Holomorphic sheaf

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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.

[edit] Definition

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

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

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

Given a coherent such I, the quotient sheaf O_D / I is such that [O_D / I](U) is always a module over O_D(U); we call such a sheaf a O_D-module. It is also coherent, and its restriction to its support A is a coherent sheaf O_A of local C-algebras. Such a substructure (A,O_A) of (D,O_D) is called a closed complex subspace of D.

Given a topological space X and a sheaf O_X of local C-algebras, if for any point x in X there is an open subset V of X containing it and a subset D of Cⁿ so that the restriction (V,O_V) of (X,O_X) is isomorphic to a closed complex subspace of D, O_X is also coherent, and we call it a holomorphic sheaf.