In mathematics, particularly differential geometry and complex geometry, a complex analytic variety[note 1] or complex analytic space is a generalization of a complex manifold that allows the presence of singularities.
Complex analytic varieties are locally ringed spaces that 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
-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
Let
be the common vanishing locus of these holomorphic functions, that is,
Define a sheaf of rings on
is the sheaf of holomorphic functions on
Then the locally ringed
is a local model space.
A complex analytic variety is a locally ringed
that 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.
A structure sheaf may have nilpotent element,[1] and also, when the complex analytic space whose structure sheaf is reduced, then the complex analytic space is reduced, that is, the complex analytic space may not be reduced.
An associated complex analytic space (variety)