Complex analytic variety

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 formula_1 by formula_2. A formula_1-space is a locally ringed space formula_4 whose structure sheaf is an algebra over formula_2.
Choose an open subset formula_6 of some complex affine space formula_7, and fix finitely many holomorphic functions formula_8 in formula_6. Let formula_10 be the common vanishing locus of these holomorphic functions, that is, formula_11. Define a sheaf of rings on formula_12 by letting formula_13 be the restriction to formula_12 of formula_15, where formula_16 is the sheaf of holomorphic functions on formula_6. Then the locally ringed formula_1-space formula_4 is a local model space.
A complex analytic variety is a locally ringed formula_1-space formula_4 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.