Holomorphic sheaf
Encyclopedia
In mathematics
, more specifically complex analysis
, a holomorphic sheaf (often also called an analytic sheaf) is a natural generalization of the sheaf
of holomorphic function
s on a complex manifold
.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, more specifically complex analysis
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
, a holomorphic sheaf (often also called an analytic sheaf) is a natural generalization of the sheaf
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
of holomorphic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
s on a complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
.
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 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
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
.
An ideal I of OD is a sheaf such that I(U) is always a complex submodule of OD(U).
Given a coherent
Coherent sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a specific class of sheaves having particularly manageable properties closely linked to the geometrical properties of the underlying space. The definition of coherent sheaves is made with...
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
Support (mathematics)
In mathematics, the support of a function is the set of points where the function is not zero, or the closure of that set . This concept is used very widely in mathematical analysis...
A is a coherent sheaf OA of local
Local ring
In abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or...
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.