2012/05/14 by Dmitri V. Alekseevsky, Vicente Cortés, Alekseevsky, Dmitri V. +3
Mathematics · Physics and Astronomy · #53C26 #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #hep-th #math.DG #msc:53C26
paper · pdf · doi:10.48550/arxiv.1205.2964
conjecture replaced by reference
arxiv created 2012/07/17 · arxiv updated 2012/07/19
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,J1,J2,J3) endowed with a Killing vector field Z, Hamiltonian with respect to the Kähler form of J1 and satisfying LZJ2= -2J3, 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.