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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?

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    $\begingroup$ Note that Moishezon surfaces are not important since every smooth Moishezon surface is projective: So the higher dimension can be useful . In fact for some singular cases, the projectivity of Moishezon surface is known, for example if $S$ be a normal Moishezon surface with at worst rational singularites then it is projective $\endgroup$
    – user21574
    Commented Jul 20, 2017 at 23:24
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    $\begingroup$ A Moishezon manifold is projective if and only if it is a Kähler manifold or if and only if it has a line bundle whose curvature is semi-positive and positive in at least one point due to Siu and Demailly $\endgroup$
    – user21574
    Commented Jul 22, 2017 at 16:40
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    $\begingroup$ There is a new projectivity criterion for Moishezon 3-folds $X$ due to Kollár which says that $X$ is projective if and only if there is no irreducible curve $C⊂X$ homologous to zero and $NE(X)\cap−\overline{NE(X)}=0$, where $NE(X)$ is the cone of effective curves in the vector space of 1-cycles modulo numerical equivalence. $\endgroup$
    – user21574
    Commented Jul 22, 2017 at 16:43
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    $\begingroup$ Any complex Moishezon manifold homeomorphic to $P^n_{\mathbb C}$ is isomorphic to $P^n_{\mathbb C}$. Any complex analytic global deformation of $P^n_{\mathbb C}$ is isomorphic to $P^n_{\mathbb C}$ $\endgroup$
    – user21574
    Commented Jul 23, 2017 at 3:37
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    $\begingroup$ Moishezon manifolds are balanced $\endgroup$
    – user21574
    Commented Jul 23, 2017 at 4:17

1 Answer 1

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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 ).

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