Introduction to Compact Riemann Surfaces

Algebraic Surfaces and 4-manifolds: Some Conjectures and Speculations

Introduction. Since the time of Riemann, there has been a close interplay between the study of the geometry of complex algebraic curves (or, equivalently, compact Riemann surfaces) and the topology of 2-manifolds. These connections arise from the uniformization theorem, which asserts that every simply connected Riemann surface is conformally equivalent to either the Riemann sphere, the plane, o...

Compact Riemann Surfaces: a Threefold Categorical Equivalence

We define and prove basic properties of Riemann surfaces, which we follow with a discussion of divisors and an elementary proof of the RiemannRoch theorem for compact Riemann surfaces. The Riemann-Roch theorem is used to prove the existence of a holomorphic embedding from any compact Riemann surface into n-dimensional complex projective space Pn. Using comparison principles such as Chow’s theor...

91 10 02 v 1 2 N ov 1 99 9 Introduction to the functions on compact Riemann surfaces and Theta - functions

Klein Surfaces and Real Algebraic Function Fields by Norman Alling and Newcomb Greenleaf

Introduction. The correspondence between compact Riemann surfaces and function fields in one variable over C is well known and has been widely exploited, both in analysis and in algebraic geometry. A similar correspondence exists between function fields in one variable over R and compact "Klein surfaces," which are more general than Riemann surfaces in two respects: they need not be orientable ...

Isometry Groups of Compact Riemann Surfaces

We explore the structure of compact Riemann surfaces by studying their isometry groups. First we give two constructions due to Accola [1] showing that for all g ≥ 2, there are Riemann surfaces of genus g that admit isometry groups of at least some minimal size. Then we prove a theorem of Hurwitz giving an upper bound on the size of any isometry group acting on any Riemann surface of genus g ≥ 2...