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Concept
Algebraic geometry and analytic geometry
Summary
In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While 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.
Let X be a projective complex algebraic variety. Because X is a complex variety, its set of complex points X(C) can be given the structure of a compact complex analytic space. This analytic space is denoted Xan. Similarly, if is a sheaf on X, then there is a corresponding sheaf on Xan. This association of an analytic object to an algebraic one is a functor. The prototypical theorem relating X and Xan says that for any two coherent sheaves and on X, the natural homomorphism:
is an isomorphism. Here is the structure sheaf of the algebraic variety X and is the structure sheaf of the analytic variety Xan. In other words, 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 to . (Note in particular that itself is coherent, a result known as the Oka coherence theorem, and also, it was proved in “Faisceaux Algebriques Coherents” () that the structure sheaf of the algebraic variety is coherent.)
Another important statement is as follows: For any coherent sheaf on an algebraic variety X the homomorphisms
are isomorphisms for all qs. This means that the q-th cohomology group on X is isomorphic to the cohomology group on Xan.
The theorem applies much more generally than stated above (see the formal statement below). It and its proof have many consequences, such as Chow's theorem, the Lefschetz principle and Kodaira vanishing theorem.
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