Complex analytic space

Definition

Denote the constant sheaf on a topological space with value C by C _ . A C -space is a locally ringed space ( X , O X ) whose structure sheaf is an algebra over C _ .

Choose an open subset U of some complex affine space C n , and fix finitely many holomorphic functions f 1 , … , f k in U . Let X = V ( f 1 , … , f k ) be the common vanishing locus of these holomorphic functions, that is, X = { x ∣ f 1 ( x ) = ⋯ = f k ( x ) = 0 } . Define a sheaf of rings on X by letting O X be the restriction to X of O U / ( f 1 , … , f k ) , where O U is the sheaf of holomorphic functions on U . Then the locally ringed C -space ( X , O X ) is a local model space.

A complex analytic space is a locally ringed C -space ( X , O X ) which is locally isomorphic to a local model space.

Morphisms of complex analytic spaces are defined to be morphisms of the underlying locally ringed spaces, they are also called holomorphic maps.