Knowlpedia

Definition

Sheaf of holomorphic functions

The sheaf assigning to each open subset of a complex manifold its algebra of holomorphic functions.

Definition

Let XX be a . Its sheaf of holomorphic functions, denoted OX\mathcal O_X, assigns to every open set U⊆XU\subseteq X the commutative unital complex algebra

OX(U)={f:U→C∣f is holomorphic},\mathcal O_X(U)=\{f:U\to\mathbb C\mid f\text{ is holomorphic}\},

with restriction maps given by restricting functions to smaller open sets. Holomorphic functions that agree on overlaps glue uniquely, and holomorphicity is local, so this assignment satisfies the axioms. The pair (X,OX)(X,\mathcal O_X) is the complex manifold viewed as a .

Stalks and local coordinates

The OX,x\mathcal O_{X,x} consists of germs of holomorphic functions near xx. It is a whose unique consists of germs vanishing at xx; evaluation at xx identifies the with C\mathbb C. A near xx identifies this stalk with the algebra of germs at the origin of convergent in dim⁡CX\dim_{\mathbb C}X variables. This identification depends on the chosen chart.

Functoriality

A F:X→YF:X\to Y pulls local holomorphic functions on YY back by composition: h↦h∘Fh\mapsto h\circ F. On stalks this gives local homomorphisms

OY,F(x)⟶OX,x.\mathcal O_{Y,F(x)}\longrightarrow\mathcal O_{X,x}.

Conversely, the compatibility of a with these structure sheaves is an intrinsic way to formulate holomorphicity. This ringed-space viewpoint makes local analytic equations, , and analytic subspaces accessible without fixing coordinates.

Conventions and scope
References
  1. Robert C. Gunning and Hugo Rossi, Analytic Functions of Several Complex Variables, Prentice-Hall, 1965; AMS Chelsea reprint, 2009. Publisher record. Relevant: Chapter I, holomorphic functions, germs, and local rings.
  2. Daniel Huybrechts, Complex Geometry: An Introduction, Universitext, Springer, 2005. Publisher record. Relevant: Chapter 1, complex manifolds and their structure sheaves.