Moishezon manifolds

Let X be a compact Moishezon manifold which becomes projective after blowing up a smooth subvariety Y subset of X. We assume also that there exists a proper map rho : X --> X' onto a projective variety X' with rho(Y) a point, such that Pic(X/X') = Z and K-X is rho-big. We prove some inequalities between the dimensions of Y and X and we construct examples which shows the optimality of the inequalities. In the last section we discuss some differential geometry properties of these examples which lead to a conjecture.