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.
"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