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Given a hypercomplex manifold with a rotating vector field (and additionaldata), we construct a conical hypercomplex manifold. As a consequence, weassociate a quaternionic manifold to a hypercomplex manifold of the samedimension with a rotating vector field. This is a generalization of theHK/QK-correspondence. As an application, we show that a quaternionic manifoldcan be associated to a conical special complex manifold of half its dimension.Furthermore, a projective special complex manifold (with a canonicalc-projective structure) associates with a quaternionic manifold. The latter isa generalization of the supergravity c-map. We do also show that the tangentbundle of any special complex manifold carries a canonical Ricci-flathypercomplex structure, thereby generalizing the rigid c-map.

The H/Q-correspondence and a generalization of the supergravity c-map