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.

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 with value \mathbb{C} by \underline{\mathbb{C}}. A \mathbb{C}-space is a locally ringed space (X, \mathcal{O}_X), whose structure sheaf is an algebra over \underline{\mathbb{C}}.

Choose an open subset U of some complex affine space \mathbb{C}^n, and fix finitely many holomorphic functions f_1,\dots,f_k in U. Let X=V(f_1,\dots,f_k) be the common vanishing locus of these holomorphic functions, that is, X=\{x\mid f_1(x)=\cdots=f_k(x)=0\}. Define a sheaf of rings on X by letting \mathcal{O}_X be the restriction to X of \mathcal{O}_U/(f_1, \ldots, f_k), where \mathcal{O}_U is the sheaf of holomorphic functions on U. Then the locally ringed \mathbb{C}-space (X, \mathcal{O}_X) is a local model space.

A complex analytic variety is a locally ringed \mathbb{C}-space (X, \mathcal{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;[1] if the structure sheaf is reduced, then the complex analytic space is called reduced.

An associated complex analytic space (variety) X_h is such that:[1]

Let X be a scheme of finite type over \mathbb{C}, and cover X with open affine subsets Y_i = \operatorname{Spec} A_i (X =\cup Y_i) (Spectrum of a ring). Then each A_i is an algebra of finite type over \mathbb{C}, and A_i \simeq \mathbb{C}[z_1, \dots, z_n]/(f_1,\dots, f_m), where f_1,\dots, f_m are polynomials in z_1, \dots, z_n, which can be regarded as a holomorphic functions on \mathbb{C}. Therefore, their set of common zeros is the complex analytic subspace (Y_i)_h \subseteq \mathbb{C}. Here, the scheme X is obtained by glueing the data of the sets Y_i, and then the same data can be used for glueing the complex analytic spaces (Y_i)_h into a complex analytic space X_h, so we call X_h an associated complex analytic space with X. The complex analytic space X is reduced if and only if the associated complex analytic space X_h is reduced.[2]

Note

  1. ^ Hartshorne 1977, p. 439.
  2. ^ Grothendieck & Raynaud (2002) (SGA 1 §XII. Proposition 2.1.)

Annotation

  1. ^ A complex analytic variety is sometimes required to be irreducible and (or) reduced.

References

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