Conification of Kähler and hyper-Kähler manifolds
Abstract
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.