15 September 2003 Compactifications defined by arrangements, II: Locally symmetric varieties of type IV
Eduard Looijenga
Duke Math. J. 119(3): 527-588 (15 September 2003). DOI: 10.1215/S0012-7094-03-11933-X


We define a new class of completions of locally symmetric varieties of type IV which interpolates between the Baily-Borel compactification and Mumford's toric compactifications. An arithmetic arrangement in a locally symmetric variety of type IV determines such a completion canonically. This completion admits a natural contraction that leaves the complement of the arrangement untouched. The resulting completion of the arrangement complement is very much like a Baily-Borel compactification: it is the Proj of an algebra of meromorphic automorphic forms. When that complement has a moduli-space interpretation, then what we get is often a compactification obtained by means of geometric invariant theory. We illustrate this with several examples: moduli spaces of polarized K3 and Enriques surfaces and the semiuniversal deformation of a triangle singularity.

We also discuss the question of when a type IV arrangement is definable by an automorphic form.


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Eduard Looijenga. "Compactifications defined by arrangements, II: Locally symmetric varieties of type IV." Duke Math. J. 119 (3) 527 - 588, 15 September 2003. https://doi.org/10.1215/S0012-7094-03-11933-X


Published: 15 September 2003
First available in Project Euclid: 23 April 2004

zbMATH: 1079.14045
MathSciNet: MR2003125
Digital Object Identifier: 10.1215/S0012-7094-03-11933-X

Primary: 14J15
Secondary: 32S22

Rights: Copyright © 2003 Duke University Press


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Vol.119 • No. 3 • 15 September 2003
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