Let $X$ be a smooth complex manifold and
$\phi:\; X \mapsto Y$ a proper holomorphic
map which is birational ("birational contraction"),
and $Z= \phi^{-1}(y)$ its fiber in a point $y$.
The variety $Y$ is not assumed to be smooth.
In this case I think that $Z$ is Moishezon.


I would be very grateful for a reference or a simple argument.


Misha



fibers of birational contraction for complex manifolds - are they Moishezon?
Up to taking a log resolution of $(X,Z)$ (which works in the analytic category), you may as well assume that $Z$ is a simple normal crossings divisor. Now I expect that the "usual" argument proves that every irreducible component $Z\_i$ of $Z$ is Moishezon. Iteratively do the following. First, blow up $y$ in $Y$ to obtain $Y\_1\to Y$. The exceptional locus is projective (by construction). The morphism $\phi$ extends to a bimeromorphism that is regular on an open subset whose complement has codimension $\geq 2$. If $Z\_i$ maps generically finitely to an exceptional locus, then $Z\_i$ is Moishezon. If $Z\_i$ does not map generically finitely, then the closure of the image of $Z\_i$ is a closed analytic subvariety of the exceptional divisor. By Chow's Theorem, this is an algebraic subvariety of the exceptional divisor. Thus, blow up this algebraic subvariety and repeat. Everytime that we blow up, we "fit" another coherent sheaf between the locally free sheaf $\Omega\_{X/\text{Spec}(\mathbb{C})}$ and its coherent, torsion quotient $\Omega\_{X/Y}$, i.e., we produce an ascending chain of coherent subsheaves of $\Omega\_{X/\text{Spec}(\mathbb{C})}$ containing the image of $\phi^\*\Omega\_{Y/\text{Spec}(\mathbb{C})}$. This should terminate, so that finally $Z\_i$ admits a generically finite, meromorphic transformation to a projective complex analytic variety. 


