Let $X$ be a smooth complex manifold and $\phi:\; X \mapsto Y$ be 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 (a) $Z$ is rationally connected, (b) $Z$ has finite fundamental group. (c) $Z$ is Moishezon.

Of course, (a)+(c) implies (b) (this is due to Campana and Kollar I guess).

I think that there is a version of (a) in "Letters of birationalist" by Shokurov, but it was for algebraic manifolds. I would be very grateful for a reference or a simple argument.

Misha