Editing Riemann surface (section)

== Analytic vs. algebraic ==
The existence of non-constant meromorphic functions can be used to show that any compact Riemann surface is a [[projective variety]], i.e. can be given by [[polynomial]] equations inside a [[projective space]]. Actually, it can be shown that every compact Riemann surface can be [[immersion (mathematics)|embedded]] into [[complex projective space|complex projective 3-space]]. This is a surprising theorem: Riemann surfaces are given by locally patching charts. If one global condition, namely compactness, is added, the surface is necessarily algebraic. This feature of Riemann surfaces allows one to study them with either the means of [[analytic geometry|analytic]] or [[algebraic geometry]]. The corresponding statement for higher-dimensional objects is false, i.e. there are compact complex 2-manifolds which are not algebraic. On the other hand, every projective complex manifold is necessarily algebraic, see [[Algebraic geometry and analytic geometry#Chow.27s theorem|Chow's theorem]].

As an example, consider the torus ''T'' := {{nowrap|'''C''' / ('''Z''' + ''τ'''''Z''')}}. The [[Weierstrass elliptic function|Weierstrass function]] ℘<sub>''τ''</sub>(''z'') belonging to the lattice {{nowrap|'''Z''' + ''τ'''''Z'''}} is a [[meromorphic function]] on ''T''. This function and its derivative ℘<sub>''τ''</sub>′(''z'') [[Generating set|generate]] the function field of&nbsp;''T''. There is an equation
: <math>[\wp'(z)]^2=4[\wp(z)]^3-g_2\wp(z)-g_3,</math>
where the coefficients ''g''<sub>2</sub> and ''g''<sub>3</sub> depend on ''τ'', thus giving an elliptic curve ''E''<sub>''τ''</sub> in the sense of algebraic geometry. Reversing this is accomplished by the [[j-invariant|''j''-invariant]] ''j''(''E''), which can be used to determine ''τ'' and hence a torus.