MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Nowadays the standard reference for Riemann's Existence theorem is SGA1, where the proof heavily relies on Serre's GAGA. I imagine that the theorem is much older, as its name suggests, and that its original proof is quite different. I thought it would be instructive for me to look at how this theorem was viewed in the pre-SGA era.


Where does a proof of Riemann's Existence Theorem original appear? Or better yet, where is a readable summary of it in English (or somewhat less preferably in French)?

share|cite|improve this question
You might want to state which version of the theorem you are asking about. Of course, there is a theorem in Riemann's work, which modulo the appeal to the dodgy Dirichlet's principle, is proved there. I guess the analysis bits were fixed by Weierstrass. It says that a compact Riemann surface has a non-constant meromorphic function and, therefore, is a complex algebraic curve. There is a French translation of Riemann's collected works. – Felipe Voloch Dec 10 '11 at 20:06
I meant the following: any topological cover of an algebraic variety defined over the complex numbers can be given an algebraic variety structure such that the covering map is algebraic. – James D. Taylor Dec 10 '11 at 20:10
As far as I remember, in SGA 1 Grothendieck uses the version of Grauert and Remmert, which was for normal spaces, and applies descent theory to extend it to the general case. – Angelo Dec 10 '11 at 21:38
up vote 9 down vote accepted

The original source for the modern version is

H.Grauert and R. Remmert, Komplexe Räume, Math.Ann. 136 (1958), 245–318.

I don't know of an exposition in English.

The version due to Riemann was just for algebraic curves. This is covered in many sources; my favorite is Narasimhan's book "Compact Riemann Surfaces".

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.