In mathematics, in particular differential geometry and complex geometry, a complex analytic variety or complex analytic space is a generalization of a complex manifold that allows the presence of singularities. Complex analytic varieties are locally ringed spaces that 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. A \mathbb{C}-space is a locally ringed space, whose structure sheaf is an algebra over. Choose an open subset U of some complex affine space, and fix finitely many holomorphic functions in U. Let be the common vanishing locus of these holomorphic functions, that is,. 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 \mathbb{C}-space is a local model space. A complex analytic variety is a locally ringed \mathbb{C}-space that 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. A structure sheaf may have nilpotent element, and also, when the complex analytic space whose structure sheaf is reduced, then the complex analytic space is reduced, that is, the complex analytic space may not be reduced. An associated complex analytic space (variety) X_h is such that;

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