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Complex analytic variety

Generalization of a complex manifold that allows the use of singularities

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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.

01Definition

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; 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:

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.

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This article is adapted from the Wikipedia article “Complex analytic variety”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Complex analytic variety

01Definition

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Related topics

Analytic space

Complex algebraic variety

Algebraic geometry and analytic geometry