Duke Mathematical Journal

Compactifications defined by arrangements, II: Locally symmetric varieties of type IV

Eduard Looijenga

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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.

Article information

Duke Math. J., Volume 119, Number 3 (2003), 527-588.

First available in Project Euclid: 23 April 2004

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14J15: Moduli, classification: analytic theory; relations with modular forms [See also 32G13]
Secondary: 32S22: Relations with arrangements of hyperplanes [See also 52C35]


Looijenga, Eduard. Compactifications defined by arrangements, II: Locally symmetric varieties of type IV. Duke Math. J. 119 (2003), no. 3, 527--588. doi:10.1215/S0012-7094-03-11933-X. https://projecteuclid.org/euclid.dmj/1082744772

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See also

  • See also: Eduard Looijenga. Compactifications defined by arrangements, I: The ball quotient case. Duke Math. J. Vol. 118, No. 1 (2003), pp. 151-187.