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.

Definition

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

Choose an open subset U {\displaystyle U} of some complex affine space C n {\displaystyle \mathbb {C} ^{n}} , and fix finitely many holomorphic functions f 1 , … , f k {\displaystyle f_{1},\dots ,f_{k}} in U {\displaystyle U} . Let X = V ( f 1 , … , f k ) {\displaystyle X=V(f_{1},\dots ,f_{k})} be the common vanishing locus of these holomorphic functions, that is, X = { x ∣ f 1 ( x ) = ⋯ = f k ( x ) = 0 } {\displaystyle X=\{x\mid f_{1}(x)=\cdots =f_{k}(x)=0\}} . Define a sheaf of rings on X {\displaystyle X} by letting O X {\displaystyle {\mathcal {O}}_{X}} be the restriction to X {\displaystyle X} of O U / ( f 1 , … , f k ) {\displaystyle {\mathcal {O}}_{U}/(f_{1},\ldots ,f_{k})} , where O U {\displaystyle {\mathcal {O}}_{U}} is the sheaf of holomorphic functions on U {\displaystyle U} . Then the locally ringed C {\displaystyle \mathbb {C} } -space ( X , O X ) {\displaystyle (X,{\mathcal {O}}_{X})} is a local model space.

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

Let X be a scheme of finite type over C {\displaystyle \mathbb {C} } , and cover X with open affine subsets Y i = Spec ⁡ A i {\displaystyle Y_{i}=\operatorname {Spec} A_{i}} ( X = ∪ Y i {\displaystyle X=\cup Y_{i}} ) (Spectrum of a ring). Then each A i {\displaystyle A_{i}} is an algebra of finite type over C {\displaystyle \mathbb {C} } , and A i ≃ C [ z 1 , … , z n ] / ( f 1 , … , f m ) {\displaystyle A_{i}\simeq \mathbb {C} [z_{1},\dots ,z_{n}]/(f_{1},\dots ,f_{m})} , where f 1 , … , f m {\displaystyle f_{1},\dots ,f_{m}} are polynomials in z 1 , … , z n {\displaystyle z_{1},\dots ,z_{n}} , which can be regarded as a holomorphic functions on C {\displaystyle \mathbb {C} } . Therefore, their set of common zeros is the complex analytic subspace ( Y i ) h ⊆ C {\displaystyle (Y_{i})_{h}\subseteq \mathbb {C} } . Here, the scheme X is obtained by glueing the data of the sets Y i {\displaystyle Y_{i}} , and then the same data can be used for glueing the complex analytic spaces ( Y i ) h {\displaystyle (Y_{i})_{h}} into a complex analytic space X h {\displaystyle X_{h}} , so we call X h {\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 X h {\displaystyle X_{h}} is reduced.^[2]

See also

• Analytic space
• Complex algebraic variety
• GAGA – Two closely related mathematical subjectsPages displaying short descriptions of redirect targets
• Rigid analytic space – Analogue of a complex analytic space over a nonarchimedean field

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