Conification of Kähler and Hyper-Kähler Manifolds

• Published in: Communications in mathematical physics Vol. 324; no. 2; pp. 637 - 655
• Main Authors: Alekseevsky, D. V., Cortés, V., Mohaupt, T.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2013
• Subjects: Classical and Quantum Gravitation; Complex Systems; Mathematical and Computational Physics; Mathematical Physics; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Theoretical; Manifold; Positive Scalar Curvature; Riemannian Submersion; Horizontal Lift; Scalar Curvature
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-013-1812-0

Given a Kähler manifold M endowed with a Hamiltonian Killing vector field Z , we construct a conical Kähler manifold M ^ such that M is recovered as a Kähler quotient of M ^ . Similarly, given a hyper-Kähler manifold ( M , g , J 1 , J 2 , J 3 ) endowed with a Killing vector field Z , Hamiltonian with respect to the Kähler form of J 1 and satisfying L Z J 2 = - 2 J 3 , we construct a hyper-Kähler cone M ^ such that M is a certain hyper-Kähler quotient of 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.