Algebraic Geometry I: Riemann Surfaces

Course guide:

Main text: R. Narasimhan, "Compact Riemann Surfaces", Birkhauser, 1996.

Content:

This is a course on Riemann surfaces and complex projective varieties, and can be taken as a second course after complex variables. We will only assume a rigorous background in real analysis and one variable complex analysis, and cover any additional prerequisites during the lectures. The topics to be covered are a classic meeting ground of complex analysis and algebraic geometry. In the spring semester, Math 607 will study complex manifolds, analytic spaces, and complex algebraic geometry in any dimension.

After discussing the basic facts and examples of Riemann surfaces, we will proceed to more advanced topics: the Riemann surface of an algebraic function, cohomology of line bundles, divisors and the Riemann-Roch theorem, the canonical bundle and Serre duality. Our proofs will be analytic, for example Serre duality will be proved using a regularity theorem for the d-bar operator. We will show that compact Riemann surfaces are smooth algebraic curves and use this to transition to algebraic geometry in higher dimensions. Further topics include Abel's theorem and the Jacobian, theta functions, the theta divisor, and Riemann's theorem about meromorphic functions and theta. If time permits we will discuss Torelli's theorem and prove the uniformization theorem.

Homework:

I plan to distribute some homework problems during the course.

Other useful book references:

- O. Forster, "Lectures on Riemann Surfaces", Springer-Verlag 1999.

- R. Miranda, "Algebraic Curves and Riemann Surfaces", American Math. Society, 1995.

- Griffiths and Harris, "Principles of Algebraic Geometry", Wiley-Interscience, 1994.