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

Abstract

Abstract. 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.

Moduli of double EPW-sextics

“…Our results suggest that the period map (0.0.2) may be understood via Looijenga's compactifications of hyperplane arrangements [17] i.e. M might be isomorphic to Looijenga's compactification of the complement of 3 specific hyperplanes in D.…”

“…Let L q : V ∼ −→ V ∨ be the isomorphism defined by q and (7.4.15) (See (1.3.1) for the definition of δ V .) Tables (16) and (17) list the values of R q on the monomials v i ∧ v j ∧ v k (denoted (ijk)): they give that R q maps each of A ± (U ) to itself and R q | A+(U) = Id A+(U) , R q | A−(U) = − Id A−(U) . (7.4.16)…”

The moduli of curves of genus six and K3 surfaces

“…cit. (see §8,[Lo2]). In case of moduli spaces of polarized K3 surfaces, the geometric compactification is known only in the case of degree two and four (J. Shah [Sh1], [Sh2]).…”

“…Secondly, an arithmetic quotient of a bounded symmetric domain has several compactifications, that is, Satake-Baily-Borel's, Mumford's toroidal and Looijenga's compactifications [Lo1], [Lo2]. A general theory explaining the relations between these types of compactifications and those of geometric nature (i.e.…”

Invariants of quartic plane curves as automorphic forms

“…But if the period map is not surjective, then some modifications on the automorphic side are in order and it is precisely for this purpose that I developed the geometric theory of meromorphic automorphic forms in [5] and [6]. What I want to do here is to illustrate that theory in the (other) case that logically comes after our guiding example, namely that of quartic plane curves.…”