Compactifying Moduli Spaces for Abelian Varieties (Lecture Notes in Mathematics, 1958) - Softcover

Olsson, Martin C.

Softcover

ISBN 10: 354070518X ISBN 13: 9783540705185

Publisher: Springer, 2008

Synopsis

The problem of compactifying the moduli spaceA of principally polarized g abelian varieties has a long and rich history. The majority of recent work has focusedonthe toroidal compacti?cations constructed over C by Mumford and his coworkers, and over Z by Chai and Faltings. The main drawback of these compacti?cations is that they are not canonical and do not represent any r- sonable moduli problem on the category of schemes. The starting point for this work is the realization of Alexeev and Nakamura that there is a canonical compacti?cation of the moduli space of principally polarized abelian varieties. Indeed Alexeev describes a moduli problem representable by a proper al- braic stack over Z which containsA as a dense open subset of one of its g irreducible components. In this text we explain how, using logarithmic structures in the sense of Fontaine, Illusie, and Kato, one can de?ne a moduli problem carving out the main component in Alexeev s space. We also explain how to generalize the theory to higher degree polarizations and discuss various applications to moduli spaces for abelian varieties with level structure.

About this title

From the Back Cover

This volume presents the construction of canonical modular compactifications of moduli spaces for polarized Abelian varieties (possibly with level structure), building on the earlier work of Alexeev, Nakamura, and Namikawa. This provides a different approach to compactifying these spaces than the more classical approach using toroical embeddings, which are not canonical. There are two main new contributions in this monograph: (1) The introduction of logarithmic geometry as understood by Fontaine, Illusie, and Kato to the study of degenerating Abelian varieties; and (2) the construction of canonical compactifications for moduli spaces with higher degree polarizations based on stack-theoretic techniques and a study of the theta group.

Edition details

• Publisher: Springer
• Publication date: 2008
• Language: English
• ISBN 10: 354070518X
• ISBN 13: 9783540705185
• Binding: Paperback
• Number of pages: 294