# 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:

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