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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. == Main statement == 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 ''X''an. Similarly, if is a sheaf on ''X'', then there is a corresponding sheaf on ''X''an. This association of an analytic object to an algebraic one is a functor. The prototypical theorem relating ''X'' and ''X''an says that for any two coherent sheaves and on ''X'', the natural homomorphism: : is the structure sheaf of the algebraic variety ''X'' and to . (Note in particular that on an algebraic variety ''X'' the homomorphism : are isomorphism for all ''qs. This means that the ''q''-th cohomology group on ''X'' are isomorphic to the cohomology group on ''X''an. 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Algebraic geometry and analytic geometry」の詳細全文を読む スポンサード リンク
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