In mathematics, specifically complex analysis, Riemann's existence theorem says, in modern formulation, that the category of compact Riemann surfaces is equivalent to the category of complex complete algebraic curves.
Sometimes, the theorem also refers to a generalization (a theorem of Grauert–Remmert),[1] which says that the category of finite topological coverings of a complex algebraic variety is equivalent to the category of finite étale coverings of the variety.
Original statement
Let X be a compact Riemann surface, distinct points in X and complex numbers. Then there is a meromorphic function on X such that for each i.[2]
Proof
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For now, see SGA 1, Expose XII, Théorème 5.1., or SGA 4, Expose XI. 4.3.
Consequences
There are a number of consequences.
By definition, if X is a complex algebraic variety, the étale fundamental group of X at a geometric point x is the projective limit
over all finite Galois coverings of . By the existence theorem, we have Hence, is exactly the profinite completion of the usual topological fundamental group of X at x.[3]
↑ Theorem 1.2. in Ishan Levy, Galois theory and Riemann surfaces.
↑ Milne, A subsection called "Varieties over " after Remark 3.3. harvnb error: no target: CITEREFMilne (help)
Works
Harbater, David. "Riemann’s existence theorem." The Legacy of Bernhard Riemann After 150 (2015) (ed. by L. Ji, F. Oort, S.-T. Yau), Beijing-Boston: Higher Education Press and International Press, ISBN 978-1571463180
M. Artin, A. Grothendieck, J.-L. Verdier, SGA 4, Théorie des topos et cohomologie étale des schémas, 1963–1964, Tomes 1 à 3, Avec la participation de N. Bourbaki, P. Deligne, B. Saint-Donat, version: c46c8b4 2018-12-20 13:39:00 +0100
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