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)?
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1$\begingroup$ 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. $\endgroup$– Jérôme PoineauCommented Oct 23, 2013 at 6:31
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$\begingroup$ Just a nitpick: it's probably better to say "projective variety" rather than "projective space"; by convention the latter really means some $\mathbf{P}^n$. $\endgroup$– user5117Commented Oct 23, 2013 at 7:47
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$\begingroup$ And every (proper) morphism of projective varieties is in fact projective. $\endgroup$– nafCommented Oct 23, 2013 at 9:21
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$\begingroup$ @ArtiePrendergast-Smith: Thank you. That was what I meant of course. $\endgroup$– Jérôme PoineauCommented Oct 23, 2013 at 9:23
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$\begingroup$ @JérômePoineau: just to clarify, my comment was really meant for the OP, not you! $\endgroup$– user5117Commented Oct 23, 2013 at 12:28
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1 Answer
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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.
