Vol. 14, No. 10, 2020

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Moduli spaces of symmetric cubic fourfolds and locally symmetric varieties

Chenglong Yu and Zhiwei Zheng

Vol. 14 (2020), No. 10, 2647–2683
Abstract

We realize the moduli spaces of cubic fourfolds with specified group actions as arithmetic quotients of complex hyperbolic balls or type IV symmetric domains, and study their compactifications. We prove the geometric ( GIT) compactifications are naturally isomorphic to the Hodge theoretic (Looijenga, in many cases Baily–Borel) compactifications. The key ingredients of the proof are the global Torelli theorem by Voisin, the characterization of the image of the period map given by Looijenga and Laza independently, and the functoriality of Looijenga compactifications proved in the Appendix.

Keywords
cubic fourfold, locally symmetric space, Looijenga compactification
Mathematical Subject Classification 2010
Primary: 14D23
Secondary: 14D07
Milestones
Received: 4 May 2019
Revised: 14 April 2020
Accepted: 4 June 2020
Published: 19 November 2020
Authors
Chenglong Yu
University of Pennsylvania
Philadelphia, PA
United States
Zhiwei Zheng
Max Planck Institute for Mathematics
Bonn
Germany