MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
There is a chapter on modifications in the book "Several complex variables VII: sheaf-theoretical methods in complex analysis" by Grauert and Peternell. they don't give the proofs but refer to the english translation of Moishezon's paper that Georges mentions in his answer. – Gjergji Zaimi Jan 11 '11 at 12:01
up vote 5 down vote accepted

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 ).

share|cite|improve this answer
Thanks, Georges. I'll find the translation. – shenghao Jan 11 '11 at 19:36

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.