Riemann's existence theorem

In mathematics, specifically complex analysis, Riemann's existence theorem states that the category of compactRiemann 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.

Let ${\displaystyle X}$ be a compact Riemann surface, ${\displaystyle p_{1},\cdots ,p_{s}}$ distinct points in ${\displaystyle X}$ and ${\displaystyle a_{1},\cdots ,a_{s}}$ complex numbers. Then there is a meromorphic function${\displaystyle f}$ on ${\displaystyle X}$ such that ${\displaystyle f(p_{i})=a_{i}}$ for all ${\displaystyle 1\leq i\leq s}$.^[2]

For now, see SGA 1, Expose XII, Théorème 5.1., or SGA 4, Expose XI. 4.3.

By definition, if ${\displaystyle X}$ is a complex algebraic variety, the étale fundamental group of ${\displaystyle X}$ at a geometric point ${\displaystyle x}$ is the projective limit

${\displaystyle \pi _{1}^{\textrm {et}}(X,x)=\varprojlim \operatorname {Aut} _{X}(Y)}$

over all finite Galois coverings ${\displaystyle Y}$ of ${\displaystyle X}$. By the existence theorem, we have

${\displaystyle \operatorname {Aut} _{X}(Y)=\operatorname {Aut} _{X^{an}}(Y^{an})}$.

Hence, ${\displaystyle \pi _{1}^{\textrm {et}}(X,x)}$ is exactly the profinite completion of the usual topological fundamental group${\displaystyle \pi _{1}(X^{\textrm {an}},x)}$ of ${\displaystyle X}$ at ${\displaystyle x}$.^[3]

• Algebraic geometry and analytic geometry

• 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

• Riemann's existence theorem (Mathoverflow)
• Finite Covers of Complex Varieties (Mathoverflow)
• Riemann's existence theorem (nLab)