Does there exist a spin (i.e. $\frac{c_{1}}{2} \in H^{2}(M,\mathbb{Z})$) smooth complex projective $6$-fold with signature $\pm 16$?
The motivation is the Rochlin-Ochanine theorem, which says that $16$ is the minimal positive integer which can occur as the signature. I know a smooth manifold which satisfies all the other conditiions but it is far from complex projective (i.e. $S \times \mathbb{HP}^2$, $S$ a K3 surface, then reverse the orientation).
Note that I am now actually asking two questions since an answer for signature $-16$ doesn't provide an answer for signature $16$ and vice versa. But, with the original motivation of the Rochlin Ochanine theorem in mind, I will accept an answer for either. Ideally it would be great to know about both.
