Selected Topics in Riemann Surfaces

Course guide:

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

Content:

This is a course on Riemann surface theory from the point of view of complex geometry, and can be taken as a second course after complex variables. We will assume a rigorous background in real and one variable complex analysis, including some familiarity with point set topology and the theory covering spaces, and provide any other prerequisites during the lectures. The topics to be covered are a classic meeting ground of complex analysis and algebraic geometry, but the algebraic aspect of the theory will lie mostly in the background.

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 then discuss Abel's theorem and the Jacobian, and continue with a treatment of theta functions, the theta divisor, and Riemann's theorem about meromorphic functions and theta. Other possible topics include Torelli's theorem, the Dirichlet problem on Riemann surfaces, the uniformization theorem, and complex analytic geometry in higher dimensions.

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 Mathematical Society, 1995.

- D. Varolin, "Riemann Surfaces by Way of Complex Analytic Geometry", American Mathematical Society, 2011.

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