Moishezon manifold

{{Short description|Compact complex manifold in algebraic geometry}}

In mathematics, a Moishezon manifold {{mvar|M}} is a compact complex manifold such that the field of meromorphic functions on each component {{mvar|M}} has transcendence degree equal the complex dimension of the component:

: $\dim_\mathbf{C}M=a(M)=\operatorname{tr.deg.}_\mathbf{C}\mathbf{C}(M).$

Complex algebraic varieties have this property, but the converse is not true: Hironaka's example gives a smooth 3-dimensional Moishezon manifold that is not an algebraic variety or scheme. {{harvtxt|Moishezon|1967|loc=Chapter I, Theorem 11|authorlink=Boris Moishezon}} showed that a Moishezon manifold is a projective algebraic variety if and only if it admits a Kähler metric. {{harvtxt|Artin|1970}} showed that any Moishezon manifold carries an algebraic space structure; more precisely, the category of Moishezon spaces (similar to Moishezon manifolds, but are allowed to have singularities) is equivalent with the category of algebraic spaces that are proper over {{math|Spec(C)}}.

References

{{Citation | last1=Artin | first1=M. |authorlink=Michael Artin| title=Algebraization of formal moduli, II. Existence of modification | year=1970 | journal=Ann. of Math. | volume=91 | issue=1 | pages=88–135|jstor=1970602|doi=10.2307/1970602}}
{{cite book | last1=Moishezon | first1=B.G.| title=Seven Papers on Algebra, Algebraic Geometry and Algebraic Topology |chapter=On n-dimensional compact varieties with n algebraically independent meromorphic functions, I, II and III (1966) (English translation version)|series=American Mathematical Society Translations: Series 2 |year=1967 |volume=63 |isbn=9780821844335|doi=10.1090/trans2/063 }}
{{cite journal | last1=Moishezon | first1=B.G. |url=http://mi.mathnet.ru/eng/izv/v30/i1/p133|title=B. G. Moishezon, "On n-dimensional compact complex manifolds having n algebraically independent meromorphic functions. I"|pages = 133–174 | year=1966 | journal=Izv. Akad. Nauk SSSR Ser. Mat. | volume=30|issue=1}}
{{cite journal | last1=Moishezon | first1=B.G. |url=http://mi.mathnet.ru/eng/izv/v30/i2/p345|title=B. G. Moishezon, "On n-dimensional compact complex manifolds having n algebraically independent meromorphic functions. II"|pages = 345–386 | year=1966 | journal=Izv. Akad. Nauk SSSR Ser. Mat. | volume=30|issue=2}}
{{cite journal | last1=Moishezon | first1=B.G. |url=http://mi.mathnet.ru/eng/izv/v30/i3/p621|title=B. G. Moishezon, "On n-dimensional compact complex manifolds having n algebraically independent meromorphic functions. III"|pages = 621–656 | year=1966 | journal=Izv. Akad. Nauk SSSR Ser. Mat. | volume=30|issue=3}}
{{citation|first=B.|last=Moishezon|chapter=Algebraic varieties and compact complex spaces|title=Proc. Internat. Congress Mathematicians (Nice, 1970)|volume=2|publisher=Gauthier-Villars|year=1971|pages=643–648|mr=0425189|chapter-url=https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1970.2/ICM1970.2.ocr.pdf|url=http://ada00.math.uni-bielefeld.de/ICM/ICM1970.2/|access-date=2013-06-14|archive-url=https://web.archive.org/web/20150213112226/http://ada00.math.uni-bielefeld.de/ICM/ICM1970.2/|archive-date=2015-02-13|url-status=dead}}

Category:Algebraic geometry

Category:Analytic geometry