Moduli spaces of abelian surfaces: compactification, degenerations, and theta functions
type: Item
instance of: scholarly article
title: Moduli spaces of abelian surfaces: compactification, degenerations, and theta functions
zbMATH Open document ID: 0809.14035
author: Klaus Hulek
author: Steven H. Weintraub
author: Constantin P. Kahn
published in: De Gruyter Expositions in Mathematics
publication date: 1994-02-27T00:00:00Z
review text: This book deals with the problem of constructing compactifications of moduli spaces of abelian varieties. The problem has been attacked by many people and in many different ways. The first solution was given by Satake. The disadvantage of his construction is the fact that the boundary of his compactification has many singularities. Another way of attacking the problem is Mumford's very general method of toroidal compactifications of quotients of bounded symmetric spaces. A special case of this method gives Igusa's compactification. -- In this book the authors use Mumford's method to construct a compactification of the moduli space \({\mathcal A} (1,p)\) of abelian surfaces with a polarization of type \((1,p)\) where \(p\) is an odd prime or \(p=1\). The book has three main parts. Part I gives a description of the space \({\mathcal A} (1,p)\) and its compactification \({\mathcal A}^* (1,p)\). The Igusa compactification corresponds with the special case \(p=1\). -- In the chapters 1 and 2 a description of the moduli problem and of Mumford's theory of torus embeddings is given. In chapter 3 the authors give a detailed description of the toroidal compactification \({\mathcal A}^* (1,p)\) and its boundary. -- A good compactification of a moduli space of abelian varieties should have the property that boundary points correspond with degenerations of the objects parametrized by the given moduli space. In general toroidal compactifications do not have this property. The aim of part II is to associate to each boundary point of \({\mathcal A}^* (1,p)\) a degenerated abelian surface. The basic tool is Mumford's construction of degenerating abelian varieties over complete local rings. In particular the ``final example'' of Mumford's paper will be important for this correspondence. In the chapters 1 and 2 Mumford's theory and his ``final example'' are recalled. In chapter 3 a degenerated abelian surface is constructed for each point of \({\mathcal A}^* (1,1) \backslash {\mathcal…
Mathematics Subject Classification ID: 14K10
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 32J05
Mathematics Subject Classification ID: 14K25
zbMATH DE Number: 520405
zbMATH Keywords: compactifications of moduli spaces of abelian varieties
zbMATH Keywords: polarization
zbMATH Keywords: toroidal compactifications
zbMATH Keywords: degenerated abelian surface
zbMATH Keywords: Horrocks-Mumford bundle
reviewed by: Guido Van Steen
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 520405