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 varietyComplex analytic variety (or just variety) is sometimes required to be irreducible

and (or) reduced 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,{{sfn|Hartshorne|1977|p=439}}

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;{{sfn|Hartshorne|1977|p=439}}

:Let X be 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 polynomial 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 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.{{harvtxt|Grothendieck|Raynaud|2002}} (SGA 1 §XII. Proposition 2.1.)

Note

Annotation

References

{{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. |last4=Dolbeault |first4=P. |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. Variety (Sec. B. Analytic covers)|chapter-url={{Google books|wsqFAwAAQBAJ|page=101|plainurl=yes}}| title=Analytic Functions of Several Complex Variables | isbn=9780821821657 | last1=Gunning | first1=Robert Clifford | last2=Rossi | first2=Hugo | year=2009 |publisher=American Mathematical Soc. }}
{{cite book |chapter=Chapter V. Analytic spaces|chapter-url={{Google books|wsqFAwAAQBAJ|page=150|plainurl=yes}}| title=Analytic Functions of Several Complex Variables | isbn=9780821821657 | last1=Gunning | first1=Robert Clifford | last2=Rossi | first2=Hugo | year=2009 |publisher=American Mathematical Soc. }}
{{cite journal |doi=10.1007/BF01362011|title=Komplexe Räume|year=1958|last1=Grauert|first1=Hans|last2=Remmert|first2=Reinhold|journal=Mathematische Annalen|volume=136|issue=3|pages=245–318|s2cid=121348794|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002286920}}
{{cite book |isbn=978-3-642-69582-7|title=Coherent Analytic Sheaves|last1=Grauert|first1=H.|last2=Remmert|first2=R.|date=6 December 2012|publisher=Springer |url={{Google books|title=Coherent Analytic Sheaves|blPxCAAAQBAJ|plainurl=yes}}}}
{{cite book |isbn=978-3-662-09873-8|title=Several Complex Variables VII: Sheaf-Theoretical Methods in Complex Analysis|last1=Grauert|first1=H.|last2=Peternell|first2=Thomas|last3=Remmert|first3=R.|date=9 March 2013|publisher=Springer }}
{{cite book |arxiv=math/0206203|last1=Grothendieck|first1=Alexander|author1link = Alexander Grothendieck|author2link = Michèle Raynaud|last2=Raynaud|first2=Michèle|title=Revêtements étales et groupe fondamental (SGA 1)|chapter =Revêtements étales et groupe fondamental§XII. Géométrie algébrique et géométrie analytique|year=2002|isbn=978-2-85629-141-2|chapter-url=https://link.springer.com/chapter/10.1007%2FBFb0058667|doi=10.1007/BFb0058656|language=fr}}
{{cite book | last1=Hartshorne | first1=Robin|doi=10.1007/BFb0067839|title=Ample Subvarieties of Algebraic Varieties |series=Lecture Notes in Mathematics |year=1970 |volume=156 |isbn=978-3-540-05184-8| url={{Google books|PC58CwAAQBAJ|plainurl=yes|page=221}}}}
{{Cite book| last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=Algebraic Geometry | series=Graduate Texts in Mathematics | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-0-387-90244-9 | mr=0463157 | zbl=0367.14001 | year=1977 | volume=52 | url={{Google books|7z4mBQAAQBAJ|Algebraic Geometry|page=438|plainurl=yes}}|doi=10.1007/978-1-4757-3849-0| s2cid=197660097 }}
{{cite journal |last1=Huckleberry |first1=Alan |title=Hans Grauert (1930–2011) |journal=Jahresbericht der Deutschen Mathematiker-Vereinigung |year=2013 |volume=115 |pages=21–45 |doi=10.1365/s13291-013-0061-7|arxiv=1303.6933|s2cid=119685542 }}
{{cite journal |last1=Remmert |first1=Reinhold |title=From Riemann Surfaces to Complex Spaces |journal=Séminaires et Congrès |date=1998|zbl=1044.01520}}
{{cite journal | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Géométrie algébrique et géométrie analytique | url=http://www.numdam.org/numdam-bin/item?id=AIF_1956__6__1_0 | mr=0082175 | year=1956 | journal=Annales de l'Institut Fourier | issn=0373-0956 | volume=6 | pages=1–42 | doi=10.5802/aif.59| doi-access=free }}
{{cite book |isbn=978-3-642-10944-7|title=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|last1=Tognoli|first1=A.|editor1-first=A|editor1-last=Tognoli|date=2 June 2011|url={{Google books|title=Singularities of Analytic Spaces: Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Bressanone (Bolzano)|MVck0twHKSIC|page=163|plainurl=yes}}|doi=10.1007/978-3-642-10944-7}}
{{cite book |doi=10.2969/msjmemoirs/01401C020|chapter=Chapter II. 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 }}

Future reading

{{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 }}

External links

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}}

Category:Algebraic geometry

Category:Several complex variables

Category:Complex geometry