Riemann surfaces
Mastermath course in spring 2016
Place and time
Wednesdays, 10.15-13.00 SP C1.112
Aim
This course gives an introduction into the theory of Riemann surfaces, both from an analytical point of view as well as from an algebraic point of view.
Description
In this course we will approach the theory of Riemann surfaces from a complex geometric angle, using complex function theory as the basis. Topics we will discuss include, sheaves and their cohomologies, differential forms and residues. After that we come to the major results in the field, the Riemann-Roch theorem and Serre duality. We will also discuss covering spaces and the Riemann-Hurwitz formula. As an application of these results, we make contact with the algebraic point of view on Riemann surfaces (algebraic curves). If time permits, we will also treat the Jacobian and the Abel-Jacobi map.
Organization
Two hour lecture + 1 hour exercise class.
Examination
Written Exam, homework counts as 20%.
Literature
O. Forster: Lectures on Riemann surfaces. Graduate Texts in Mathematics, 81. Springer-Verlag, New York-Berlin, Further literature will be given during the course.
Prerequisites
Complex analysis, topology. Please make sure you understand the following topics from the algebraic topology: fundamental group, covering spaces, Deck transformations and the main classification theorem of covering spaces using the fundamental group. (see e.g. the book "Algebraic Topology" by Hatcher, sections 1.1 & 1.3)
Week by week schedule
- Lecture 1: Introduction, definition of a Riemann surface, holomorphic and meromorphic functions. (1.1 of the book). Exercises: 1.2, 1.3, 1.4, 1.5.
- Lecture 2: Elementary properties of holomorphic maps, branched coverings. (1.2, 1.4 ) Exercises: 2.1, 2.5, 4.4, 4.5
- Lecture 3 : Galois and universal coverings, Sheaves, Analytic continuation (1.5,1.6 and 1.7) Exercises:
5.3, 5.4, 5.5, 5.7, 6.1, 6.2, 6.3 Homework for next week (March 2): 4.4. & 5.3 Remark: in 4.4.c) arctan_k(0)=k\pi.
- Lecture 4 : (analytic?) continuation of analytic continuation, algebraic functions (1.7 & 1.8) Exercises: 7.1, 7.2, 8.1, 8.2.
- Lecture 5 : Algebraic functions continued, differential forms. (8 & 9) Exercises: 9.1, 9.2, 9.3.
- Lecture 6 : Residues and integration of forms. (10) Exercises: 10.1, 10.2, 10.3, 10.4. Homework for next week (March 23): Extra exercises
- Lecture 7 : Residues of meromorphic 1-forms (finally!) Sheaf cohomology (12) Exercises: 12.1, 12.2, 12.3
- Lecture 8 : Sheaf cohomology, long exact sequence & Dolbeault's Lemma (12, 13 &15) Exercises: 13.1, 13.2
- Lecture 9 : Proof of Dolbeault's theorem, Divisors (13, 15, 16) Exercises: 15.1, 15.2, 15.3, 15.4. Homework for next week (March 23): Extra exercises 2
- Lecture 10 : The Rieman-Roch theorem (16) Exercises: 16.1, 16.2, 16.3, 16.4
- Lecture 11 : Serre duality, the Riemann--Hurwitz theorem & hyperelliptic Riemann surfaces (17) Exercises: 17.1, 17.2, 17.3, 17.4
- Lecture 12 : hyperelliptic Riemann surfaces & embedding into projective space (17) Exercises: 17.5, 17.6, 17.7.
- Lecture 13 : embedding into projective space (proof), harmonic forms Exercises: remaining exercises of last week. Homework for next week (March 18): Extra exercises 3
- Lecture 14 : harmonic forms continued, the Hodge-de Rham theorem (19), the Jacobian and Abel's theorem (20&21) Exercises: 19.1, 19.2.
- Lecture 15 : Test exam Disclaimer: question 3d & 4e are difficult/a lot of work and are above exam-level!