Compact Riemann Surfaces (MAGIC006)
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The purpose of the course is to present enough material on compact Riemann surfaces for students to be able to read literature where ideas such as meromorphic differentials, Abel's map and the Jacobi variety, divisor classes and divisor line bundles are used. Compact Riemann surfaces are also the simplest examples of Kaehler manifolds, and every complete smooth algebraic curve is a compact Riemann surface, so they provide an entry into complex manifold theory as well as algebraic geometry. While sheaf theory provides an elegant way of treating many of the topics covered, it will not be explicitly invoked but we will take an approach (and use notation) which is in the spirit of analytic sheaf theory.
Spring 2016 (Monday, January 11 to Friday, March 18)
The approach we take will rely on a good grounding in complex analysis and a little point set topology. Some experience of the differential geometry of surfaces will be helpful.
Riemann surface as a complex manifold (motivated by multi-valued functions); vector fields and differential forms; basics of integration and singular homology for curves on surfaces; the Abel-Jacobi map and Abel's theorem; the Riemann-Roch theorem; (maybe get as far as Weierstrass points).
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Assessment will be by a single take home examination during the Spring assessment period. The exam will have three questions, of which you will be asked to attempt two. To obtain a pass will require the equivalent of one complete correct answer to one question.
MAGIC006 exam 2016
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