Question

I'm looking for references on the proof (due to B. Moishezon, I guess) that any Moishezon space becomes a projective smooth complex variety after a finite number of blow-ups (called a modification?) His own articles on this that I could find are mainly in Russian, except a few survey papers (in English) without too much proof. I don't read Russian.

Could anyone give some references in English/French? Also, since I didn't read Moishezon's original paper, I don't know the precise statement for this result (e.g. if one can impose some constraints on the subspaces that we blow up). Can someone give the precise statement?

Answer 1

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 ).