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

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

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

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