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.

中文翻译：

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对称三次四重和局部对称变体的模空间

在本文中，我们实现了具有指定自同构群的三次四重的模空间作为复双曲球或 IV 型对称域的算术商，并研究了它们的紧化。我们的结果主要依赖于关于三次四重模空间的著名著作，包括由 Voisin ([Voi86]) 证明的全局 Torelli 定理和由 Laza ([Laz09, Laz10]) 和 Looijenga ([Loo09]) 独立。我们研究紧化的关键输入是 Looijenga 紧化的函数性，我们在附录（A 部分）中对其进行了表述。附录也可用于研究奇异 K3 曲面和三次四重曲面的模空间，这将在后续论文中出现。