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