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.