Cubic fourfolds with symplectic automorphisms

We determine projective equations of smooth complex cubic fourfolds with symplectic automorphisms by classifying 6-dimensional projective representations of Laza and Zheng’s 34 groups. In particular, we determine the number of irreducible components for moduli spaces of cubic fourfolds with symplectic actions by these groups. We also discuss the fields of definition of cubic fourfolds in six maximal cases.

💡 Research Summary

The paper “Cubic fourfolds with symplectic automorphisms” by Kenji Koike addresses the explicit classification of smooth complex cubic fourfolds that admit symplectic actions of finite groups. Building on the complete group-theoretic classification of possible symplectic automorphism groups carried out by Laza and Zheng (2022), which identified 34 finite groups (including six maximal cases: A₆, A₇, 3¹⁺⁴:2, 2²·22, M₁₀, and L₂(11)), the author seeks to determine concrete projective equations for the corresponding cubic fourfolds and to understand the geometry of the associated moduli spaces.

The central methodology consists of three intertwined steps:

1. Schur Cover Lifting – Any projective representation ρ: G → PGL₅(ℂ) can be lifted to a linear representation of a central extension G* (a Schur covering group) with multiplier M ≅ H²(G,ℂˣ). The author systematically identifies, for each of the 34 groups, an appropriate Schur cover (often the “triple cover” when M is non‑trivial) and restricts to linear representations where the central subgroup M acts by scalar matrices. This guarantees that the induced projective action on ℙ⁴ is faithful.

2. Character‑Table Filtering – Using GAP’s character tables, the author examines absolute traces |Tr g| of group elements. Since symplectic automorphisms of cubic fourfolds are conjugate to a short list of diagonal normal forms (orders 1–8, see Table 1), only those representations whose character values match the absolute traces of these normal forms can possibly arise. This dramatically reduces the search space.

3. Invariant Cubic Forms and Moduli Dimension – For each admissible linear representation ˜ρ: G* → GL₆(ℂ), the space V₃(˜ρ) of ˜ρ‑invariant homogeneous cubic polynomials is computed via Molien’s formula (implemented in GAP). The dimension of V₃(˜ρ) must exceed the expected moduli dimension 20 − rank S_G (where S_G is the invariant lattice orthogonal to H⁴(X,ℤ)^G). If dim V₃ ≤ 20 − rank S_G, the representation cannot yield a smooth cubic fourfold (Lemma 2.7). When the inequality is strict, a general cubic F∈V₃ defines a smooth fourfold X_F, and the author verifies that Aut⁽ˢ⁾(X_F)=G by comparing dimensions of the normalizer N_˜ρ,χ(G*) and the centralizer C_˜ρ(G*).

Through this systematic analysis, the author obtains the following key results:

• Irreducible Components of Moduli – For 22 of the 34 groups the moduli space of cubic fourfolds with a symplectic G‑action is irreducible. For the remaining 12 groups, the space splits into exactly two irreducible components. The groups with two components are precisely those listed in Theorem 1.1: 3·A₃, 3·D₁₂, A₄·2, A₅·3², A₆, S₅, A₇, M₁₀, etc. In particular, A₇ and M₁₀ are the only groups for which two non‑isomorphic cubic fourfolds occur.

• Explicit Equations – For each admissible representation the author writes down a basis of invariant cubic forms, thereby producing explicit defining equations. In many cases the coefficients lie in ℚ; for the maximal groups this is true except for M₁₀, where the defining equations require the quadratic field ℚ(√6). The two M₁₀‑fourfolds X⁺ and X⁻ are Galois conjugates over ℚ, confirming Theorem 1.2(2).

• Schur Cover Phenomena – The analysis highlights the necessity of non‑split central extensions for certain groups (e.g., the triple cover 3·A₇). In these cases the projective action cannot be lifted to a linear representation of the original group, and the invariant cubic space is obtained only after passing to the cover.

• Computational Tools – The entire classification relies heavily on GAP for group data (generators, character tables, Schur multipliers) and on custom scripts to compute Molien series, normalizers, and to test smoothness of the resulting hypersurfaces (including reduction modulo small primes).

The paper concludes by emphasizing that the explicit equations and the detailed description of the moduli components fill a gap left by the purely group‑theoretic classification of Laza‑Zheng. The results provide a concrete database for further investigations into derived categories, rationality questions, and connections with hyperkähler geometry (e.g., the Beauville–Donagi construction). Moreover, the field‑of‑definition analysis for the maximal cases suggests intriguing arithmetic phenomena, especially the appearance of the quadratic field ℚ(√6) for M₁₀, which may have implications for the study of Galois actions on the cohomology of cubic fourfolds.

Overall, Koike’s work delivers a complete, computationally verified catalogue of smooth cubic fourfolds admitting symplectic finite group actions, clarifies the structure of their moduli spaces, and opens new avenues for arithmetic and geometric exploration.