Compactifications of moduli of abelian varieties: an introduction

Summary

We survey the various approaches to compactifying moduli stacks of polarized abelian varieties. To motivate the different approaches to compactifying, we first discuss three different points of view of the moduli stacks themselves. Then we explain how each point of view leads to a different compactification. Throughout we emphasize maximal degenerations which capture much of the essence of the theory without many of the technicalities.

1. Introduction

A central theme in modern algebraic geometry is to study the degenerations of algebraic varieties, and its relationship with compactifications of moduli stacks. The standard example considered in this context is the moduli stack of genus g curves (where) and the Deligne-Mumford compactification [Deligne and Mumford 1969]. The stack has many wonderful properties:

• (1) It has a moduli interpretation as the moduli stack of stable genus g curves.

• (2) The stack is smooth.

• (3) The inclusion is a dense open immersion and is a divisor with normal crossings in.

Unfortunately the story of the compactification is not reflective of the general situation. There are very few known instances where one has a moduli stack classifying some kind of algebraic varieties and a compactification with the three properties above.

After studying moduli of curves, perhaps to next natural example to consider is the moduli stack of principally polarized abelian varieties of a fixed dimension g. Already here the story becomes much more complicated, though work of several people has led to a compactification which enjoys the following properties:

• (1) The stack is the solution to a natural moduli problem.

• (2’) The stack has only toric singularities.

• (3’) The inclusion is a dense open immersion, and the complement defines an fs-log structure (in the sense of Fontaine and Illusie [Kato 1989]) on such that is log smooth over Spec.

Our aim in this paper is to give an overview of the various approaches to compactifying, and to outline the story of the canonical compactification. In addition, we also consider higher degree polarizations.

What one considers a ‘natural’ compactification of depends to a large extent on one's view of itself.