Algebraic geometry and analytic geometry

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In mathematics, algebraic geometry and analytic geometry are closely related subjects, where analytic geometry is the theory of complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. A (mathematical) theory of the relationship between the two was put in place during the early part of the 1950s, as part of the business of laying the foundations of algebraic geometry to include, for example, techniques from Hodge theory. (NB While analytic geometry as use of Cartesian coordinates is also in a sense included in the scope of algebraic geometry, that is not the topic being discussed in this article.)

The major paper consolidating the theory was Géometrie Algébrique et Géométrie Analytique by Serre, now usually referred to as GAGA. A GAGA-style result would now mean any theorem of comparison, allowing passage between a category of objects from algebraic geometry, and their morphisms, and a well-defined subcategory of analytic geometry objects and holomorphic mappings. Since it is usually simple to pass from polynomials over the complex numbers to holomorphic functions, the interest is typically in going the other way. Results of this kind were proved first in the nineteenth century.

For example, it is easy to characterise polynomials as analytic functions from the Riemann sphere to itself: an entire function that is not a polynomial must have an essential singularity at the point at infinity. In particular it cannot satisfy a bound on its growth such as O(|z|N).

Riemann surface theory shows that a compact Riemann surface has enough meromorphic functions on it, to make 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.

In the twentieth century, the Lefschetz principle was cited, to justify the use of topological techniques for algebraic geometry over any algebraically closed field K of characteristic 0, by treating K as if it were the complex number field. This has a basis in logic: unless K is of very large transcendence degree, it can be embedded into the complex numbers somehow; and a given geometric problem will in fact relate to a finitely-generated subfield of K.

Chow's theorem is an example of the most immediately useful kind of comparison available. It states that an analytic subspace of complex projective space that is closed in the strong topology is a subvariety (closed for the Zariski topology). This allows quite a free use of complex-analytic methods within the classical parts of algebraic geometry.

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