250067 VO Riemann surfaces (2018S)

Riemann surfaces were originally conceived in complex analysis in order to deal with multivalued functions. The analytic continuation of a given holomorphic function element along different paths leads in general to different branches of that function. Riemann replaced the domain of the function by a multiple-sheeted covering of the complex
plane to get a single valued function on the covering space.

Abstract Riemann surfaces are by definition connected complex one-dimensional manifolds. They are the natural domains of definitions of holomorphic functions in one variable.

First we will introduce Riemann surfaces and discuss basic properties. We develop the fundamentals of the theory of topological covering spaces including the fundamental group, the universal covering, and deck transformations. It will turn out that non-constant holomorphic maps between Riemann surfaces are covering maps, possibly with branch points.

Then we get acquainted with the language of sheaves. It proves very useful in the construction of Riemann surfaces which arise from the analytic continuation of germs of holomorphic functions. Some attention is devoted to the Riemann surfaces of algebraic functions, i.e., functions which satisfy a polynomial equation with meromorphic
coefficients.

For the further study of Riemann surfaces we need the calculus of differential forms. We also briefly discuss periods and summands of automorphy.

Another important tool for the investigation of the geometry of Riemann surfaces is Cech cohomology. We will develop the basics of this theory. We shall only need the cohomology groups of zeroth and first order. The long exact cohomology sequence will prove useful for the computation of cohomology groups. On Riemann surfaces we
prove versions of Dolbeault's and deRham's theorem.

Next we will focus on compact Riemann surfaces. We present and prove the main classical results, like the Riemann--Roch theorem, Abel's theorem, and the Jacobi inversion problem. Following Serre, all the main theorems are derived from the fact that the first cohomology group with coefficients in the sheaf of holomorphic functions is a finite dimensional complex vector space. The proof of this fact
is based on a functional-analytic result due to Schwartz. Its dimension is the genus of the Riemann surface. By means of the Serre duality theorem we will see that the genus equals the maximal number of linearly independent holomorphic one-forms on the compact Riemann surface. Eventually, it will turn out that the genus is a topological invariant. Much of this part is concerned with the existence of meromorphic functions on compact Riemann
surfaces with prescribed principal parts or divisors.

Non-compact Riemann surfaces are studied next. The function theory of non-compact Riemann surfaces has many similarities with the one
on regions in the complex plane. In contrast to compact Riemann surfaces, there are analogues of Runge's theorem, the Mittag--Leffler theorem, and the Weierstrass' theorem. The solution of the Dirichlet problem, based on Perron's method, will provide a further existence theorem. It will lead to a proof of Rado's theorem that every Riemann surface has a countable topology. We shall also prove the uniformization theorem for Riemann surfaces: any simply connected Riemann surface is isomorphic to one of three normal forms, i.e,
the Riemann sphere, the complex plane, or the unit disk. Evidently, this is a generalization of the Riemann mapping theorem. As a consequence we get the classification of Riemann surfaces: every Riemann surface is isomorphic to the quotient of one of the three normal forms by a group of Möbius transformations isomorphic to the fundamental group of the Riemann surface which acts discretely and fixed point freely.

Apart from some familiarity with basic complex analysis, general topology, and basic algebra no other prerequisites are demanded. All necessary tools will be developed when needed.