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 ${\displaystyle {\underline {\mathbb {C} }}}$. A ${\displaystyle \mathbb {C} }$-space is a locally ringed space ${\displaystyle (X,{\mathcal {O}}_{X})}$, whose structure sheaf is an algebra over ${\displaystyle {\underline {\mathbb {C} }}}$.

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 ${\displaystyle {\mathcal {O}}_{X}}$ be the restriction to ${\displaystyle X}$ of ${\displaystyle {\mathcal {O}}_{U}/(f_{1},\ldots ,f_{k})}$, where ${\displaystyle {\mathcal {O}}_{U}}$ is the sheaf of holomorphic functions on ${\displaystyle U}$. Then the locally ringed ${\displaystyle \mathbb {C} }$-space ${\displaystyle (X,{\mathcal {O}}_{X})}$ is a local model space.

A complex analytic variety is a locally ringed ${\displaystyle \mathbb {C} }$-space ${\displaystyle (X,{\mathcal {O}}_{X})}$ 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) ${\displaystyle X_{h}}$ is such that:¹

Let X be a scheme of finite type over ${\displaystyle \mathbb {C} }$, and cover X with open affine subsets ${\displaystyle Y_{i}=\operatorname {Spec} A_{i}}$ (${\displaystyle X=\cup Y_{i}}$) (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