# Abelian varieties and Fourier-Mukai transforms

Meets: W 13.15-15.00 in von Neumann 1.023.

Starts: 15.10.2014.

## Description

During the semester we'll develop the theories of abelian varieties and derived categories as a geometric invariant in parallel; about halfway through, the two theories will converge, and we'll see that many of the classical results on abelian varieties are more naturally expressed and proved using the derived category. This will culminate in a classification due to Orlov and Polishchuk of those abelian varieties that have isomorphic derived categories, as well as a complete understanding of the group of derived autoequivalences.

The main references we'll be using are:

• A. Polishchuk. Abelian varieties, Theta functions, and the Fourier-Mukai transform.
• D. Mumford. Abelian varieties.
• D. Huybrechts. Fourier-Mukai transforms in algebraic geometry.
The following may also prove of use:
• J. S. Milne. Abelian varieties, available at www.jmilne.org/math/.
• S. Gelfand and Y. Manin. Methods of homological algebra.
There is a seminar at Bonn this semester on the derived category (though not focusing on abelian varieties) that will be covering much of what we're doing, and might also prove a useful reference:

## Lecture Topics and References

15.10.14: Overview (Ben). Notes.
22.10.14: Triangulated categories (Nicolas). Notes.
29.10.14: Complex tori (Emre). Notes.
05.11.14: Derived categories (Giovanni). Notes.
12.11.14: Abelian varieties (Angela). Notes.
19.11.14: Derived categories of varieties (Gabriel).
26.11.14: Line bundles on abelian varieties (Irfan). Notes.
03.12.14: Exact functors (Rostislav). Notes.
10.12.14: The dual abelian variety (Niels). Notes.
17.12.14: Mukai's theorem (Ana). Notes.
07.01.15: Orlov's criterion (Gregor). Notes.
14.01.15: The derived category of an elliptic curve (Irene).
21.01.15: Derived equivalences of abelian varieties I (Daniele). Notes.
28.01.15: Derived equivalences of abelian varieties II (Ignacio). Notes.
04.02.15: Autoequivalences of abelian varieties (Andre). Notes.
11.02.15: Further topics (Ben).