Complex analytic variety

A cone is not a complex manifold, but it is a 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 with value $ℂ$ by $\underset{\_}{ℂ}$. A $ℂ$-space is a locally ringed space $(X,{𝒪}_X)$, whose structure sheaf is an algebra over $\underset{\_}{ℂ}$.

Choose an open subset $U$ of some complex affine space ${ℂ}^n$, 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 ${𝒪}_X$ be the restriction to $X$ of ${𝒪}_U/(f_1,\dots ,f_k)$, where ${𝒪}_U$ is the sheaf of holomorphic functions on $U$. Then the locally ringed $ℂ$-space $(X,{𝒪}_X)$ is a local model space.

A complex analytic variety is a locally ringed $ℂ$-space $(X,{𝒪}_X)$ 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;Template:Sfn 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:Template:Sfn

Let X be a scheme of finite type over $ℂ$, and cover X with open affine subsets $Y_i=\mathrm{Spec}{A}_i$ ($X=\cup Y_i$) (Spectrum of a ring). Then each $A_i$ is an algebra of finite type over $ℂ$, and $A_i\simeq ℂ[z_1,\dots ,z_n]/(f_1,\dots ,f_m)$, where $f_1,\dots ,f_m$ are polynomials in $z_1,\dots ,z_n$, which can be regarded as a holomorphic functions on $ℂ$. Therefore, their set of common zeros is the complex analytic subspace $(Y_i{)}_h\subseteq ℂ$. 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.

See also

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References

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Future reading

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External links

• Kiran Kedlaya. 18.726 Algebraic Geometry (LEC # 30 - 33 GAGA)Spring 2009. Massachusetts Institute of Technology: MIT OpenCourseWare Creative Commons BY-NC-SA.
• Tasty Bits of Several Complex Variables (p. 137) open source book by Jiří Lebl BY-NC-SA.
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