Algebraic varieties are locally defined as the common zero sets of polynomials and since polynomials over the complex numbers are holomorphic functions, algebraic varieties over C can be interpreted as analytic spaces. Similarly, regular morphisms between varieties are interpreted as holomorphic mappings between analytic spaces. Somewhat surprisingly, it is often possible to go the other way, to interpret analytic objects in an algebraic way.
For example, it is easy to prove that the analytic functions from the Riemann sphere to itself are precisely the polynomial functions (an extension of Liouville's theorem). For if we look at such a function f on C (as a subset of the Riemann sphere), it's an entire function. As the point at infinity must be a pole, we have f(z) < C|z|n for some natural number n and z in a neighbourhood of infinity, say |z| > R. Using Cauchy's integral formula over a circle centered at 0 and with radius r > R, we obtain an upper bound for the Taylor coefficients of f at 0: |ak| < C rn-k. By letting r go to infinity we see that ak = 0 for k > n, so f has to be a polynomial. This fact shows there is no essential difference between the complex projective line as an algebraic variety, or as the Riemann sphere.
Riemann surface theory shows that a compact Riemann surface has enough meromorphic functions on it, making it an algebraic curve. Under the name Riemann's existence theorem a deeper result on ramified coverings of a compact Riemann surface was known: such finite coverings as topological spaces are classified by permutation representations of the fundamental group of the complement of the ramification points. Since the Riemann surface property is local, such coverings are quite easily seen to be coverings in the complex-analytic sense. It is then possible to conclude that they come from covering maps of algebraic curves — that is, such coverings all come from finite extensions of the function field.
This principle allowed to carry over results obtained using analytic or topological methods for algebraic varieties over C to other algebraically closed ground fields of characteristic 0.
Nowadays the phrase GAGA-style result is used for any theorem of comparison, allowing passage between a category of objects from algebraic geometry, and their morphisms, to a well-defined subcategory of analytic geometry objects and holomorphic mappings.
1. Let a scheme of finite type over . Then there is a topological space which as a set consists of the closed points of with a continuous inclusion map . The topology on is called "complex topology" (and is not the subspace topology in general).
2. Suppose is a morphism of schemes of locally finite type over . Then there exists a continuous map such .
3. There is a sheaf on such that is a ringed space and becomes a map of ringed spaces. The space is called the "analytifiction" of and is an analytic space. For every the map defined above is a mapping of analytic spaces. Furthermore, the map maps open immersions into open immersions. If then and for every polydisc is a suitable quotient of the space of holomorphic functions on .
4. For every sheaf on (called algebraic sheaf) there is a sheaf on (called analytic sheaf) and a map of sheaves of -modules . The sheaf is defined as . The correspondence defines an exact functor from the category of sheaves over to the category of sheaves of .
The following two statements are the heart of Serre's GAGA theorem (as extended by Grothendieck, Neeman et al.)
5. If is an arbitrary morphism of schemes of finite type over and is coherent then the natural map is injective. If is proper then this map is an isomorphism. One also has isomorphisms of all higher direct image sheaves in this case.
6. Now assume that is hausdorff and compact. If are two coherent algebraic sheaves on and if is a map of sheaves of modules then there exists a unique map of sheaves of modules with . If is a coherent analytic sheaf of modules over then there exists a coherent algebraic sheaf of -modules and an isomorphism .
A Moishezon manifold M is a compact connected complex manifold such that the field of meromorphic functions on M has transcendence degree equal to the complex dimension of M. Complex algebraic varieties have this property, but the converse is not (quite) true. The converse is true in the setting of algebraic spaces. Moishezon in 1967 showed that a Moishezon manifold is a projective algebraic variety if and only if it admits a Kähler metric.