[Paper Review] The geometry of Siegel modular varieties

Klaus Hulek, G. K. Sankaran|ArXiv.org|Oct 27, 1998

Algebraic Geometry and Number Theory87 references3 citations

TL;DR

This survey paper explores the geometry of Siegel modular varieties over the complex numbers, focusing on moduli of abelian varieties—particularly surfaces—through compactification, Kodaira classification, and lifting of Jacobi forms to modular forms. It provides projective models of special 3-folds and constructs degenerating families, offering a comprehensive overview of key geometric and arithmetic structures in the theory.

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.

Motivation & Objective

• To provide a comprehensive survey of the geometry of Siegel modular varieties over the complex numbers from the moduli-theoretic perspective.
• To analyze the compactification of Siegel modular varieties and classify them by Kodaira dimension.
• To investigate moduli of abelian surfaces, especially through the lifting of Jacobi forms to Siegel modular forms.
• To construct and study projective models of special Siegel modular 3-folds.
• To examine non-principally polarized abelian surfaces and degenerating families of abelian varieties.

Proposed method

• Utilizes moduli-theoretic methods to study Siegel modular varieties as parameter spaces of abelian varieties.
• Applies the theory of Jacobi forms and their lifting to modular forms to analyze the geometry of moduli spaces.
• Employs compactification techniques to extend the moduli spaces and study their boundary behavior.
• Uses projective geometry to construct explicit models of certain Siegel modular 3-folds.
• Analyzes degenerating families of abelian varieties via geometric and arithmetic methods.
• Applies classification tools from algebraic geometry, including Kodaira dimension, to compactified varieties.

Experimental results

Research questions

• RQ1 How do Siegel modular varieties compactify, and what are the geometric properties of their compactifications?
• RQ2 How can Jacobi forms be lifted to modular forms, and what does this imply for the geometry of moduli spaces of abelian surfaces?
• RQ3 What are the projective models of special Siegel modular 3-folds, and how are they constructed?
• RQ4 What role do non-principally polarized abelian surfaces play in the geometry of these moduli spaces?
• RQ5 How can degenerating families of abelian varieties be systematically constructed and analyzed?

Key findings

• The paper provides a detailed classification of compactified Siegel modular varieties by Kodaira dimension, offering insight into their birational geometry.
• The lifting of Jacobi forms to Siegel modular forms is shown to be instrumental in constructing and understanding modular forms on moduli spaces of abelian surfaces.
• Explicit projective models are constructed for certain Siegel modular 3-folds, revealing their geometric structure and embedding properties.
• The study of non-principally polarized abelian surfaces is shown to extend the scope of moduli-theoretic constructions beyond the principally polarized case.
• Degenerating families of abelian varieties are systematically constructed, with geometric and arithmetic data analyzed at the boundary of the compactified moduli space.
• The survey establishes connections between number theory (modular forms) and algebraic geometry (moduli spaces), highlighting deep interplay in the theory of Siegel modular varieties.