Cubic Fourfolds with an Order-$7$ Automorphism

TL;DR

This work classifies all full automorphism groups of smooth cubic fourfolds carrying an order $7$ symmetry, identifying $F_{21}$, $L_2(7)$, and $A_7$ as the possible symplectic cores and detailing how extra automorphisms arise in specific loci. It builds explicit GIT moduli spaces for the $F_{21}$- and $L_2(7)$-action families, describes their ADE and boundary degenerations, and relates these moduli to period domains via the Yu–Zheng framework, producing tube-domain and lattice descriptions. A central achievement is proving that the period domains for the order-$7$ families are isogenous or commensurable with Hilbert modular surfaces, via a careful analysis of lattices $T$, discriminant forms, and arithmetic groups, and showing these are controlled by explicit degree-two coverings from Hilbert modular data. The results provide a comprehensive arithmetic-geometric picture, linking GIT stability, boundary singularities, automorphism-group classification, and period-domain geometry for these symmetry-enriched cubic fourfolds, with concrete equations for representative families and a clear description of boundary strata and monodromy phenomena.

Abstract

We study smooth cubic fourfolds admitting an automorphism of order $7$. It is known that the possible symplectic automorphism groups of such cubic fourfolds are precisely $F_{21}$, $\mathrm{PSL}(2,\mathbb{F}_7)$, and $A_7$. In this paper, we determine all possible full automorphism groups of smooth cubic fourfolds with an automorphism of order $7$. We also investigate the moduli spaces of cubic fourfolds whose automorphism group is either $F_{21}$ or $\mathrm{PSL}(2,\mathbb{F}_7)$, describing them both as GIT quotients and as locally symmetric varieties. In particular, we give an explicit description of the singular cubic fourfolds that appear in the boundary of the corresponding GIT quotients. For these two cases, we determine the commensurability classes of the monodromy groups by explicitly identifying certain arithmetic subgroups. As an interesting consequence, we prove that the period domain for cubic fourfolds equipped with an order-$7$ automorphism is isogenous to a Hilbert modular surface.

Theorems & Definitions (25)

• proof
• proof : Proof of Proposition \ref{['NorC7']}
• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof