Abelian Varieties WS 09/10

Dates: Tue, 10-12, IEM 09; Wed, 10-12, IEM 09. The course starts on Wednesday, October 14.

Prerequisites: Hartshorne, Algebraic Geometry, Chapters I-III.

References:

  • O. Debarre, Complex Tori and Abelian Varieties, SMF/AMS Texts and Monographs
  • G. van der Geer, B. Moonen, Abelian Varieties (Book in preparation).
  • J. Milne, Abelian varieties.
  • D. Mumford, Abelian Varieties, Oxford University Press.

Plan of the course:

  1. Introduction
    1. Elliptic curves
    2. Overview of the course
  2. Definitions and basic properties
    1. Definition and examples
    2. Rigidity
    3. Rational maps into abelian varieties
  3. Abelian varieties over the complex numbers
    1. Complex tori
    2. Line bundles on a complex torus
    3. Algebraizability of tori
  4. Group schemes
    1. Definitions
    2. Elementary properties
    3. Quotients by finite group schemes
    4. Finite group schemes over a field
  5. The theorem of the cube
    1. Cohomology and base change
    2. Proof of the theorem of the cube
    3. Abelian varieties are projective
  6. The dual abelian variety
    1. Isogenies
    2. The Picard functor
    3. The dual abelian variety
  7. Cohomology of line bundles
  8. Tate modules and p-divisible groups