Dear Shenghao,

I) I am happy to report that according to Ueno's Classification of Algebraic Varieties and Compact Complex Spaces, Springer LNM 439, 1975, Moishezon's papers have been translated into English : AMS Translation Ser.2, 63 (1967), 51-177.

II) Here is the precise statement you request. Given a Moishezon variety $X$, you can obtain a smooth projective manifold $\tilde X$ from it by a finite succession of blowing-ups and the canonical morphism between their meromorphic function fields $\mathcal M (X) \to \mathcal M (\tilde {X})$ is an isomorphism of fields. If $X$ is smooth, the blow-ups can be taken with smooth centers.

III) And finally two facts you probably already know.

a) A Moishezon manifold is projective algebraic iff it is Kähler.

b) A smooth Moishezon surface (i.e. manifold of dimension two) is automatically projective. This follows from the fact that if you blow-up a point in a compact smooth manifold and get a projective manifold, then the original manifold was actually already projective.

Dear Shenghao,

I) I am happy to report that according to Ueno's Classification of Algebraic Varieties and Compact Complex Spaces, Springer LNM 439, 1975, Moishezon's papers have been translated into English : AMS Translation Ser.2, 63 (1967), 51-177.

II) Here is the precise statement you request. Given a Moishezon variety $X$, you can obtain a smooth projective manifold $\tilde X$ from it by a finite succession of blowing-ups and the canonical morphism between their meromorphic function fields $\mathcal M (X) \to \mathcal M (\tilde {X})$ is an isomorphism of fields. If $X$ is smooth, the blow-ups can be taken with smooth centers.

III) And finally two facts you probably already know.

a) A Moishezon manifold is projective algebraic iff it is Kähler.

b) A smooth Moishezon surface , i.e. manifold of dimension two, is automatically projective. This is a theorem by Chow and Kodaira ( proved long before Moishezon introduced his general definition ).