# Complex analytic variety

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In mathematics, and in particular differential geometry and complex geometry, a complex analytic variety or complex analytic space is a generalization of a complex manifold which allows the presence of singularities. Complex analytic varieties 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 variety is a locally ringed $\mathbb {C}$ -space $(X,{\mathcal {O}}_{X})$  which is locally isomorphic to a local model space.

Morphisms of complex analytic varieties are defined to be morphisms of the underlying locally ringed spaces, they are also called holomorphic maps.