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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 C by  . A C-space is a locally ringed space  whose structure sheaf is an algebra over .
Choose an open subset U of some complex affine space Cn, and fix finitely many holomorphic functions f1, ..., fk in U. Let X = V(f1, ..., fk) be the common vanishing locus of these holomorphic functions, that is, X = {x|f1(x) = ... = fk(x) = 0}. Define a sheaf of rings on X by letting  be the restriction to X of  , where  is the sheaf of holomorphic functions on U. Then the locally ringed C-space is a local model space.
A complex analytic space is a locally ringed C-space 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
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