Compactifying Moduli Spaces for Abelian Varieties

By: Martin C. Olsson

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Paperback | 25 August 2008

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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.

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"No doubt, the work presented in this research monograph is a fundamental contribution to the compactification theory of moduli spaces in general, and of moduli spaces for abelian varieties in particular. ... the present monograph is written in a very lucid, comprehensive, largely self-contained and enlightening style, including numerous additional remarks and hints." (Werner Kleinert, Zentralblatt MATH, Vol. 1165, 2009)

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Summary	p. VII
Introduction	p. 1
A Brief Primer on Algebraic Stacks	p. 7
S-groupoids	p. 7
Stacks	p. 17
Comparison of Topologies	p. 25
Coarse Moduli Spaces	p. 26
Rigidification of Stacks	p. 26
Preliminaries	p. 31
Abelian Schemes and Torsors	p. 31
Biextensions	p. 34
Logarithmic Geometry	p. 43
Summary of Alexeev's Results	p. 51
Moduli of Broken Toric Varieties	p. 57
The Basic Construction	p. 57
Automorphisms of the Standard Family over a Field	p. 67
Deformation Theory	p. 70
Algebraization	p. 75
Approximation	p. 76
Automorphisms over a General Base	p. 78
The Stack K[subscript Q]	p. 80
Moduli of Principally Polarized Abelian Varieties	p. 85
The Standard Construction	p. 85
Automorphisms over a Field	p. 93
Deformation Theory	p. 98
Isomorphisms over Artinian Local Rings	p. 110
Versal Families	p. 113
Definition of the Moduli Problem	p. 121
The Valuative Criterion for Properness	p. 121
Algebraization	p. 125
Completion of Proof of 4.6.2	p. 130
Moduli of Abelian Varieties with Higher Degree Polarizations	p. 135
Rethinking A[subscript g,d]	p. 135
The Standard Construction	p. 138
Another Interpretation of P to P	p. 142
The Theta Group	p. 144
Deformation Theory	p. 158
Isomorphisms without Log Structures	p. 160
Algebraization of Formal Log Structures	p. 163
Description of the Group H[subscript S superscript gp]	p. 166
Specialization	p. 178
Isomorphisms in J[subscript g,d]	p. 201
Rigidification	p. 202
Example: Higher Dimensional Tate Curve	p. 207
The Case g = 1	p. 225
Level Structure	p. 231
First Approach Using Kummer etale Topology	p. 21
Second Approach using the Theta Group	p. 237
Resolving Singularities of Theta Functions	p. 241
References	p. 273
Index of Terminology	p. 277
Index of Notation	p. 279
Table of Contents provided by Ingram. All Rights Reserved.

Compactifying Moduli Spaces for Abelian Varieties

At a Glance

Paperback

Industry Reviews

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How to return your order

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