Deformations of geometric structures on a subclass of LVM threefolds

We study deformations of geometric structures on some LVM manifolds of complex dimension $3$. More precisely, we study resonant structures, a particular type of $(G,X)$-structures, via the Ehresmann-Thurston principle, and their link with the deformation of the complex structure of the LVM manifold in the sense of Kodaira, Spencer and Kuranishi. We describe the Kuranishi family of these manifolds and construct a family containing all of them and complete at every point.