Instructor: Chad Schoen
Complex manifolds of dimension one; elliptic functions; the basic theorems of Riemann surface theory in the special case of one dimensional complex tori; holomorphic maps between compact Riemann surfaces; maps to projective space; linear systems; embedding criteria; sheaves; differential forms; Cech, De Rham, Dolbeault cohomology; existence of non-constant meromorphic functions on compact Riemann surfaces; Hurwitz formula; Riemann-Roch and Serre Duality; applications of the aforementioned theorems; the Hodge decomposition theorem; the Jacobian variety; the Abel-Jacobi map.
After defining Riemann surfaces we begin the study of complex tori via the theory of elliptic functions. This allows us to give elementary proofs of the main theorems in the subject in the special case of (one dimensional) complex tori. The remainder of the course is devoted to extending these results to the case of arbitrary compact connected Riemann surfaces. For this we introduce and study carefully a number of important technical tools, most prominently sheaves, differential forms and Cech cohomology. Considerable time and effort will be spent acquiring the necessary familiarity with these basic tools before applying them to prove the theorems above.
This is a standard foundational course on a topic of fundamental importance in many areas of mathematics. It supplies important background knowledge for students who intend to write a dissertation with many of the pure mathematicians at Duke. More advanced courses on complex manifolds build directly on this material. The course may be viewed as an introduction to algebraic geometry in the sense that there is a parallel theory of algebraic curves. Several crucial results about algebraic curves are much harder to prove than the analogous results about Riemann surfaces. Thus the study of Riemann surfaces is an important source of intuition in algebraic geometry. Riemann surfaces play a very important role in string theory. There are numerous connections with differential geometry (metrics of constant curvature, minimal surfaces, the Hodge theorem). Finally there are substantial connections to algebraic topology and to number theory. The latter arise in two quite separate ways, one being through the theory of elliptic functions and modular forms, the other via the analogy of number fields to fields of meromorphic functions.