Complex analytic variety

Short description: Generalization of a complex manifold that allows the use of singularities

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.

## Contents

* 1 Definition
* 2 See also
* 3 Note
* 4 Annotation
* 5 References
* 6 Future reading
* 7 External links

## Definition

Denote the constant sheaf on a topological space with value ℂ by ℂ_. A ℂ-space is a locally ringed space (X,𝒪X), whose structure sheaf is an algebra over ℂ_.

Choose an open subset U of some complex affine space ℂn, and fix finitely many holomorphic functions f1,…,fk in U. Let X=V(f1,…,fk) be the common vanishing locus of these holomorphic functions, that is, X={x∣f1(x)=⋯=fk(x)=0}. Define a sheaf of rings on X by letting 𝒪X be the restriction to X of 𝒪U/(f1,…,fk), where 𝒪U is the sheaf of holomorphic functions on U. Then the locally ringed ℂ-space (X,𝒪X) is a local model space.

A complex analytic variety is a locally ringed ℂ-space (X,𝒪X) 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) Xh is such that:[1]

Let X be scheme of finite type over ℂ, and cover X with open affine subsets Yi=Spec⁡Ai (X=∪Yi) (Spectrum of a ring). Then each Ai is an algebra of finite type over ℂ, and Ai≃ℂ[z1,…,zn]/(f1,…,fm), where f1,…,fm are polynomials in z1,…,zn, which can be regarded as a holomorphic functions on ℂ. Therefore, their set of common zeros is the complex analytic subspace (Yi)h⊆ℂ. Here, the scheme X is obtained by glueing the data of the sets Yi, and then the same data can be used for glueing the complex analytic spaces (Yi)h into a complex analytic space Xh, so we call Xh an associated complex analytic space with X. The complex analytic space X is reduced if and only if the associated complex analytic space Xh is 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
* Rigid analytic space - An analogue of a complex analytic space over a nonarchimedean field

## Note

1. ↑ 1.0 1.1 Hartshorne 1977, p. 439.
2. ↑ (Grothendieck Raynaud) (SGA 1 §XII. Proposition 2.1.)

## Annotation

1. ↑ Complex analytic variety (or just variety) is sometimes required to be irreducible and (or) reduced

## 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. https://books.google.com/books?id=rw92DwAAQBAJ&pg=PA6.
* Bloom, Thomas; Herrera, Miguel (1969). "De Rham cohomology of an analytic space". Inventiones Mathematicae 7 (4): 275–296. doi:10.1007/BF01425536. Bibcode: 1969InMat...7..275B. https://www.researchgate.net/publication/226554588.
* Cartan, H.; Bruhat, F.; Cerf, Jean.; Dolbeault, P.; Frenkel, Jean.; Hervé, Michel; Malatian.; Serre, J-P.. "Séminaire Henri Cartan, Tome 4 (1951-1952)". http://www.numdam.org/volume/SHC_1951-1952__4/. (no.10-13)
* Fischer, G. (14 November 2006). Complex Analytic Geometry. Springer. ISBN 978-3-540-38121-1. https://books.google.com/books?id=jR56CwAAQBAJ.
* Gunning, Robert Clifford; Rossi, Hugo (2009). "Chapter III. Variety (Sec. B. Analytic covers)". Analytic Functions of Several Complex Variables. American Mathematical Soc.. ISBN 9780821821657. https://books.google.com/books?id=wsqFAwAAQBAJ&pg=PA101.
* Gunning, Robert Clifford; Rossi, Hugo (2009). "Chapter V. Analytic spaces". Analytic Functions of Several Complex Variables. American Mathematical Soc.. ISBN 9780821821657. https://books.google.com/books?id=wsqFAwAAQBAJ&pg=PA150.
* Grauert, Hans; Remmert, Reinhold (1958). "Komplexe Räume". Mathematische Annalen 136 (3): 245–318. doi:10.1007/BF01362011. http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002286920.
* Grauert, H.; Remmert, R. (6 December 2012). Coherent Analytic Sheaves. Springer. ISBN 978-3-642-69582-7. https://books.google.com/books?id=blPxCAAAQBAJ.
* Grauert, H.; Peternell, Thomas; Remmert, R. (9 March 2013). Several Complex Variables VII: Sheaf-Theoretical Methods in Complex Analysis. Springer. ISBN 978-3-662-09873-8.
* Grothendieck, Alexander; Raynaud, Michèle (2002). "Revêtements étales et groupe fondamental§XII. Géométrie algébrique et géométrie analytique" (in fr). Revêtements étales et groupe fondamental (SGA 1). doi:10.1007/BFb0058656. ISBN 978-2-85629-141-2. https://link.springer.com/chapter/10.1007/BFb0058667.
* Hartshorne, Robin (1970). Ample Subvarieties of Algebraic Varieties. Lecture Notes in Mathematics. 156. doi:10.1007/BFb0067839. ISBN 978-3-540-05184-8. https://books.google.com/books?id=PC58CwAAQBAJ&pg=PA221.
* Hartshorne, Robin (1977). Algebraic Geometry. Graduate Texts in Mathematics. 52. Berlin, New York: Springer-Verlag. doi:10.1007/978-1-4757-3849-0. ISBN 978-0-387-90244-9. https://books.google.com/books?id=7z4mBQAAQBAJ&pg=PA438.
* Huckleberry, Alan (2013). "Hans Grauert (1930–2011)". Jahresbericht der Deutschen Mathematiker-Vereinigung 115: 21–45. doi:10.1365/s13291-013-0061-7.
* Remmert, Reinhold (1998). "From Riemann Surfaces to Complex Spaces". Séminaires et Congrès.
* 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. http://www.numdam.org/numdam-bin/item?id=AIF_1956__6__1_0.
* 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. https://books.google.com/books?id=MVck0twHKSIC&pg=PA163.
* "Chapter II. Preliminaries". Zariski-decomposition and Abundance. Mathematical Society of Japan Memoirs. 14. Mathematical Society of Japan. 2004. pp. 13–78. doi:10.2969/msjmemoirs/01401C020. ISBN 978-4-931469-31-0. http://projecteuclid.org/euclid.msjm/1389986108.
* 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. doi:10.5802/afst.1582.

## Future reading

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

## 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.
* Hazewinkel, Michiel, ed. (2001), "Analytic space", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page
* Hazewinkel, Michiel, ed. (2001), "Analytic set", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page

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