Consistent Warped - Space Kaluza - Klein Reductions , Half - Maximal Gauged Supergravities and CP n Constructions

We obtain new consistent Kaluza-Klein embeddings of the gauged supergravities with half of maximal supersymmetry in dimensions D = 7, 6, 5 and 4. They take the form of warped embeddings in type IIA, type IIB, M-theory and type IIB respectively, and are obtained by performing Kaluza-Klein circle reductions or T-duality transformations on Hopf fibres in S 3 submanifolds of the previously-known sphere reductions. The new internal spaces are in some sense " mirror manifolds " that are dual to the original internal spheres. The vacuum AdS solutions of the gauged supergravities then give rise to warped products with these internal spaces. As well as these embeddings, which have singularities, we also construct new non-singular warped Kaluza-Klein embeddings for the D = 5 and D = 4 gauged supergravities. The geometry of the internal spaces in these cases leads us to study Fubini-Study metrics on complex projective spaces in some detail.