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In mathematics, algebraic geometry and analytic geometry are two closely related subjects. Where algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. The deep relation between these subjects has numerous applications in which algebraic techniques are applied to analytic spaces and analytic techniques to algebraic varieties.
Introduction
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 rational functions (quotients of polynomials). In particular, an entire function that is not a polynomial must have an essential singularity at infinity. In particular it cannot satisfy a bound on its growth such as O(|z|N). Hence, there is no essential difference between the complex projective line as an algebraic variety, and the Riemann sphere.
Important results
There is a long history of comparison results between algebraic geometry and analytic geometry, beginning in the nineteenth century and still continuing today. Some of the more important advances are listed here in chronological order.
Riemann's existence theorem
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.
The Lefschetz principle
In the twentieth century, the Lefschetz principle, named for Solomon Lefschetz, was cited in algebraic geometry 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. It roughly asserts that true statements in algebraic geometry over C are true over any algebraically closed field K of characteristic zero. The precise principle and its proof are due to Tarski and are based in mathematical logic.This principle allowed to carry over results obtained using analytic or topological methods for algebraic varieties over C to other algebraically closed base fields of characteristic 0.
Chow's theorem
Chow's theorem, proved by W. L. Chow. 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 an algebraic subvariety (closed for the Zariski topology). This allows quite a free use of complex-analytic methods within the classical parts of algebraic geometry.Serre's GAGA
Foundations for the many relations between the two theories were 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. The major paper consolidating the theory was Géometrie Algébrique et Géométrie Analytique by Serre, now usually referred to as GAGA. It proves general results that relate classes of algebraic varieties, regular morphisms and sheaves with classes of analytic spaces, holomorphic mappings and sheaves.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, and a well-defined subcategory of analytic geometry objects and holomorphic mappings.
Moishezon manifolds
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 true (quite). Moishezon in 1967 showed that a Moishezon manifold is a projective algebraic variety if and only if it admits a Kähler metric.
Reference
- J. P. Serre (1956), "Géométrie algébrique et géométrie analytique." Annales de l'Institut Fourier 6, 1-42.
"Algebraic geometry and analytic geometry"