Moishezon manifold

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

[math]\displaystyle{ \dim_\mathbf{C}M=a(M)=\operatorname{tr.deg.}_\mathbf{C}\mathbf{C}(M). }[/math]

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. (Moishezon 1967) showed that a Moishezon manifold is a projective algebraic variety if and only if it admits a Kähler metric. (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 Spec(C).

References

• Artin, M. (1970), "Algebraization of formal moduli, II. Existence of modification", Ann. of Math. 91: 88–135, doi:10.2307/1970602 
• Moishezon, B.G. (1967). "On n-dimensional compact varieties with n algebraically independent meromorphic functions, I, II and III (1966) (English translation version)". Seven Papers on Algebra, Algebraic Geometry and Algebraic Topology. American Mathematical Society Translations: Series 2. 63. doi:10.1090/trans2/063. ISBN 9780821844335. 
  • Moishezon, B.G. (1966). "B. G. Moishezon, "On n-dimensional compact complex manifolds having n algebraically independent meromorphic functions. I"". Izv. Akad. Nauk SSSR Ser. Mat. 30 (1): 133–174. http://mi.mathnet.ru/eng/izv/v30/i1/p133. 
  • Moishezon, B.G. (1966). "B. G. Moishezon, "On n-dimensional compact complex manifolds having n algebraically independent meromorphic functions. II"". Izv. Akad. Nauk SSSR Ser. Mat. 30 (2): 345–386. http://mi.mathnet.ru/eng/izv/v30/i2/p345. 
  • Moishezon, B.G. (1966). "B. G. Moishezon, "On n-dimensional compact complex manifolds having n algebraically independent meromorphic functions. III"". Izv. Akad. Nauk SSSR Ser. Mat. 30 (3): 621–656. http://mi.mathnet.ru/eng/izv/v30/i3/p621. 
• Moishezon, B. (1971), "Algebraic varieties and compact complex spaces", Proc. Internat. Congress Mathematicians (Nice, 1970), 2, Gauthier-Villars, pp. 643–648, https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1970.2/ICM1970.2.ocr.pdf, retrieved 2013-06-14