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

Choose an open subset U of some complex affine space Cⁿ, and fix finitely many holomorphic functions f₁, ..., f_k in U. Let X = V(f₁, ..., f_k) be the common vanishing locus of these holomorphic functions, that is, X = {x|f₁(x) = ... = 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 C-space $(X, \mathcal{O}_X)$ is a local model space.

A complex analytic space is a locally ringed C-space $(X, \mathcal{O}_X)$ which is locally isomorphic to a local model space.

References

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

See also

• Analytic space

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