MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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

Is there something like the Chow lemma for complex spaces (stating that every compact complex space is birational to a projective variety, or some variant of this, like: every proper morphism of complex spaces is birational to a projective morphism)?

share|cite|improve this question
For projective spaces, the transcendence degree of the field of meromorphic functions is equal to the dimension. For arbitrary compact analytic spaces, it can be smaller. So asking for a birational map is too much. – Jérôme Poineau Oct 23 '13 at 6:31
Just a nitpick: it's probably better to say "projective variety" rather than "projective space"; by convention the latter really means some $\mathbf{P}^n$. – user5117 Oct 23 '13 at 7:47
And every (proper) morphism of projective varieties is in fact projective. – ulrich Oct 23 '13 at 9:21
@ArtiePrendergast-Smith: Thank you. That was what I meant of course. – Jérôme Poineau Oct 23 '13 at 9:23
@JérômePoineau: just to clarify, my comment was really meant for the OP, not you! – user5117 Oct 23 '13 at 12:28
up vote 4 down vote accepted

As pointed out in the comments, if a complex compact manifold $X$ of dimension $n$ is bimeromorphic to a projective variety, its field of meromorphic functions must have transcendence degree $n$ (the maximum possible) -- one says that $X$ is a Moisezon manifold. Conversely, a deep theorem of Moisezon asserts that any compact Moisezon manifold becomes projective after a finite number of blowing ups with smooth centers.

share|cite|improve this answer

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.