Generalization of a complex manifold that allows the use of singularities
In mathematics, and in particular 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.
Definition
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
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,
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)
is such that;
- Let X be schemes finite type over
, and cover X with open affine subset
(
) (Spectrum of a ring). Then each
is an algebra of finite type over
, and
. Where
are polynomial in
, which can be regarded as a holomorphic function on
. Therefore, their common zero of the set is the complex analytic subspace
. Here, scheme X obtained by glueing the data of the set
, and then the same data can be used to glueing the complex analytic space
into an complex analytic space
, so we call
a associated complex analytic space with X. The complex analytic space X is reduced if and only if the associated complex analytic space
reduced.[2]