Algebraic geometry and analytic geometry

In mathematics, algebraic geometry and analytic geometry are two closely related subjects. Where algebraic geometry studies algebraic variety, 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.

: Note: 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.

= 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 characterise polynomials as analytic functions from the .

= 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 ==

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== 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 . 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 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.

== 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 Sheaf (mathematics) 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.

=Reference=

*J. P. Serre (1956), Géométrie algébrique et géométrie analytique. Annales de l Institut Fourier 6, 1-42.