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.

Definition[edit]

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

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

Given a topological space X and a sheaf OX 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,OV) of (X,OX) is isomorphic to a closed complex subspace of D, OX is also coherent, and we call it a holomorphic sheaf.