Complex analytic variety

A cone is not a complex manifold, but it is a complex analytic variety.

In mathematics, particularly differential geometry and complex geometry, a complex analytic variety 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 $\mathbb {C}$ by ${\underline {\mathbb {C} }}$. A $\mathbb {C}$-space is a locally ringed space $(X,{\mathcal {O}}_{X})$, whose structure sheaf is an algebra over ${\underline {\mathbb {C} }}$.

Choose an open subset $U$ of some complex affine space $\mathbb {C} ^{n}$, 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, $X=\{x\mid f_{1}(x)=\cdots =f_{k}(x)=0\}$. Define a sheaf of rings on $X$ by letting ${\mathcal {O}}_{X}$ be the restriction to $X$ of ${\mathcal {O}}_{U}/(f_{1},\ldots ,f_{k})$, where ${\mathcal {O}}_{U}$ is the sheaf of holomorphic functions on $U$. Then the locally ringed $\mathbb {C}$-space $(X,{\mathcal {O}}_{X})$ is a local model space.

A complex analytic variety is a locally ringed $\mathbb {C}$-space $(X,{\mathcal {O}}_{X})$ 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; if the structure sheaf is reduced, then the complex analytic space is called reduced.

An associated complex analytic space (variety) $X_{h}$ is such that:

Let X be a scheme of finite type over $\mathbb {C}$, and cover X with open affine subsets $Y_{i}=\operatorname {Spec} A_{i}$ ($X=\cup Y_{i}$) (Spectrum of a ring). Then each $A_{i}$ is an algebra of finite type over $\mathbb {C}$, and $A_{i}\simeq \mathbb {C} [z_{1},\dots ,z_{n}]/(f_{1},\dots ,f_{m})$, where $f_{1},\dots ,f_{m}$ are polynomials in $z_{1},\dots ,z_{n}$, which can be regarded as a holomorphic functions on $\mathbb {C}$. Therefore, their set of common zeros is the complex analytic subspace $(Y_{i})_{h}\subseteq \mathbb {C}$. Here, the scheme X is 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.

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

References

• Aroca, José Manuel; Hironaka, Heisuke; Vicente, José Luis (3 November 2018). Complex Analytic Desingularization. doi:10.1007/978-4-431-49822-3. ISBN 978-4-431-49822-3.
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• Huckleberry, Alan (2013). "Hans Grauert (1930–2011)". Jahresbericht der Deutschen Mathematiker-Vereinigung. 115: 21–45. arXiv:1303.6933. doi:10.1365/s13291-013-0061-7. S2CID 119685542.
• Remmert, Reinhold (1998). "From Riemann Surfaces to Complex Spaces". Séminaires et Congrès. Zbl 1044.01520.
• Serre, Jean-Pierre (1956). "Géométrie algébrique et géométrie analytique". Annales de l'Institut Fourier. 6: 1–42. doi:10.5802/aif.59. ISSN 0373-0956. MR 0082175.
• Tognoli, A. (2 June 2011). Tognoli, A (ed.). Singularities of Analytic Spaces: Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Bressanone (Bolzano), Italy, June 16-25, 1974. doi:10.1007/978-3-642-10944-7. ISBN 978-3-642-10944-7.
• "Chapter II. Preliminaries". Zariski-decomposition and Abundance. Mathematical Society of Japan Memoirs. Vol. 14. Mathematical Society of Japan. 2004. pp. 13–78. doi:10.2969/msjmemoirs/01401C020. ISBN 978-4-931469-31-0.
• Flores, Arturo Giles; Teissier, Bernard (2018). "Local polar varieties in the geometric study of singularities". Annales de la Faculté des Sciences de Toulouse: Mathématiques. 27 (4): 679–775. arXiv:1607.07979. doi:10.5802/afst.1582. S2CID 119150240.

Future reading

• Huckleberry, Alan (2013). "Hans Grauert (1930–2011)". Jahresbericht der Deutschen Mathematiker-Vereinigung. 115: 21–45. arXiv:1303.6933. doi:10.1365/s13291-013-0061-7. S2CID 256084531.

External links

• Kiran Kedlaya. 18.726 Algebraic Geometry (LEC # 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.
• Onishchik, A.L. (2001) [1994], "Analytic space", Encyclopedia of Mathematics, EMS Press
• El'kin, A.G. (2001) [1994], "Analytic set", Encyclopedia of Mathematics, EMS Press