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