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

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 ${\mathcal {O}}_{X}$ be the restriction to $X$ of ${\mathcal {O}}_{U}/(f_{1},\ldots ,f_{k})$, where ${\mathcal {O}}_{U}$ is the sheaf of holomorphic functions on $U$. Then the locally ringed $\mathbb {C}$-space $(X,{\mathcal {O}}_{X})$ is a local model space.

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