Kaluza-Klein consistency, Killing vectors and Kahler spaces

We make a detailed investigation of all spaces Q(n1...nN)(q1...qN) of the form of U(1) bundles over arbitrary products Pi (i) CPni of complex projective spaces, with arbitrary winding numbers qi over each factor in the base. Special cases, including Q(11)(11) (sometimes known as T-11), Q(111)(111) and Q(21)(32), are relevant for compactifications of type IIB and D = 11 supergravity. Remarkable 'conspiracies' allow consistent Kaluza-Klein S-5, S-4 and S-7 Sphere reductions of these theories that retain all the Yang-Mills fields of the isometry group in a massless truncation. We prove that such conspiracies do not occur for the reductions on the Q(n1...nN)(q1...qN) spaces, and that it is inconsistent to make a massless truncation in which the non-Abelian SU(n(i) + 1) factors in their isometry groups are retained. In the course of proving this we derive many properties of the spaces Q(n1...nN)(q1...qN) of more general utility. In particular, we show that they always admit Einstein metrics, and that the spaces where q(i) = (n(i) + 1)/l all admit two Killing spinors. We also obtain an iterative construction for real metrics on CPn, and construct the Killing vectors on Q(n1...nN)(q1...qN) in terms of scalar eigenfunctions on CPni. We derive bounds that allow us to prove that certain Killing-vector identities an spheres, necessary for consistent Kaluza-Klein reductions, are never satisfied on Q(n1...nN)(q1...qN).