Math 206 - Complex Abelian Varieties

Welcome to Math 206!

Instructor: Dragos Oprea, doprea "at" math.you-know-where.edu

Lectures: MW 2:00-3:20, APM B412

Course description:

Abelian varieties form a well-studied chapter of classical and modern mathematics. They arise from the study of elliptic integrals and elliptic functions in the 19th century. The study of abelian varieties is also intimately linked with the study of curves. Since the topic is vast, we will make several choices regarding the material to be covered. In the first part of the course, we will establish basic properties of abelian varieties. As we progress, we will explore additional topics which may include theta functions, line bundles on abelian varieties, the moduli space of abelian varieties, the tautological ring. (These goals may change slightly as we go.)

Office hours: I am available for questions after lecture or by appointment.

Prerequisites:

I will make an attempt to be as self-contained as the topic permits. However, I will assume working knowledge of algebraic geometry at least at the level of Math 203, and some complex analysis.

Grading:

The course is intended for graduate students in algebraic geometry, though everybody with the right prerequisites is welcome. There are no homeworks or exams for the course.

Textbook: None.

Important dates:

First class: Monday, January 8
University Holiday: Monday, January 15
University Holiday: Monday, February 19
Last class: Wednesday, March 13

There will be no class on Wednesday, January 10.

Lecture Summaries

Lecture 1: Complex tori of dimension 1. Meromorphic functions on 1-diml tori. Theta functions.

Lecture 2: Theta functions as sections of line bundles. Systems of multiplieries. Theta functions with characteristics.

Lecture 3: Theta functions and projective embeddings of 1-dimensional tori. Higher dimensional tori. Cohomology. Hodge decomposition. Picard group.
Lecture 4: Appell-Humbert theorem. Theorem of the square. Pullbacks by multiplication maps.

Lecture 5: The Poincare bundle and the dual torus. Maps between the torus and its dual induced by line bundles on the torus. Polarizations. Principal polarizations. Periods.

Lecture 6: Dimension of the space of theta functions. Lefschetz' theorem and projective embeddings.

Lecture 7: Proof of Lefschetz's theorem. A short discussion of Kummer varieties.

Lecture 8: Analytic construction of the moduli space of abelian varieties.

Lecture 9: Weil form. Level structures. Moduli space of abelian varieties with level structure. The Hodge bundle. Siegel modular forms. Satake compactification.

Lecture 10: Cohomology of line bundles on abelian varieties. Riemann-Roch. Seesaw theorem. Cohomology of line bundles of degree 0. Mumford's line bundle. Cohomology of the Poincare bundle.
Lecture 11: Cohomology of the Poincare bundle (continued). Cohomology of nondegenerate line bundles. Index of nondegenerate line bundles.

Lecture 12: Theta groups. Interpretation of the Weil pairing via commutator in the theta group. Lagrangian subgroups. Stone von-Neumann. Schrodinger representation of the Heisenberg group via theta functions.
Lecture 13: More on theta groups and embeddings of abelian varieties in projective spaces. Endomorphisms of abelian varieties. Abelian varieties with real multiplication.

Lecture 14: Isogenies. Isogenies form an equivalence relation. Poincare reducibility theorem. Endomorphism algebra of a simple abelian variety. The Rosati involution.
Lecture 15: Positivity of the Rosati involution. Albert's classification of division algebras over number fields with positive involutions. Examples of abelian varieties with complex/quaternionic multiplication.

Lecture 16-17: Hodge bundle and the tautological ring of the moduli space. Lagrangian Grassmannians as the compact dual of the moduli space. Hirzebruch-Mumford proportionality principle. Compact subvarieties of the moduli space.