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