Moduli Spaces of Symmetric Cubic Fourfolds and Locally Symmetric Varieties

TL;DR

This paper describes the structure of moduli spaces of cubic fourfolds with automorphisms as arithmetic quotients of complex hyperbolic or type IV domains, and explores their compactifications using advanced geometric techniques.

Contribution

It provides a new realization of these moduli spaces as arithmetic quotients and develops a functorial approach to their compactifications, extending previous work on cubic fourfolds.

Findings

Moduli spaces are realized as arithmetic quotients of complex hyperbolic balls or type IV domains.

The study introduces a functorial framework for compactifications of these moduli spaces.

Results depend on and extend classical theorems by Voisin, Laza, and Looijenga.

Abstract

In this paper we realize the moduli spaces of cubic fourfolds with specified automorphism groups as arithmetic quotients of complex hyperbolic balls or type IV symmetric domains, and study their compactifications. Our results mainly depend on the well-known works about moduli space of cubic fourfolds, including the global Torelli theorem proved by Voisin ([Voi86]) and the characterization of the image of the period map, which is given by Laza ([Laz09, Laz10]) and Looijenga ([Loo09]) independently. The key input for our study of compactifications is the functoriality of Looijenga compactifications, which we formulate in the appendix (section A). The appendix can also be applied to study the moduli spaces of singular K3 surfaces and cubic fourfolds, which will appear in a subsequent paper.