Complex analytic variety

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

In mathematics, and in particular differential geometry and complex geometry, a complex analytic variety 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, 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;

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.

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

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. ISBN 978-3-540-38121-1. https://books.google.com/books?id=jR56CwAAQBAJ. 
• Gunning, Robert Clifford; Rossi, Hugo (2009). "Chapter III. Variety (Sec. B. Anlytic cover)". Analytic Functions of Several Complex Variables. ISBN 9780821821657. https://books.google.com/books?id=wsqFAwAAQBAJ&pg=PA101. 
• Gunning, Robert Clifford; Rossi, Hugo (2009). "Chapter V. Anlytic space". Analytic Functions of Several Complex Variables. 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. 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. ISBN 978-3-662-09873-8. 
• Grothendieck, Alexander; Raynaud, Michele (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%2FBFb0058667. 
• 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