Conification of Kahler and Hyper-Kahler Manifolds

Given a Kahler manifold M endowed with a Hamiltonian Killing vector field Z, we construct a conical Kahler manifold such that M is recovered as a Kahler quotient of . Similarly, given a hyper-Kahler manifold (M, g, J (1), J (2), J (3)) endowed with a Killing vector field Z, Hamiltonian with respect to the Kahler form of J (1) and satisfying , we construct a hyper-Kahler cone such that M is a certain hyper-Kahler quotient of . In this way, we recover a theorem by Haydys. Our work is motivated by the problem of relating the supergravity c-map to the rigid c-map. We show that any hyper-Kahler manifold in the image of the c-map admits a Killing vector field with the above properties. Therefore, it gives rise to a hyper-Kahler cone, which in turn defines a quaternionic Kahler manifold. Our results for the signature of the metric and the sign of the scalar curvature are consistent with what we know about the supergravity c-map.