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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)|0 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, it is also called holomorphic maps.