Complex analytic variety

In mathematics, and in particular differential geometry and complex geometry, a complex analytic variety ^{[note 1]} or complex analytic space is a generalization of a complex manifold which allows the presence of singularities. Complex analytic varieties are locally ringed spaces which 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.

Definition

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

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

A complex analytic variety is a locally ringed [math]\displaystyle{ \mathbb{C} }[/math]-space [math]\displaystyle{ (X, \mathcal{O}_X) }[/math] which 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 element,^[1] and also, when the complex analytic space whose structure sheaf is reduced, then the complex analytic space is reduced, that is, the complex analytic space may not be reduced.

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

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

See also

• Algebraic variety - Roughly speaking, an (complex) analytic variety is a zero locus of a set of an (complex) analytic function, while an algebraic variety is a zero locus of a set of a polynomial function and allowing singular point.
• Analytic space
• Complex algebraic variety
• GAGA
• Rigid analytic space

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