Cubic Fourfolds with an Order-$7$ Automorphism

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.