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), 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 {\displaystyle X} be a compact Riemann surface, p 1 , ⋯ , p s {\displaystyle p_{1},\cdots ,p_{s}} distinct points in X {\displaystyle X} and a 1 , ⋯ , a s {\displaystyle a_{1},\cdots ,a_{s}} complex numbers. Then there is a meromorphic function f {\displaystyle f} on X {\displaystyle X} such that f ( p i ) = a i {\displaystyle f(p_{i})=a_{i}} for all 1 ≤ i ≤ s {\displaystyle 1\leq i\leq s} .
Proof
For now, see SGA 1, Expose XII, Théorème 5.1., or SGA 4, Expose XI. 4.3.
Consequences By definition, if X {\displaystyle X} is a complex algebraic variety, the étale fundamental group of X {\displaystyle X} at a geometric point x {\displaystyle x} is the projective limit
π 1 et ( X , x ) = lim ← Aut X ( Y ) {\displaystyle \pi _{1}^{\textrm {et}}(X,x)=\varprojlim \operatorname {Aut} _{X}(Y)}
over all finite Galois coverings Y {\displaystyle Y} of X {\displaystyle X} . By the existence theorem, we have
Aut X ( Y ) = Aut X a n ( Y a n ) {\displaystyle \operatorname {Aut} _{X}(Y)=\operatorname {Aut} _{X^{an}}(Y^{an})} . Hence, π 1 et ( X , x ) {\displaystyle \pi _{1}^{\textrm {et}}(X,x)} is exactly the profinite completion of the usual topological fundamental group π 1 ( X an , x ) {\displaystyle \pi _{1}(X^{\textrm {an}},x)} of X {\displaystyle X} at x {\displaystyle x} .
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. Grothendieck, Alexander; Raynaud, Michèle (2003) [1971], Revêtements étales et groupe fondamental (SGA 1), Documents Mathématiques (Paris) [Mathematical Documents (Paris)], vol. 3, Paris: Société Mathématique de France, arXiv:math/0206203, Bibcode:2002math......6203G, ISBN 978-2-85629-141-2, MR 2017446 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 Danilov, V. I. (1996). "Cohomology of Algebraic Varieties". Algebraic Geometry II. Encyclopaedia of Mathematical Sciences. Vol. 35. pp. 1–125. doi:10.1007/978-3-642-60925-1_1. ISBN 978-3-642-64607-2. 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)