I would like to know if there exist eight-dimensional manifolds such that:


* It has SU(4)-structure.
* It is locally conformal Kahler.
* It is not a Calabi-Yau four-fold.


A weaker question that also interests me is if there exist an eight-dimensional manifold such that: 


* It is spin
* It is locally conformal Kahler.
* It is not a Calabi-Yau four-fold.


Finally, a related question would be:


* Are all spin manifolds simply connected?


Thanks.



Locally conformal Kahler manifolds with SU(4) structure
Here's an example: $M^8 = S^1\times S^7$.


This manifold is parallelizable, so it has an $\mathrm{SU}(4)$-structure.


It is diffeomorphic to the quotient of $\mathbb{C}^4\setminus\{0\}$ divided by the $\mathbb{Z}$-action generated by $z\mapsto 2z$, which preserves the standard Kähler structure up to a constant multiple, so it is locally conformally Kähler.


It is not a Kähler manifold, so it can't be Calabi-Yau, since it is compact and has vanishing second deRham cohomology group.


Because it is parallelizable, it is spinnable, but it is not simply connected.


