Complex analytic variety

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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.


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 alsoEdit


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