Holomorphic sheaf

In mathematics, more specifically complex analysis, a holomorphic sheaf (often also called an analytic sheaf) is a natural generalization of the sheaf of holomorphic functions on a complex manifold.

Related Topics:
Mathematics - Complex analysis - Sheaf - Holomorphic function - Complex manifold

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It takes a rather involved string of definitions to state more precisely what a holomorphic sheaf is.

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Given a simply connected open subset D of mathbb C^n, there is an associated sheaf OD of holomorphic functions on D. Throughout, U is any open subset of D. Then the set OD(U) of holomorphic functions from U to mathbb C has a natural (componentwise) mathbb C-algebra structure and one can collate sections that agree on intersections to create larger sections; this is outlined in more detail at sheaf.

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An ideal I of OD is a sheaf such that I(U) is always a complex submodule of OD(U).

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Given a coherent such I,

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the quotient sheaf OD / I

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is such that (U)

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is always a module over OD(U);

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we call such a sheaf a OD-module.

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It is also coherent, and its restriction to its support A

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is a coherent sheaf OA of local mathbb C-algebras.

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Such a substructure

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(A,OA) of (D,OD) is called a closed complex subspace of D.

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Given a topological space X and a sheaf OX of local mathbb C-algebras,

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if for any point x in X

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there is an open subset V of X containing it

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and a subset D of mathbb C^n so that the restriction

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(V,OV) of (X,OX) is isomorphic to a closed complex subspace of D,

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OX is also coherent, and we call it a holomorphic sheaf.

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