I am interested to know an example of a simply connected smooth projective 3-fold $X$ (over $\mathbb{C}$) satisfying the following two constraints:


1. $X$ has the same Betti numbers as $\mathbb{C}\mathbb{P}^{3}$ i.e. $b\_{1}(X) = b\_{3}(X) = 0$ and $b\_{2}(X) = 1$ and all of its cohomology groups are torsion-free.
2. $\mathrm{Kod}(X) \geq 0$.


($\mathrm{Kod}(X)$ denotes the Kodaira dimension). 



3-folds with "simple" Betti numbers and positive Kodaira dimension
Let me just mention that the non-existence of such a threefold is an immediate consequence of Yau's inequality. First, as explained in the above comment, the conditions $b\_2=1$ and $\mathrm{Kod}(X)\geq 0$ imply that $K\_X$ is ample. Then Yau gives $c\_1^3\geq \frac{8}{3}c\_1c\_2 $, which is equivalent by Riemann-Roch to $K\_X^3\leq 64 \chi (K\_X)$. But the conditions on the Betti numbers imply $H^i(X,\mathcal{O}\_X)=0$ for $i>0$, hence $\chi (K\_X)=-\chi (\mathcal{O}\_X)=-1$, a contradiction.


