  The compact complex manifolds considered in this article are principal torus
bundles over a torus. We consider the Kodaira Spencer map of the complete
Appell Humbert family (introduced by the first author in Part I) and are able
to show that we obtain in this way a connected component of the space of
complex structures each time that the base dimension is two, the fibre
dimension is one, and a suitable topological condition is verified.
