# Masters Seminar: The Analytic Theory of Abelian Varieties

## Seminar Overview

This is a seminar on abelian varieties over the complex numbers, looked at from the point of view of complex analysis. An abelian variety is perhaps the simplest compact complex manifold, being simply of the form **C**^{n}/Λ, where Λ=**Z**^{2n} is a lattice in **C**^{n}, a so-called complex torus. The case of dimension 1 is the well-known example of elliptic curves. In general, the interplay of the complex function theory on complex torii with the abstract condition that the torus is algebraic gets translated into a fascinating mixture of linear algebra and arithmetic that makes abelian varieties a fertile testing ground and collection of examples for many important phenomena in algebraic geometry, including the study of line bundles, the Hodge theory on the singular cohomology, questions of embeddings into projective spaces, groups of automorphisms, and numerous other areas of study. Thus, through the study of abelian varieties, many of the fundamental properties of algebraic varieties are illustrated in a concrete and down to earth manner.

The prerequisites for this seminar are: theory of complex functions of 1-variable (Funktiontheorie), Analysis I, II, Linear Algebra I, II, Algebra I. In addition, basic knowledge of differential forms and integration of differential forms will be needed. Some knowledge of complex functions of several variables, singular cohomology and the de Rham theorem will be helpful, but not required. Algebraic geometry will not be required.

The program will follow closely the book

**[1] Analytic theory of abelian varieties**, H.P.F. Swinnerton-Dyer,

with some additional material taken from chapter 1 of

**[2] Abelian varieties**, D. Mumford,

and some applications taken from

**[3] Curves and their Jacobians**, D. Mumford,

if time permits, We will discuss the program of lectures (by the participants) in the first meeting, on Friday, 13.10.2017. The lectures will be given by the seminar participants, in English or German as desired, following a prepared program.

## Program

Lecture 1. (20.10) Alessandro Danelon. Compact Riemann surfaces and algebraic curves. [1, §1]

Lecture 2. (27.10) Elena Bonan. Doubly periodic functions [1, §2]

Lecture 3. (03.11/10.11) Marc Levine. Background on differentiable manifolds

Lecture 4. (17.11/24.11) Francesco Bruzzesi. Functions of several complex variables [1,§3]

Lecture 5. (01.12) Alessandro Biagi. Theta functions and Riemann forms: reduction to theta functions [1, §4]

Lecture 6. (08.12) Francesco Chiatti. Consistency conditions and Riemann forms [1, §5]

Lecture 6. (15.12) Francesco Chiatti. Consistency conditions and Riemann forms (continuation)

Lecture 7. (22.12) Alessandro Biagi. Construction of theta functions

Lecture 8. (12.01.2018) Alessandro Danelon. Morphisms of abelian varieties [1,§7]

Lecture 9. (19.01) Francesco Chiatti. Duality of abelian varieties [1,§8]

Lecture 10. (26.01) Elena Bonan. Representations of End_{0}(`A`) [1,§9]

Lectures 11. (02.02) Francesco Bruzzesi. The structure of End_{0}(`A`) [1,§10]