Moduli spaces of symmetric cubic fourfolds and locally symmetric varieties

Yu, Chenglong; Zheng, Zhiwei

doi:10.2140/ant.2020.14.2647

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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

DOI: 10.2140/ant.2020.14.2647

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.

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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

Vol. 14, No. 10, 2020