Communications in Mathematical Physics

, Volume 324, Issue 2, pp 637–655 | Cite as

Conification of Kähler and Hyper-Kähler Manifolds



Given a Kähler manifold M endowed with a Hamiltonian Killing vector field Z, we construct a conical Kähler manifold \({\hat{M}}\) such that M is recovered as a Kähler quotient of \({\hat{M}}\). Similarly, given a hyper-Kähler manifold (M, g, J1, J2, J3) endowed with a Killing vector field Z, Hamiltonian with respect to the Kähler form of J1 and satisfying \({\mathcal{L}_ZJ_2 = -2J_3}\), we construct a hyper-Kähler cone \({\hat{M}}\) such that M is a certain hyper-Kähler quotient of \({\hat{M}}\). 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-Kähler manifold in the image of the c-map admits a Killing vector field with the above properties. Therefore, it gives rise to a hyper-Kähler cone, which in turn defines a quaternionic Kähler 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.


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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institute for Information Transmission ProblemsMoscowRussia
  2. 2.Masaryk UniversityBrnoCzech Republic
  3. 3.Department of Mathematics and Center for Mathematical PhysicsUniversity of HamburgHamburgGermany
  4. 4.Department of Mathematical SciencesUniversity of LiverpoolLiverpoolUK

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