Geometric model for complex non-Kanhler manifolds with SU (3) structure

For a given complex n-fold M we present an explicit construction of all complex (n + 1)-folds which are principal holomorphic T-2-fibrations over M. For physical applications we consider the case of M being a Calabi-Yau 2-fold. We show that for such M, there is a subclass of the 3-folds that we construct, which has natural families of non-Kahler SU(3)-structures satisfying the conditions for N = 1 supersymmetry in the heterotic string theory compactified on the 3-folds. We present examples in the aforementioned subclass with M being a K3-surface and a 4-torus.