Completeness of projective special K\"ahler and quaternionic K\"ahler manifolds

TL;DR

This paper proves that certain classes of projective special K"ahler manifolds are complete and can generate families of complete quaternionic K"ahler manifolds, expanding understanding of their geometric structures.

Contribution

It establishes conditions under which projective special K"ahler manifolds are complete and constructs associated quaternionic K"ahler families, including non-trivial deformations.

Findings

Every projective special K"ahler manifold with regular boundary behaviour is complete.

Complete projective special K"ahler manifolds with cubic prepotential produce quaternionic K"ahler families.

Examples include deformations of non-compact symmetric quaternionic K"ahler manifolds.

Abstract

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