Complex analytic variety explained

In mathematics, particularly differential geometry and complex geometry, a complex analytic variety[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

C

by

\underline{C

}. A

C

-space is a locally ringed space

(X,l{O}X)

, whose structure sheaf is an algebra over

\underline{C

}.

Choose an open subset

U

of some complex affine space

Cn

, and fix finitely many holomorphic functions

f1,...,fk

in

U

. Let

X=V(f1,...,fk)

be the common vanishing locus of these holomorphic functions, that is,

X=\{x\midf1(x)= … =fk(x)=0\}

. Define a sheaf of rings on

X

by letting

l{O}X

be the restriction to

X

of

l{O}U/(f1,\ldots,fk)

, where

l{O}U

is the sheaf of holomorphic functions on

U

. Then the locally ringed

C

-space

(X,l{O}X)

is a local model space.

A complex analytic variety is a locally ringed

C

-space

(X,l{O}X)

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 elements;if the structure sheaf is reduced, then the complex analytic space is called reduced.

An associated complex analytic space (variety)

Xh

is such that:

Let X be scheme of finite type over

C

, and cover X with open affine subsets

Yi=\operatorname{Spec}Ai

(

X=\cupYi

) (Spectrum of a ring). Then each

Ai

is an algebra of finite type over

C

, and

Ai\simeqC[z1,...,zn]/(f1,...,fm)

, where

f1,...,fm

are polynomials in

z1,...,zn

, which can be regarded as a holomorphic functions on

C

. Therefore, their set of common zeros is the complex analytic subspace

(Yi)h\subseteqC

. Here, the scheme X is obtained by glueing the data of the sets

Yi

, and then the same data can be used for glueing the complex analytic spaces

(Yi)h

into a complex analytic space

Xh

, so we call

Xh

an associated complex analytic space with X. The complex analytic space X is reduced if and only if the associated complex analytic space

Xh

is reduced.[2]

See also

References

Future reading

External links

Notes and References

  1. Complex analytic variety (or just variety) is sometimes required to be irreducible and (or) reduced
  2. (SGA 1 §XII. Proposition 2.1.)

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