Complex analytic variety

In mathematics, in particular 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.

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 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 schemes finite type over $\mathbb {C}$, and cover X with open affine subset $Y_{i}=\operatorname {Spec} A_{i}$ ($X=\cup Y_{i}$) (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 polynomial in $z_{1},\dots ,z_{n}$, which can be regarded as a holomorphic function on $\mathbb {C}$. Therefore, their common zero of the set is the complex analytic subspace $(Y_{i})_{h}\subseteq \mathbb {C}$. Here, scheme X obtained by glueing the data of the set $Y_{i}$, and then the same data can be used to glueing the complex analytic space $(Y_{i})_{h}$ into an complex analytic space $X_{h}$, so we call $X_{h}$ a associated complex analytic space with X. The complex analytic space X is reduced if and only if the associated complex analytic space $X_{h}$ reduced.

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 – locally ringed space glued together from analytic varietiesPages displaying wikidata descriptions as a fallback
• Complex algebraic variety
• GAGA – Two closely related mathematical subjectsPages displaying short descriptions of redirect targets
• Rigid analytic space – An 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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• Fischer, G. (14 November 2006). Complex Analytic Geometry. Springer. ISBN 978-3-540-38121-1.
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• Gunning, Robert Clifford; Rossi, Hugo (2009). "Chapter V. Anlytic space". Analytic Functions of Several Complex Variables. American Mathematical Soc. ISBN 9780821821657.
• Grauert, Hans; Remmert, Reinhold (1958). "Komplexe Räume". Mathematische Annalen. 136 (3): 245–318. doi:10.1007/BF01362011. S2CID 121348794.
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• Hartshorne, Robin (1970). Ample Subvarieties of Algebraic Varieties. Lecture Notes in Mathematics. Vol. 156. doi:10.1007/BFb0067839. ISBN 978-3-540-05184-8.
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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. 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