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 GrauertâÂÂRemmert), 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 X be a compact Riemann surface, distinct points in X and complex numbers. Then there is a meromorphic function on X such that for each i.
A standard proof of the theorem uses the theory of divisors and meromorphic functions on compact Riemann surfaces. Given distinct points in and prescribed complex values , one considers the divisor and applies the RiemannâÂÂRoch theorem to show that the space of meromorphic functions with divisor at least is nontrivial. This yields a meromorphic function on satisfying for each . A detailed proof can be found in SGA 1, Exposé XII, Théorème 5.1, or in SGA 4, Exposé XI, ç4.3.
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
over all finite Galois coverings of . By the existence theorem, we have Hence, is exactly the profinite completion of the usual topological fundamental group of X at x.