From cubic polynomials to complete quaternionic Kähler manifolds

I will explain two supergravity constructions, which allow to construct certain special Riemannian manifolds starting with other special Riemannian manifolds. We show that the resulting manifolds are complete if the original manifolds are. By composition of the two constructions we obtain complete quaternionic Kähler manifolds out of certain cubic hypersurfaces. At the end I will formulate two open problems concerning such hypersurfaces.
The talk is based on arXiv:1101.5103 (hepth, mathdg).