Completeness of projective special Kähler and quaternionic Kähler manifolds

Abstract

We prove that every projective special Kähler manifold with \emph{regular boundary behaviour} is complete and defines a family of complete quaternionic Kähler manifolds depending on a parameter $c\ge 0$. We also show that, irrespective of its boundary behaviour, every complete projective special Kähler manifold with \emph{cubic prepotential} gives rise to such a family. Examples include non-trivial deformations of non-compact symmetric quaternionic Kähler manifolds.

Comments are welcome, V2: revised after review, accepted for publication in the proceedings of the workshop "New perspectives in differential geometry: special metrics and quaternionic geometry" in honour of Simon Salamon