TITLE: Simply connected threefold with nef $K_X$ and $b_3=0$
QUESTION [5 upvotes]: Is there a simply connected smooth projective threefold $X$ with nef canonical divisor and vanishing third Betti number?
$X$ cannot have the same integral cohomology ring as the projective space (Fujita). $X$ cannot be of general type. The discussion in the introduction to this paper seems relevant.

REPLY [6 votes]: Such a threefold does not exist, even assuming only $b_1(X)=0$ instead of $\pi_1(X)=0$. Indeed the Miyaoka-Yau inequality holds for projective manifolds with $K$ nef -- see this paper.
For threefolds, this reads $K_X^3\leq -64\chi (\mathscr{O}_X)$, hence $\chi (\mathscr{O}_X)\leq 0$ because $K_X$ is nef.
Since $h^1(\mathscr{O}_X)=0$, this implies $h^3(\mathscr{O}_X)\geq 1$, hence $b_3>0$.