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Mathematics, Algebraic Geometry
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.
