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

Proof

A standard proof of the theorem uses the theory of divisors and meromorphic functions on compact Riemann surfaces. Given distinct points ${\displaystyle p_{1},\ldots ,p_{s}}$ in ${\displaystyle X}$ and prescribed complex values ${\displaystyle a_{1},\ldots ,a_{s}}$, one considers the divisor ${\displaystyle D=\sum _{i=1}^{s}p_{i}}$ and applies the Riemann–Roch theorem to show that the space of meromorphic functions with divisor at least ${\displaystyle -D}$ is nontrivial. This yields a meromorphic function ${\displaystyle f}$ on ${\displaystyle X}$ satisfying ${\displaystyle f(p_{i})=a_{i}}$ for each ${\displaystyle i}$. A detailed proof can be found in SGA 1, Exposé XII, Théorème 5.1, or in SGA 4, Exposé 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

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

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

See also

• Algebraic geometry and analytic geometry

References

1. SGA 1, Expose XII, Théorème 5.1.
2. Theorem 1.2. in Ishan Levy, Galois theory and Riemann surfaces.
3. Milne , A subsection called "Varieties over ${\displaystyle \mathbb {C} }$" 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
• 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 (NcatLab)