Complex analytic variety

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.

Denote the constant sheaf on a topological space with value ${\displaystyle \mathbb{ℂ}}$ by ${\displaystyle \underset{\_}{\mathbb{ℂ}}}$. A ${\displaystyle \mathbb{ℂ}}$-space is a locally ringed space ${\displaystyle (X,{{𝒪}}_X)}$, whose structure sheaf is an algebra over ${\displaystyle \underset{\_}{\mathbb{ℂ}}}$.

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

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

An associated complex analytic space (variety) ${\displaystyle X_h}$ is such that:^[1]

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

• Analytic space
• Complex algebraic variety
• Rigid analytic space – An analogue of a complex analytic space over a nonarchimedean field

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