- Holomorphic sheaf
In
mathematics , more specificallycomplex analysis , a holomorphic sheaf (often also called an analytic sheaf) is a natural generalization of the sheaf ofholomorphic function s on acomplex manifold .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 C"n", 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 C"n" 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.
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