-symplectic involution (equivalently, order of G is odd).
We now switch our attention to the case of anti-symplectic involutions of order 3 and 4. The main point here is that in these cases T (X) has a decomposition into two conjugate eigenspaces, and in fact it acquires the structure of a (Hermitian) lattice over the Eisenstein Z[ω] or respectively Gaussian Z[i] integers. This fact is the starting point of multiple works by Kondō (e.g. [DK07]) and Allcock-Carlson-Toledo (e.g. [ACT11]). In our situation, T (X) is of rank 2, and thus of rank 1 as Eisenstein/Gaussian lattice. This allows us to obtain the following simple criterion for |G| to be a multiple of 3 or 4.
Lemma 6.11. Let T be a positive definite rank 2 even lattice. Then T admits an automorphism of order 3 if and only if then there exists a positive integer a such that T ∼ = A 2 (a), and T admits an automorphism of order 4 if and only if there exists a positive integer a such that T ∼ = A 2 1 (2a).
Proof. By Theorem 1.8, we know the transcendental lattices of the 8 cubic fourfolds. By Lemma 6.11, we identify the cubic fourfolds which have order 3 or 4 non-symplectic automorphisms. Combining with Corollary 6.10 we conclude the proposition.
Corrolary 6.14. The maximal possible order for automorphism groups of smooth cubic fourfolds is 174, 960, which is reached only by the Fermat cubic fourfold.
Proof. The order of automorphism group G for a smooth cubic fourfold is given by the product of |G s | and n = |G|. The value of n is bounded by (6.1). The claim follows by a straightforward inspection of Theorem 1.2, Theorem 1.8, and Proposition 6.12.
B.3. Extraspecial group. For p prime, recall that a p-group is a finite group with order a power of p.
Definition B.2. An extraspecial group is a non-abelian p-group G with center Z(G) ∼ = p and the quotient G/Z(G) elementary abelian.
Every extraspecial group has order p 1+2k with k a positive integer. Conversely, for any prime number p and positive integer k, there exist two extraspecial groups of order p 1+2k . By convention, the symbol p 1+2k represent for an extraspecial group of order p 1+2k . For p = 2 and k = 1, the two extraspecial groups 2 1+2 are the dihedral group D 8 and quaternion group Q 8 . B.4. Linear and projective groups over finite fields. Linear and projective groups over a field K refer to Zariski-closed subgroups of GL(n, K) or PGL(n, K). When K is a finite field, these groups are finite and play an important role in the classification of finite simple groups. In the final section we collect such kinds of groups related to our classifications.
We introduce the unitary groups over finite fields. For a finite group F q 2 where q = p r and p is a prime number, there is an F q -linear involution α : F q 2 −→ F q 2 sending x to x q (this is the r-th power of the Frobenius automorphism of F q ). Let V be an n dimensional vector space over F q 2 , then there is a unique F q -bilinear form (called Hermitian form over finite field) H : V × V −→ F q 2 satisfying H(w, v) = α(H(v, w)) and H(v, cw) = cH(v, w) for any c ∈ F q 2 . Explicitly, The unitary group 9 U (n, q) represents for the automorphism group of the Hermitian space (V, H). We note that the projective special unitary group PSU(3, F 2 ) is isomorphic to the Mathieu group M 9 , and appears as symplectic automorphism group of a degree 2 K3 surface. The group PSL(2, F 11 ) is simple and appears as the automorphism group of the Klein cubic threefold V (x 2 1 x 2 + x 2 2 x 3 + x 2 3 x 4 + x 2 4 x 5 + x 2 5 x 1 ) (see [Adl78]). As shown in Theorem 1.8, there is a unique cubic fourfold with an order 11 automorphism which is a triple cover of P 4 branched along the Klein cubic threefold.