# 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 ${\displaystyle \mathbb {C} }$  by ${\displaystyle {\underline {\mathbb {C} }}}$ . A ${\displaystyle \mathbb {C} }$ -space is a locally ringed space ${\displaystyle (X,{\mathcal {O}}_{X})}$  whose structure sheaf is an algebra over ${\displaystyle {\underline {\mathbb {C} }}}$ .

Choose an open subset ${\displaystyle U}$  of some complex affine space ${\displaystyle \mathbb {C} ^{n}}$ , and fix finitely many holomorphic functions ${\displaystyle f_{1},\dots ,f_{k}}$  in ${\displaystyle U}$ . Let ${\displaystyle X=V(f_{1},\dots ,f_{k})}$  be the common vanishing locus of these holomorphic functions, that is, ${\displaystyle X=\{x\mid f_{1}(x)=\cdots =f_{k}(x)=0\}}$ . Define a sheaf of rings on ${\displaystyle X}$  by letting ${\displaystyle {\mathcal {O}}_{X}}$  be the restriction to ${\displaystyle X}$  of ${\displaystyle {\mathcal {O}}_{U}/(f_{1},\ldots ,f_{k})}$ , where ${\displaystyle {\mathcal {O}}_{U}}$  is the sheaf of holomorphic functions on ${\displaystyle U}$ . Then the locally ringed ${\displaystyle \mathbb {C} }$ -space ${\displaystyle (X,{\mathcal {O}}_{X})}$  is a local model space.

A complex analytic variety is a locally ringed ${\displaystyle \mathbb {C} }$ -space ${\displaystyle (X,{\mathcal {O}}_{X})}$  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.