Mirror symmetry and generalized complex manifolds: Part II. Integrability and the transform for torus bundles

In this paper we continue the development of a relative version of T-duality in generalized complex geometry which we propose as a manifestation of mirror symmetry. We discuss the integrability of the transform from Part I in terms of data on the base manifold. We work with semi-flat generalized complex structures on real n-torus bundles with section over an n-dimensional base and use the transform on vector bundles developed in Part I of this paper to discuss the bijective correspondence between semi-flat generalized complex structures on pairs of dual torus bundles. We give interpretations of these results in terms of relationships between the cohomology of torus bundles and their duals. We comment on the ways in which our results generalize some well established aspects of mirror symmetry. Along the way, we give methods of constructing generalized complex structures on the total spaces of the bundles we consider.