TITLE: 3-folds with "simple" Betti numbers and positive Kodaira dimension
QUESTION [5 upvotes]: I am interested to know an example of a simply connected smooth projective 3-fold $X$ (over $\mathbb{C}$) satisfying the following two constraints:

$X$ 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.
$\mathrm{Kod}(X) \geq 0$.

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

REPLY [8 votes]: 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.