The Riemann-Roch theorem for compact Riemannian surfaces

Prerequisites: MAT213 Functions of a Complex Variable and MAT220 Algebra. Also, MAT243 Manifolds (which can be taken in parallel) can be an advantage. Furthermore, knowledge from the courses MAT224 Commutative Algebra and MAT321 Algebraic Geometry, which may be taken in parallel, will be used to adapt the project to suit.

Description: A Riemannian surface is a two-dimensional real manifold with a complex structure on it (which basically means that the manifold can be covered by open sets that are subsets of the complex numbers C). These are essential objects in analytic geometry and algebraic geometry over C.

One of the most important and well-known results on compact Riemannian surfaces is the Riemann-Roch theorem, and the student's project wil be to present the fundamental definitions, concepts and results on Reimannian surfaces, as well as the proof of the Riemann-Roch theorem. One idea is to use §1 and §6 (and part of §9) of chapter 1 and §12-16 of chapter 2 from the book [1] below as a starting point.

The background material for this project has much in common with the background material for the project Bundle cohomology, but this project will focus on a more specific case and result than the other project. It can be advantageous if the two projects are taken by a pair of students who want to work together on the background material.

This project is suitable for students who intend to take a master's degree in algebraic geometry and already want to immerse themselves in a bachelor's project in a topic that is central to algebraic geometry, but which is not taught in a regular course.

References:

[1] Forster, Otto; Lectures on Riemann surfaces. Graduate Texts in Mathematics, 81. Springer-Verlag, New York-Berlin, 1981.
[2] Narasimhan, Raghavan; Compact Riemann surfaces. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1992.