Topics on Abelian Varieties

An episode of GAGLeS organized by Raymond Cheng during the spring of 2018.

The first half of this seminar aims to develop the basic theory of abelian varieties, mainly following Mumford’s beautiful book mentioned below. The second half of this seminar is a scattering of topics around the geometric theory of abelian varieties.


Abelian varieties are heavily studied objects and there are many great references for the subject. Some notable general references on abelian varieties include:

  1. David Mumford, Abelian Varieties. This book is a beautiful classic. Mumford treats abelian varieties first from a complex analytic point of view, before moving onto an old-style variety-theoretic manner, before finally dealing with the modern scheme-theoretic language.

  2. Christina Birkenhake and Herbert Lange, Complex abelian varieties. If there is something about complex analytic abelian varieties you would like to know, this book probably contains it. In particular, this book is a rich source on theta functions.

  3. Bas Edixhoven, Gerard van der Geer, and Ben Moonen, Abelian Varieties. This is book-in-progress which, along with the basic theory of abelian varieties, touches on many interesting arithmetic topics. See also Ben Moonen’s research page for (possibly) updated veresions of chapters of this book.

  4. James S. Milne, Abelian Varieties. Notes on abelian varieties by Milne. I don’t think any more has to be said.

  5. Brian Conrad with notes by Tony Feng, Math 249C: Abelian Varieties. These are live-\(\TeX\)ed notes for a topics course on abelian varieties taught by Brian Conrad at Stanford in 2015.

  6. Bhargav Bhatt with notes by Matt Stevenson, Math 731: Topics in Algebraic Geometry I – Abelian Varieties, These are notes from a topics course on abelian varieties taught by Bhargav Bhatt at the University of Michigan during the Fall of 2017. Beyond the basic theory of abelian varieties, these notes discuss derived properties of abelian varieties, such as the Fourier–Mukai equivalence between the derived category of an abelain variety and its dual.


We meet Wednesdays in Mathematics Room 507 between 2:30PM and 4:00PM.

Raymond Cheng
Introduction and Organization
Semon Rezchnikov
Line bundles on abelian varieties: Theorem of the Cube and applications
Song Yu
More line bundles and projectivity of abelian varieties
Raymond Cheng
Duality of abelian varieties
Dmitrii Pirozhkov
Derived categories of abelian varieties
Carl Lian
Cohomology of line bundles on abelian varieties
Monica Marinescu
Jacobian varieties
Raymond Cheng
The Theta Divisor
No Seminar
Spring Recess
No Seminar
Raymond in Italy
Raymond Cheng
Cubic Threefolds, their Intermediate Jacobians, and Prym Varieties
Carl Lian
Kuga—Satake Construction
Dmitrii Pirozhkov
Weil Conjectures for K3 Surfaces