  We make a detailed investigation of all spaces Q_{n_1... n_N}^{q_1... q_N} of
the form of U(1) bundles over arbitrary products \prod_i CP^{n_i} of complex
projective spaces, with arbitrary winding numbers q_i 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_{n_1...
n_N}^{q_1... q_N} 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_{n_1... n_N}^{q_1... q_N} of more general utility. In particular, we
show that they always admit Einstein metrics, and that the spaces where
q_i=(n_i+1)/\ell all admit two Killing spinors. We also obtain an iterative
construction for real metrics on CP^n, and construct the Killing vectors on
Q_{n_1... n_N}^{q_1... q_N} in terms of scalar eigenfunctions on CP^{n_i}. We
derive bounds that allow us to prove that certain Killing-vector identities on
spheres, necessary for consistent Kaluza-Klein reductions, are never satisfied
on Q_{n_1... n_N}^{q_1... q_N}.
