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](https://mathoverflow.net/a/358661/178279) be of general type. The discussion in the introduction to [this paper](https://arxiv.org/abs/1801.10489) seems relevant.



Simply connected threefold with nef $K_X$ and $b_3=0$
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](https://arxiv.org/pdf/1611.05981.pdf).
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 $b\_3>0$.


