The geometry of Siegel modular varieties

TL;DR

This survey explores the complex geometry, compactification, classification, and moduli of Siegel modular varieties, emphasizing abelian surfaces and their degenerations, providing a comprehensive overview of their structure and applications.

Contribution

It compiles and discusses recent advances in the understanding of Siegel modular varieties, including their compactifications, classifications, and moduli spaces, with a focus on abelian surfaces.

Findings

Classification of compactified varieties by Kodaira dimension

Construction of degenerating families of abelian varieties

Applications of Jacobi forms to moduli of abelian surfaces

Abstract

This is a survey article about Siegel modular varieties over the complex numbers. It is written mostly from the point of view of moduli of abelian varieties, especially surfaces. We cover compactification of Siegel modular varieties; classification of the compactified varieties by Kodaira dimension, etc.; moduli of abelian surfaces and especially applications of the lifting of Jacobi forms to modular forms; projective models of some special Siegel modular 3-folds; non-principally polarized abelian surfaces; and constructing degenerating families of abelian varieties.