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