In mathematics, specifically complex analysis, Riemann's existence theorem says, in modern formulation, that the category of compact

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...

s is equivalent to the category of complex

complete algebraic curve In algebraic geometry, a complete algebraic curve is an algebraic curve that is complete variety, complete as an algebraic variety. A projective curve, a dimension-one projective variety, is a complete curve. A complete curve (over an algebraicall ...

s. Sometimes, the theorem also refers to a generalization (a theorem of Grauert–Remmert), which says that the category of finite topological coverings of a complex

algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, ...

is equivalent to the category of finite étale coverings of the variety.

Original statement

Let ''X'' be a compact Riemann surface,

p_1, \cdots, p_s

distinct points in ''X'' and

a_1, \cdots, a_s

complex numbers. Then there is a

meromorphic function In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are ''poles'' of the function. ...

f

on ''X'' such that

f(p_i) = a_i

for each ''i''.

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 The étale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces. Topological analogue/informal discussion In algebraic topology, the fundamental group \pi_1(X,x) of ...

of ''X'' at a geometric point ''x'' is the projective limit :

\pi_1^(X, x) = \varprojlim \operatorname_X(Y)

over all finite Galois coverings

Y

X

. By the existence theorem, we have

\operatorname_X(Y) = \operatorname_(Y^).

Hence,

\pi_1^(X, x)

is exactly the profinite completion of the usual topological

fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...

\pi_1(X^, x)

of ''X'' at ''x''.

References

Works

* Harbater, David.
Riemann’s existence theorem
" The Legacy of Bernhard Riemann After 150 (2015) (ed. by L. Ji, F. Oort, S.-T. Yau), Beijing-Boston: Higher Education Press and International Press, ISBN 978-1571463180 * Ryan Patrick Catullo
Riemann Existence Theorem

slide
for the paper. * * M. Artin, A. Grothendieck, J.-L. Verdier, SGA 4, Théorie des topos et cohomologie étale des schémas, 1963–1964, Tomes 1 à 3, Avec la participation de N. Bourbaki, P. Deligne, B. Saint-Donat, version : c46c8b4 2018-12-20 13:39:00 +0100 * *Remmert, Reinhold (1998), From Riemann surfaces to complex spaces, France, Paris: S´emin. Congr., 3, Soc. Math *J. S. Milne (2008).
Lectures on Étale Cohomology
'

Original statement

Proof

Consequences

See also

References

Works

Further reading