1. Background

The subject of compact Riemann surfaces or algebraic curves has its origin going back to the work of Riemann. Its development requires ideas from analysis, PDE, differential geometry, complex geometry, algebra, and topology e.t.c. This course serves as a introductory course to more general theories of complex manifold and higher dimensional algebraic geometry.

2. Outline

In this course, we will introduce the notion of Riemann surfaces, holomorphic functions, meromorphic functions, differential forms on Riemann surfaces, maps between Riemann surfaces, Riemann-Roch Theorem, Abel-Jacobi theorem. We might discuss the existence of meromorphic functions and the uniformization theorem if time allows.

Tentative plan:

1. Formal definition of a Riemann surface: From CP^1 to real/complex manifolds.
2. Examples of Riemann surfaces: quotient, graph, affine curves, projective spaces and projective curves
3. Holomorphic and meromorphic functions
4. Rational polynomials and theta functions
5. Holomorphic maps of Riemann surfaces
6. Hurwitz formula: proof and applications
7. Differential forms, Stoke Theorem, and Residue Theorem
8. Holomorphic one forms and Abel-Jacobi map
9.  Theory of divisors
10.  Finiteness theorem and Riemann-Roch formula
11.  Applications of Riemann-Roch formula
12.  Residue map and Serre duality  
13.  Existence of meromorphic functions (from calculus of variations, harmonic functions, to constructions of meromorphic fucntions)
14.  A proof of Riemann-Roch formula
15.  Uniformisaztion Theorem
    **2022/1/5 [ in-person class] Presentation in NCKU (國立成功大學)
    **2022/1/12 [ in-person class] Presentation in NCKU (國立成功大學)

3. Evaluation

1. Attendance and Discussion 20%
2. Homework 30%
3. Final project 50%

4. Reference:

1. Rick Miranda, Algebraic Curves and Riemann Surfaces, (Graduate Studies in Msrhematics, Volume 5)
2. Simon Donaldson, Riemann Surfaces, 2011.

5. Credit: 3