Prerequisites
Algebraic Geometry 1 (the notion of varieties and how to work with them); Algebraic Geometry 2 (schemes, coherent sheaves; we will use cohomology of sheaves and the basics will be recalled); Algebraic Topology (homology and cohomology groups); Complex Analysis (holomorphic functions). Prior knowledge of Riemann surfaces/algebraic curves will be helpful.
Aim of the course
Abelian varieties are proper algebraic varieties that carry a group structure. They bear very rich structures and find various applications in algebraic geometry, arithmetic geometry, number theory, mathematical physics, and cryptography. In this course, the students will apply their knowledge and skills of algebraic geometry to the study of abelian varieties. 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, families of abelian varieties, etc. By the end, the students are expected to have the required knowledge so that they feel comfortable working with such objects.
Rules about Homework/Exam
There will be roughly seven in-class online quizzes (roughly once every two weeks). They may count 10% toward the final grade.
The final exam will be written, and the retake may be written or oral depending on the number of students. To pass the course, a minimum of 5 is needed in the final exam/retake, whichever the student prefers to take. The final grade will be the maximum of (10% of quizzes + 90% final/retake) and 100% of the final/retake.
Every week there will be a set of exercise questions released. The students are supposed to solve these during the tutorial or at home. The students are more than welcome to hand in some of these problems and get feedback as they would in the usual written homework setup. However, these sets will be ungraded. At least one question from these sets will appear in the final exam.
Lecture notes/Literature
D. Mumford, Abelian Varieties
Lecturers
Yajnaseni Dutta (Leiden) and Mingmin Shen (UvA)
- Docent: Yajnaseni Dutta
- Docent: Mingmin Shen