Draft: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 algebraic curves.

Sometimes, the theorem also refers to a generalization, 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.

References

• Harbater, David. "Riemann’s existence theorem." The Legacy of Bernhard Riemann After 150 (2015)
• Ryan Patrick Catullo, Riemann Existence Theorem. A slide for the paper.
• SGA 1 and 4
• 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.

Further reading

• https://mathoverflow.net/questions/80770/reference-request-riemanns-existence-theorem
• https://mathoverflow.net/questions/40791/finite-covers-of-complex-varieties-all-but-two-questions-answered
• https://ncatlab.org/nlab/show/Riemann+existence+theorem