Riemann's existence theorem

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

Proof

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

\pi _{1}^{{\acute {e}}t}(X,x)=\varprojlim \operatorname {Aut} _{X}(Y)

over all finite Galois coverings $Y$ of $X$ . By the existence theorem, we have $\operatorname {Aut} _{X}(Y)=\operatorname {Aut} _{X^{an}}(Y^{an}).$ Hence, $\pi _{1}^{{\acute {e}}t}(X,x)$ is exactly the profinite completion of the usual topological fundamental group $\pi _{1}(X^{an},x)$ of X at x.^[3]

References

^ SGA 1, Expose XII, Théorème 5.1.
^ Theorem 1.2. in Ishan Levy, Galois theory and Riemann surfaces. [1]
^ Milne, A subsection called "Varieties over $\mathbb {C}$ " after Remark 3.3.

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
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