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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.

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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)?

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    $\begingroup$ 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. $\endgroup$
    – Felipe Voloch
    Commented Dec 10, 2011 at 20:06
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    $\begingroup$ 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. $\endgroup$
    – James D. Taylor
    Commented Dec 10, 2011 at 20:10
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    $\begingroup$ 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. $\endgroup$
    – Angelo
    Commented Dec 10, 2011 at 21:38

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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".

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