Compactifying moduli spaces for abelian varieties
type: Item
instance of: scholarly article
title: Compactifying moduli spaces for abelian varieties
zbMATH Open document ID: 1165.14004
DOI: 10.1007/978-3-540-70519-2
published in: Lecture Notes in Mathematics
publication date: 2008-07-31T00:00:00Z
review text: In the past five decades, beginning with the precise definition of the concept of moduli problems in algebraic geometry introduced by David Mumford in the early 1960s, both the construction of moduli spaces for certain classes of algebro-geometric objects and the study of their geometric properties have played a central role in algebraic geometry, arithmetic geometry, complex-analytic geometry, and increasingly also in mathematical physics. In attempting to study any established moduli space \(M\), one of the basic first steps is to find a suitable compactificatfon \(\overline M\) of \(M\). Preferably, such a compactification \(\overline M\) should have reasonably good geometric properties, its boundary \(\overline M\setminus M\) should be a manageable divisor in \(\overline M\), and the space \(\overline M\) should also have a reasonable moduli interpretation, with boundary points corresponding to degenerate objects. In this general context, a prominent example is provided by the well-known Deligne-Mumford compactification \(\overline M_{g,n}\) of the moduli space \(M_{g,n}\) of smooth projective curves of genus \(g\) with \(n\) marked points. The analoguous problem of compactifying the moduli space \(A_g\) of principally polarized abelian varieties of dimension \(g\) has a just as long and rich history, but the situation in this important special case has turned out to be much more difficult than for moduli spaces of curves. In fact, there are natural compactifications of \(A_g\), namely the so-called toroidal compactifications constructed by D. Mumford and his coworkers [cf.: \textit{A. Ash}, \textit{D. Mumford}, \textit{M. Rapoport} and \textit{Y. S. Tai}, Smooth compactification of locally symmetric varieties. Lie Groups: History, Frontiers and Applications. Vol. IV. Brookline, Mass.: Math Sci Press. IV (1975; Zbl 0334.14007)] over the ground field \(\mathbb{C}\) the variant proposed by \textit{Y. Namikawa} [Toroidal compactification of Siegel spaces. Lecture N…
reviewed by: Werner Kleinert
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 14Kxx
Mathematics Subject Classification ID: 14K10
Mathematics Subject Classification ID: 14K25
Mathematics Subject Classification ID: 14A20
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 14F10
zbMATH DE Number: 5306366
zbMATH Keywords: research exposition (algebraic geometry)
zbMATH Keywords: abelian varities
zbMATH Keywords: moduli spaces
zbMATH Keywords: compactification
zbMATH Keywords: algebraic stacks
zbMATH Keywords: logarithmic geometry
zbMATH Keywords: toric varieties
author: Martin Olsson
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-540-70519-2
OpenAlex ID: W588234827
description: scientific article; zbMATH DE number 5306366