Complex analytic space

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In mathematics, a complex analytic space is a generalization of a complex manifold which allows the presence of singularities. Complex analytic spaces 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.


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 space is a locally ringed -space 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.

See also[edit]


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