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 $\mathbb{ℂ}$ by $\underset{\_}{\mathbb{ℂ}}$. A $\mathbb{ℂ}$-space is a locally ringed space $(X,{{𝒪}}_X)$ whose structure sheaf is an algebra over $\underset{\_}{\mathbb{ℂ}}$.

Choose an open subset $U$ of some complex affine space ${\mathbb{ℂ}}^n$, and fix finitely many holomorphic functions $f_1,\dots ,f_k$ in $U$. Let $X=V(f_1,\dots ,f_k)$ be the common vanishing locus of these holomorphic functions, that is, $X=\{x\mid f_1(x)=\cdots =f_k(x)=0\}$. Define a sheaf of rings on $X$ by letting ${{𝒪}}_X$ be the restriction to $X$ of ${{𝒪}}_U/(f_1,\dots ,f_k)$, where ${{𝒪}}_U$ is the sheaf of holomorphic functions on $U$. Then the locally ringed $\mathbb{ℂ}$-space $(X,{{𝒪}}_X)$ is a local model space.

A complex analytic space is a locally ringed $\mathbb{ℂ}$-space $(X,{{𝒪}}_X)$ 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