Advanced Algebraic Geometry: Abelian Varieties [2019]

Algebraic Geometry 1 & 2 (basic knowledge of varieties, schemes, coherent sheaves and cohomology); Algebraic Topology (homology and cohomology groups); Complex Analysis (holomorphic functions). The knowledge of Riemann surfaces is also preferred.

Aim of the course
Abelian varieties form an important class of varieties; they are smooth projective varieties with a group structure. They are very important in algebraic geometry and number theory. In this course, the students will apply their knowledge and skills of algebraic geometry to the study of abelian varieties. The course consists of two parts. In the first part, we will go through the basic theory of abelian varieties from both analytic and algebraic point of views. We will cover the topics such as cohomology of line bundles, the dual abelian variety, polarization, isogenies, etc. In the second part, we will focus on moduli theory of abelian varieties, algebraically and analytically. By the end, the students are expected to have the required knowledge so that they feel comfortable working with such objects.

Bas Edixhoven (UL) and Mingmin Shen (UvA)