Complex analytic variety explained

In mathematics, and in particular 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 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)

Xh

is such that;

Let X be schemes finite type over

C

, and cover X with open affine subset

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 polynomial in

z1,...,zn

, which can be regarded as a holomorphic function on

C

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

(Yi)h\subseteqC

. Here, scheme X obtained by glueing the data of the set

Yi

, and then the same data can be used to glueing the complex analytic space

(Yi)h

into an complex analytic space

Xh

, so we call

Xh

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

Xh

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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