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 by . A **-space** is a locally ringed space whose structure sheaf is an algebra over .

Choose an open subset of some complex affine space , and fix finitely many holomorphic functions in . Let be the common vanishing locus of these holomorphic functions, that is, . Define a sheaf of rings on by letting be the restriction to of , where is the sheaf of holomorphic functions on . Then the locally ringed -space is a **local model space**.

A **complex analytic variety** is a locally ringed -space 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.

## See also

## References

- Grauert and Remmert,
*Complex Analytic Spaces* - Grauert, Peternell, and Remmert,
*Encyclopaedia of Mathematical Sciences 74: Several Complex Variables VII*