A class of non-geometric M-theory compactification backgrounds

We study a particular class of supersymmetric M-theory eight-dimensional non-geometric compactification backgrounds to three-dimensional Minkowski space-time, proving that the global space of the non-geometric compactification is still a differentiable manifold, although with very different geometric and topological properties with respect to the corresponding standard M-theory compactification background: it is a compact complex manifold admitting a K\"ahler covering with deck transformations acting by holomorphic homotheties with respect to the K\"ahler metric. We show that this class of non-geometric compactifications evade the Maldacena-Nu\~nez no-go theorem by means of a mechanism originally developed by Mario Garc\'ia-Fern\'andez and the author for Heterotic Supergravity, and thus do not require $l_{P}$-corrections to allow for a non-trivial warp factor or four-form flux. We obtain an explicit compactification background on a complex Hopf four-fold that solves all the equations of motion of the theory. We also show that this class of non-geometric compactification backgrounds is equipped with a holomorphic principal torus fibration over a projective K\"ahler base as well as a codimension-one foliation with nearly-parallel $G_{2}$-leaves, making thus contact with the work of M. Babalic and C. Lazaroiu on the foliation structure of the most general M-theory supersymmetric compactifications.