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Wikipedia / Complex analytic variety

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

Complex analytic varieties are analogous to algebraic varieties. Roughly speaking, a complex analytic variety is a zero locus of a set of a complex analytic function, while an algebraic variety is a zero locus of a set of a polynomial function.

Definition

Denote the constant sheaf on a topological space. A -space is a locally ringed space, whose structure sheaf is an algebra.

of some complex affine space is a local model space.

A complex analytic variety 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) is such that:

  1. Let X be a scheme of finite type) (Spectrum of a ring is an algebra of finite type. Here, the scheme X is obtained by glueing is reduced.

See also

Note

Annotation

References

  • (no.10-13)

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