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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

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    $\begingroup$ Given a smooth surface you can blow down any curves with negative self intersection... $\endgroup$ – Darx Apr 17 '17 at 11:49
  • $\begingroup$ As darx points out, your conjectures (a) and (b) are false. You can begin with any curve in any surface, blow up a large number of closed points in the curve, and then contract the strict transform of the curve. $\endgroup$ – Jason Starr Apr 17 '17 at 12:11
  • $\begingroup$ If you impose that $Y$ has Kawamata log terminal singularities, then there are positive results. First of all, certainly $Z$ will not be rationally connected, but it is rationally chain connected if $Y$ is Moishezon. This follows from the solution of Shokurov's conjecture by Hacon and McKernan. $\endgroup$ – Jason Starr Apr 17 '17 at 12:16
  • $\begingroup$ Regarding (c), a rationally connected, Moishezon analytic space need not be simply connected: for instance a nodal plane cubic has infinite fundamental group. $\endgroup$ – Jason Starr Apr 17 '17 at 12:17
  • $\begingroup$ the question was silly - I am sorry. I should add the assumption that M is Calabi-Yau, $\endgroup$ – Misha Verbitsky Apr 17 '17 at 12:24
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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.

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This is proved in Hironaka, Flattening theorem in complex-analytic geometry, Corollary 2.

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