Riemann's existence theorem

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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.

Contents

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, distinct points in X and complex numbers. Then there is a meromorphic function on X such that 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

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. [3]

See also

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 " after Remark 3.3.

Works

Further reading