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# 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, 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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