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 [math]\displaystyle{ \mathbb{C} }[/math] by [math]\displaystyle{ \underline{\mathbb{C}} }[/math]. A [math]\displaystyle{ \mathbb{C} }[/math]-space is a locally ringed space [math]\displaystyle{ (X, \mathcal{O}_X) }[/math] whose structure sheaf is an algebra over [math]\displaystyle{ \underline{\mathbb{C}} }[/math].

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

A complex analytic space is a locally ringed [math]\displaystyle{ \mathbb{C} }[/math]-space [math]\displaystyle{ (X, \mathcal{O}_X) }[/math] 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

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