Complex analytic space

In mathematics, a complex analytic space is a generalization of a complex manifold which allows the presence of singularities. Complex analytic spaces are locally ringed spaces which are locally isomorphic to local model spaces, where a local model space is an open subset of the vanishing locus of a finite set of holomorphic functions.

## Definition

Denote the constant sheaf on a topological space with value ℂ by ℂ_. A ℂ-space is a locally ringed space (X,𝒪X) whose structure sheaf is an algebra over ℂ_.

Choose an open subset U of some complex affine space ℂn, and fix finitely many holomorphic functions f1,…,fk in U. Let X=V(f1,…,fk) be the common vanishing locus of these holomorphic functions, that is, X={x∣f1(x)=⋯=fk(x)=0}. Define a sheaf of rings on X by letting 𝒪X be the restriction to X of 𝒪U/(f1,…,fk), where 𝒪U is the sheaf of holomorphic functions on U. Then the locally ringed ℂ-space (X,𝒪X) is a local model space.

A complex analytic space is a locally ringed ℂ-space (X,𝒪X) which is locally isomorphic to a local model space.

Morphisms of complex analytic spaces are defined to be morphisms of the underlying locally ringed spaces, they are also called holomorphic maps.

## See also

* Analytic space

## References

* Grauert and Remmert, Complex Analytic Spaces
* Grauert, Peternell, and Remmert, Encyclopaedia of Mathematical Sciences 74: Several Complex Variables VII

0.00

(0 votes)

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Complex_analytic_space&oldid=41202"