complex analytic varieties

In mathematics, particularly differential geometry and complex geometry, a complex analytic varietyComplex analytic variety (or just variety) is sometimes required to be irreducible
and (or) reduced 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.

Definition

Denote the constant sheaf on a topological space with value $\mathbb$ by $\underline$. A $\mathbb$-space is a locally ringed space $(X, \mathcal_X)$, whose structure sheaf is an algebra over $\underline$.

Choose an open subset $U$ of some complex affine space $\mathbb^n$, 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=\$. Define a sheaf of rings on $X$ by letting $\mathcal_X$ be the restriction to $X$ of $\mathcal_U/(f_1, \ldots, f_k)$, where $\mathcal_U$ is the sheaf of holomorphic functions on $U$. Then the locally ringed $\mathbb$-space $(X, \mathcal_X)$ is a local model space.

A complex analytic variety is a locally ringed $\mathbb$-space $(X, \mathcal_X)$ 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 element,
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) $X_h$ is such that;

:Let X be scheme of finite type over $\mathbb$, and cover X with open affine subsets $Y_i = \operatorname A_i$ ($X =\cup Y_i$) (Spectrum of a ring). Then each $A_i$ is an algebra of finite type over $\mathbb$, and A_i \simeq \mathbb _1, \dots, z_n(f_1,\dots, f_m). Where $f_1,\dots, f_m$ are polynomial in $z_1, \dots, z_n$, which can be regarded as a holomorphic functions on $\mathbb$. Therefore, their set of common zeros is the complex analytic subspace $(Y_i)_h \subseteq \mathbb$. Here, the scheme X 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. (SGA 1 §XII. Proposition 2.1.)

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.
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Note

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References

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Future reading

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External links

* Kiran Kedlaya. 18.72
Algebraic GeometryLEC # 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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Algebraic geometry
Several complex variables
Complex geometry