2002/01/22 by Eduard Looijenga, Looijenga, Eduard · 1 citation
Mathematics · Physics and Astronomy · #14J15 #32N15 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Nonlinear Waves and Solitons #math.AG #msc:14J15 #msc:32N15
paper · pdf · doi:10.48550/arxiv.math/0201218
The section on arrangements on tube domains has beeen expanded in order to make a connection with a conjecture of Gritsenko and Nikulin. Also added: a list of notation and some references. Finally some typo's corrected and a few minor changes made in notation
openalex publication_date 2002/01/22 · arxiv created 2002/04/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 semi-universal deformation of a triangle singularity. We also discuss the question when a type IV arrangement is definable by an automorphic form.