Compactifications defined by arrangements. II: Locally symmetric varieties of type IV
type: Item
instance of: scholarly article
title: Compactifications defined by arrangements. II: Locally symmetric varieties of type IV
zbMATH Open document ID: 1079.14045
DOI: 10.1215/S0012-7094-03-11933-X
author: Eduard Looijenga
published in: Duke Mathematical Journal
publication date: 2004-05-18T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0201218
review text: Two decades separate the origins of this article from its publication, during which time the techniques it describes have acquired renewed relevance and further applications. Many examples are given here, and thus it goes well beyond (although it contains) a revision of the Nijmegen preprint of 1985 [Semi-toric partial compactifications~I, report 8520, University of Nijmegen; per bibl.] which was its first version. The main results concern the construction and description of a new class of compactifications of (open parts of) locally symmetric varieties. Here they are worked out for type~IV locally symmetric varieties, which together with the ball quotient case studied in the first part of the paper [Duke Math. J. 118, No. 1, 151--187 (2003; Zbl 1052.14036)] covers all the applications found so far. The compactifications are defined in terms of arithmetic arrangements \({\mathcal H}\) of hyperplane sections of the symmetric domain \({\mathbb D}\): that is, those that are locally finite and invariant under some arithmetic group \(\Gamma\). Thus \({\mathcal H}\) defines a hypersurface \(D\subset X=\Gamma\backslash{\mathbb D}\). But the closure of \(D\) in the Baily-Borel compactification \(X^{\text{bb}}\) of \(X\) will not in general be \({\mathbb Q}\)-Cartier. In particular it is not, in general, defined by an automorphic form. Therefore one should choose a compactification adapted to the pair \((X,D)\), i.e. to \(X^\circ=X\setminus D\), in as natural a way as possible. Such a compactification \(\widehat{X^\circ}\) of \(X^\circ\) is what is constructed here. To construct \(\widehat{X^\circ}\) one first blows up \(X^{\text{bb}}\) along the boundary so as to make each irreducible component of the closure of \(D\) be a Cartier divisor (or at least \({\mathbb Q}\)-Cartier). To do so both economically and constructively one makes use of combinatorial data supplied by \(D\). This part of the procedure is a special case of semitoric partial compactifications, described orig…
Mathematics Subject Classification ID: 14J15
Mathematics Subject Classification ID: 32N15
Mathematics Subject Classification ID: 14J27
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 32S22
zbMATH DE Number: 2065200
zbMATH Keywords: locally symmetric varieties
zbMATH Keywords: compactifications
zbMATH Keywords: modular forms
zbMATH Keywords: \(K3\) surfaces
reviewed by: G. K. Sankaran
MaRDI profile type: Publication
arXiv ID: math/0201218
cites work: Q4099744
cites work: Compactification of arithmetic quotients of bounded symmetric domains
cites work: Monstrous moonshine and monstrous Lie superalgebras
cites work: The moduli space of Enriques surfaces and the fake monster Lie superalgebra
cites work: Automorphic forms with singularities on Grassmannians
cites work: Families of K3 surfaces
cites work: Q5580322
cites work: Borcherds products on \(O(2,l)\) and Chern classes of Heegner divisors
cites work: Local Borcherds products
cites work: Q4219187
cites work: AUTOMORPHIC FORMS AND LORENTZIAN KAC–MOODY ALGEBRAS PART I
cites work: On classification of Lorentzian Kac-Moody algebras
cites work: Invariant theory for generalized root systems
cites work: The smoothing components of a triangle singularity. II
cites work: Q3762435
cites work: Compactifications defined by arrangements. I: The ball quotient case.
cites work: Families of K-3 Surfaces
cites work: A remark on discriminants of moduli of \(K3\) surfaces as sets of zeros of automorphic forms
cites work: Projective Models of K - 3 Surfaces
cites work: A complete moduli space for K3 surfaces of degree 2
cites work: Degenerations of K3 Surfaces of Degree 4
cites work: Projective degenerations of Enriques' surfaces
cites work: Compactifications of the period space of Enriques surfaces. I
cites work: Compactifications of the period space of Enriques surfaces. II
cites work: Lattices and \(K3\) surfaces of degree 6
description: scientific article