We consider the problem of finding exact, eleven-dimensional, BPS supergravity solutions in which the compaction involves a non-trivial Calabi-Yau manifold, $cY$, as opposed to simply a $T^6$. Since there are no explicitly-known metrics on non-trivial, compact Calabi-Yau manifolds, we use a non-compact “local model'' and take the compactification manifold to be $cY = cM_{GH} times T^2$ where $cM_{GH}$ is a hyper-K”ahler, Gibbons-Hawking ALE space. We focus on backgrounds with three electric charges in five dimensions and find exact families of solutions to the BPS equations that have the same four supersymmetries as the three-charge black hole. Our exact solution to the BPS system requires that the Calabi-Yau manifold be fibered over the space-time using compensators on $cY$. The role of the compensators is to ensure smoothness of the eleven-dimensional metric when the moduli of $cY$ depend on the space-time. We examine the equations of motion and discuss the brane distributions on generic internal manifolds that do not have enough symmetry to allow smearing.
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