A toric variety is a complex algebraic variety containing a dense orbit of a complex torus; its structure is completely determined by the datum of a polyhedral fan, this reduces the computation of its algebraic and topological invariants to a combinatoric exercise. When a smoothness condition is satisfied, toric varieties can be obtained as symplectic reductions of an open subset of the complex euclidean space as well; this makes them an important class of Kähler manifolds. In fact, by applying a “partial reduction“ of the same set, it is possible to associate to any smooth polyhedral fan, a holomorphic torus bundle over the corresponding toric variety. The total spaces of these bundles appear in literature with multiple names: LVM manifolds, moment-angle manifolds or generalized Calabi-Eckmann fibrations. Indeed, the well known Calabi-Eckmann manifolds are a special case of this construction. Since generalized Calabi-Eckamann manifolds are systematically non-Kähler, it is natural to wonder whether this class contains new examples of Hermitian manifolds with special metrics. In this talk I will discuss a sufficient criterion based on the computation of the Euler class of the corresponding bundle.