In mathematics, specifically complex analysis, Riemann's existence theorem states 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), SGA 1, Expose XII, Théorème 5.1. 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 be a compact Riemann surface, distinct points in and complex numbers. Then there is a meromorphic function on such that for all .Theorem 1.2. in Ishan Levy, Galois theory and Riemann surfaces.
Proof
For now, see SGA 1, Expose XII, Théorème 5.1., or SGA 4, Expose XI. 4.3.
Consequences By definition, if is a complex algebraic variety, the étale fundamental group of at a geometric point 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 at .
See also Algebraic geometry and analytic geometry
Notes
References Harbater, David. "Riemann’s existence theorem." The Legacy of Bernhard Riemann After 150 Years (2015) (ed. by L. Ji, F. Oort, S.-T. Yau), Beijing-Boston: Higher Education Press and International Press, ISBN 978-1571463180 Ryan Patrick Catullo, Riemann Existence Theorem. A slide for the paper.
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
Remmert, Reinhold (1998), From Riemann surfaces to complex spaces, France, Paris: S´emin. Congr., 3, Soc. Math J. S. Milne (2008). Lectures on Étale Cohomology
External links Riemann's existence theorem (Mathoverflow) Finite Covers of Complex Varieties (Mathoverflow) Riemann's existence theorem (nLab)