Completeness in Supergravity Constructions

Abstract

We prove that the supergravity r- and c-maps preserve completeness. As a consequence, any component $${\mathcal{H}}$$ of a hypersurface {h = 1} defined by a homogeneous cubic polynomial h such that $${-\partial^2h}$$ is a complete Riemannian metric on $${\mathcal{H}}$$ defines a complete projective special Kähler manifold and any complete projective special Kähler manifold defines a complete quaternionic Kähler manifold of negative scalar curvature. We classify all complete quaternionic Kähler manifolds of dimension less or equal to 12 which are obtained in this way and describe some complete examples in 16 dimensions.