Compact Riemann surface

Compact Riemann surface

In mathematics, a compact Riemann surface is a complex manifold of dimension one that is a compact space. Riemann surfaces are generally classified first into the compact (those that are closed manifolds) and the open (the rest, which from the point of view of complex analysis are very different, being for example Stein manifolds).

A compact Riemann surface C that is a connected space is an algebraic curve defined over the complex number field. More precisely, the meromorphic functions on C make up the function field F on the corresponding curve; F is a field extension of the complex numbers of transcendence degree equal to 1. It can in fact be generated by two functions f and g. This is a structural result on the meromorphic functions: there are enough in the sense of separating out the points of C, and any two are algebraically dependent. These facts were known in the nineteenth century (see GAGA for more in this direction).

A general compact Riemann surface is therefore a finite disjoint union of complex (non-singular) algebraic curves.


Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Riemann surface — For the Riemann surface of a subring of a field, see Zariski–Riemann space. Riemann surface for the function ƒ(z) = √z. The two horizontal axes represent the real and imaginary parts of z, while the vertical axis represents the real… …   Wikipedia

  • Riemann sphere — The Riemann sphere can be visualized as the complex number plane wrapped around a sphere (by some form of stereographic projection – details are given below). In mathematics, the Riemann sphere (or extended complex plane), named after the 19th… …   Wikipedia

  • Riemann–Roch theorem — In mathematics, specifically in complex analysis and algebraic geometry, the Riemann–Roch theorem is an important tool in the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles. It relates… …   Wikipedia

  • Surface de riemann — Pour les articles homonymes, voir Surface (homoymie). En géométrie différentielle, une surface de Riemann est une variété différentielle analytique complexe de dimension 1. Par oubli de structure, une surface de Riemann se présente comme une… …   Wikipédia en Français

  • Surface — This article discusses surfaces from the point of view of topology. For other uses, see Differential geometry of surfaces, algebraic surface, and Surface (disambiguation). An open surface with X , Y , and Z contours shown. In mathematics,… …   Wikipedia

  • Surface de Riemann — Pour les articles homonymes, voir Surface. En géométrie différentielle et géométrie analytique complexe, une surface de Riemann est une variété complexe de dimension 1. Par oubli de structure, une surface de Riemann se présente comme une variété… …   Wikipédia en Français

  • RIEMANN (B.) — Après la mort de Georg Friedrich Bernhard Riemann, son œuvre fut publiée en un seul volume, y compris les fragments posthumes, et cette brièveté ne tient pas seulement à la fin précoce du mathématicien: d’une part, ses démonstrations sont très… …   Encyclopédie Universelle

  • Grothendieck–Hirzebruch–Riemann–Roch theorem — In mathematics, specifically in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is itself a… …   Wikipedia

  • Hurwitz surface — In Riemann surface theory and hyperbolic geometry, a Hurwitz surface, named after Adolf Hurwitz, is a compact Riemann surface with precisely :84( g − 1) automorphisms, where g is the genus of the surface. This number is maximal by virtue of… …   Wikipedia

  • Algebraic surface — In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface is therefore of complex dimension two (as a complex manifold, when it is non singular)… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”