{{Short description|Generalization of a complex manifold that allows the use of singularities}} [[File:Cone intersects line.png|thumb|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'''<ref group="note">A complex analytic variety is sometimes required to be irreducible and (or) reduced.</ref> 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 <math>\mathbb{C}</math> by <math>\underline{\mathbb{C}}</math>. A '''<math>\mathbb{C}</math>-space''' is a locally ringed space <math>(X, \mathcal{O}_X)</math>, whose structure sheaf is an algebra over <math>\underline{\mathbb{C}}</math>.
Choose an open subset <math>U</math> of some complex affine space <math>\mathbb{C}^n</math>, and fix finitely many holomorphic functions <math>f_1,\dots,f_k</math> in <math>U</math>. Let <math>X=V(f_1,\dots,f_k)</math> be the common vanishing locus of these holomorphic functions, that is, <math>X=\{x\mid f_1(x)=\cdots=f_k(x)=0\}</math>. Define a sheaf of rings on <math>X</math> by letting <math>\mathcal{O}_X</math> be the restriction to <math>X</math> of <math>\mathcal{O}_U/(f_1, \ldots, f_k)</math>, where <math>\mathcal{O}_U</math> is the sheaf of holomorphic functions on <math>U</math>. Then the locally ringed <math>\mathbb{C}</math>-space <math>(X, \mathcal{O}_X)</math> is a '''local model space'''.
A '''complex analytic variety''' is a locally ringed <math>\mathbb{C}</math>-space <math>(X, \mathcal{O}_X)</math> 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;{{sfn|Hartshorne|1977|p=439}} if the structure sheaf is reduced, then the complex analytic space is called reduced.
An '''associated complex analytic space (variety)''' <math>X_h</math> is such that:{{sfn|Hartshorne|1977|p=439}}
:Let X be a scheme of finite type over <math>\mathbb{C}</math>, and cover X with open affine subsets <math>Y_i = \operatorname{Spec} A_i</math> (<math>X =\cup Y_i</math>) (Spectrum of a ring). Then each <math>A_i</math> is an algebra of finite type over <math>\mathbb{C}</math>, and <math>A_i \simeq \mathbb{C}[z_1, \dots, z_n]/(f_1,\dots, f_m)</math>, where <math>f_1,\dots, f_m</math> are polynomials in <math>z_1, \dots, z_n</math>, which can be regarded as a holomorphic functions on <math>\mathbb{C}</math>. Therefore, their set of common zeros is the complex analytic subspace <math>(Y_i)_h \subseteq \mathbb{C}</math>. Here, the scheme X is obtained by glueing the data of the sets <math>Y_i</math>, and then the same data can be used for glueing the complex analytic spaces <math>(Y_i)_h</math> into a complex analytic space <math>X_h</math>, so we call <math>X_h</math> an associated complex analytic space with X. The complex analytic space X is reduced if and only if the associated complex analytic space <math>X_h</math> is reduced.<ref>{{harvtxt|Grothendieck|Raynaud|2002}} (SGA 1 §XII. Proposition 2.1.)</ref>
==See also== *{{annotated link|Analytic space}} *{{annotated link|Complex algebraic variety}} *{{annotated link|GAGA}} *{{annotated link|Rigid analytic space}}
== Note == {{Reflist}}
== Annotation == {{Reflist|group=note}}
==References== {{refbegin}} *{{cite book |isbn=978-4-431-49822-3|title=Complex Analytic Desingularization|last1=Aroca|first1=José Manuel|last2=Hironaka|first2=Heisuke|last3=Vicente|first3=José Luis|date=3 November 2018|url={{Google books|title=Complex Analytic Desingularization|rw92DwAAQBAJ|page=6|plainurl=yes}}|doi=10.1007/978-4-431-49822-3}} *{{cite journal |doi=10.1007/BF01425536|url=https://www.researchgate.net/publication/226554588|title=De Rham cohomology of an analytic space|year=1969|last1=Bloom|first1=Thomas|last2=Herrera|first2=Miguel|journal=Inventiones Mathematicae|volume=7|issue=4|pages=275–296|bibcode=1969InMat...7..275B|s2cid=122113902}} *{{cite web |last1=Cartan |first1=H. |author1link = Henri Cartan|last2=Bruhat |first2=F.|author2link=François Bruhat |last3=Cerf |first3=Jean |author-link3=Jean Cerf |last4=Dolbeault |first4=P. |author-link4=Pierre Dolbeault |last5=Frenkel |first5=Jean |last6=Hervé |first6=Michel |last7=Malatian |last8=Serre |first8=J-P. |title=Séminaire Henri Cartan, Tome 4 (1951-1952) |url=http://www.numdam.org/volume/SHC_1951-1952__4/}} (no.10-13) *{{cite book |isbn=978-3-540-38121-1|title=Complex Analytic Geometry|last1=Fischer|first1=G.|date=14 November 2006|publisher=Springer |url={{Google books|title=Complex Analytic Geometry|jR56CwAAQBAJ|plainurl=yes}}}} *{{cite book |chapter=Chapter III. 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Preliminaries |title=Zariski-decomposition and Abundance |series=Mathematical Society of Japan Memoirs |year=2004 |volume=14 |pages=13–78 |publisher=Mathematical Society of Japan |isbn=978-4-931469-31-0 |url=http://projecteuclid.org/euclid.msjm/1389986108 }} *{{cite journal |doi=10.5802/afst.1582 |title=Local polar varieties in the geometric study of singularities |year=2018 |last1=Flores |first1=Arturo Giles |last2=Teissier |first2=Bernard |journal=Annales de la Faculté des Sciences de Toulouse: Mathématiques |volume=27 |issue=4 |pages=679–775 |s2cid=119150240 |arxiv=1607.07979 }} {{refend}}
== Future reading == {{refbegin}} *{{cite journal |doi=10.1365/s13291-013-0061-7 |title=Hans Grauert (1930–2011) |date=2013 |last1=Huckleberry |first1=Alan |journal=Jahresbericht der Deutschen Mathematiker-Vereinigung |volume=115 |pages=21–45 |s2cid=256084531 |arxiv=1303.6933 }} {{refend}}
== External links == {{refbegin}} * Kiran Kedlaya. 18.726 [https://ocw.mit.edu/courses/mathematics/18-726-algebraic-geometry-spring-2009/lecture-notes Algebraic Geometry] ([https://ocw.mit.edu/courses/mathematics/18-726-algebraic-geometry-spring-2009/lecture-notes/MIT18_726s09_lec22_gaga.pdf LEC # 30 - 33 GAGA])Spring 2009. Massachusetts Institute of Technology: MIT OpenCourseWare Creative Commons BY-NC-SA. * [https://www.jirka.org/scv/ Tasty Bits of Several Complex Variables] (p. 137) open source book by Jiří Lebl BY-NC-SA. *{{Eom| title = Analytic space| author-last1 = Onishchik| author-first1 =A.L.| oldid = 45182}} *{{Eom| title = Analytic set| author-last1 = El'kin| author-first1 =A.G. | oldid = 45180}} {{refend}}
Category:Algebraic geometry Category:Several complex variables Category:Complex geometry