More precisely, the category of coherent sheaves on the algebraic variety X is equivalent to the category of analytic coherent sheaves on the analytic variety Xan, and the equivalence is given on objects by mapping
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.
Consider the Laurent expansion at all such z and subtract off the singular part: we are left with a function on the Riemann sphere with values in C, which by Liouville's theorem is constant.
This fact shows there is no essential difference between the complex projective line as an algebraic variety, or as the Riemann sphere.
Under the name Riemann's existence theorem[4][5][6][7] 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.
A precise principle and its proof are due to Alfred Tarski and are based in mathematical logic.
It states that an analytic subspace of complex projective space that is closed (in the ordinary topological sense) is an algebraic subvariety.
This allows quite a free use of complex-analytic methods within the classical parts of algebraic geometry.
The major paper consolidating the theory was Géometrie Algébrique et Géométrie Analytique by Jean-Pierre Serre,[13] now usually referred to as GAGA.
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.
In slightly lesser generality, the GAGA theorem asserts that the category of coherent algebraic sheaves on a complex projective variety X and the category of coherent analytic sheaves on the corresponding analytic space Xan are equivalent.
The analytic space Xan is obtained roughly by pulling back to X the complex structure from Cn through the coordinate charts.
Indeed, phrasing the theorem in this manner is closer in spirit to Serre's paper, seeing how the full scheme-theoretic language that the above formal statement uses heavily had not yet been invented by the time of GAGA's publication.