Editing Complex geometry (section)

=== Stein manifolds ===
{{Main article|Stein manifold}}

Serre's GAGA theorem asserts that projective complex analytic varieties are actually algebraic. Whilst this is not strictly true for affine varieties, there is a class of complex manifolds that act very much like affine complex algebraic varieties, called [[Stein manifold]]s. A manifold <math>X</math> is Stein if it is holomorphically convex and holomorphically separable (see the article on Stein manifolds for the technical definitions). It can be shown however that this is equivalent to <math>X</math> being a complex submanifold of <math>\mathbb{C}^n</math> for some <math>n</math>. Another way in which Stein manifolds are similar to affine complex algebraic varieties is that [[Cartan's theorems A and B]] hold for Stein manifolds.

Examples of Stein manifolds include non-compact Riemann surfaces and non-singular affine complex algebraic varieties.