Calabi-Yau-threefolds with Picard number $\rho (X)=2$ and their Kaehler cone I

We describe properties of the K"ahler cone of general Calabi-Yau-threefolds with Picard number $\rho (X)=2$ and prove the rationality of the K"ahler cone, if $X$ is a Calabi-Yau-hypersurface in a ${\mathbb{P}}^2$-bundle over ${\mathbb P}^2$and$c_3(X)\le -54$. Without the latter assumption we prove the positivity of $c_2(X)$.