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:

\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. Moishezon (1966, Chapter I, Theorem 11) 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, JSTOR 1970602
Moishezon, B.G. (1966), "On n-dimensional compact varieties with n algebraically independent meromorphic functions, I, II and III", Izv. Akad. Nauk SSSR Ser. Mat., 30: 133–174 345–386 621–656 English translation. AMS Translation Ser. 2, 63 51-177
Moishezon, B. (1971), "Algebraic varieties and compact complex spaces", Proc. Internat. Congress Mathematicians (Nice, 1970), 2, Gauthier-Villars, pp. 643–648, MR 0425189

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