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. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Complex analysis is the branch of mathematics investigating holomorphic functions, i. ... In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ...

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 , 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 has a natural (componentwise) -algebra structure and one can collate sections that agree on intersections to create larger sections; this is outlined in more detail at sheaf. In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain...

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 -algebras. Such a substructure (A,O_A) of (D,O_D) is called a closed complex subspace of D. In mathematics, especially in algebraic geometry and the theory of complex manifolds, a coherent sheaf F on a locally ringed space X is a sheaf isomorphic with the cokernel of a morphism of OX_modules OXm → OXn. ... In mathematics, the support of a numerical function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, more particularly in abstract algebra, local rings are certain rings that are comparatively simple, and serve to describe the local behavior of functions defined on varieties or manifolds. ...

Given a topological space X and a sheaf O_X of local -algebras, if for any point x in X there is an open subset V of X containing it and a subset D of 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.