Asian Journal of Mathematics

Volume 25 (2021)

Number 1

A class of cubic hypersurfaces and quaternionic Kähler manifolds of co‑homogeneity one

Pages: 1 – 30

DOI: https://dx.doi.org/10.4310/AJM.2021.v25.n1.a1

Authors

V. Cortés (Department of Mathematics and Center for Mathematical Physics, University of Hamburg, Germany)

M. Dyckmanns (Department of Mathematics and Center for Mathematical Physics, University of Hamburg, Germany)

M. Jüngling (Department of Mathematics and Center for Mathematical Physics, University of Hamburg, Germany)

D. Lindemann (Department of Mathematics and Center for Mathematical Physics, University of Hamburg, Germany)

Abstract

We classify all complete projective special real manifolds with reducible cubic potential, obtaining four series. For two of the series the manifolds are homogeneous, for the two others the respective automorphism group acts with co‑homogeneity one. Complete projective special real manifolds give rise to complete quaternionic Kähler manifolds via the supergravity $q$‑map, which is the composition of the supergravity $c$‑map and $r$‑map. We develop curvature formulas for manifolds in the image of the $q$‑map. Applying the $q$‑map to one of the above series of projective special real manifolds, we obtain a series of complete quaternionic Kähler manifolds, which are shown to be inhomogeneous (of co‑homogeneity one) based on our curvature formulas.

Keywords

projective special real manifolds, projective special Kähler manifolds, quaternionic Kähler manifolds, co‑homogeneity one

2010 Mathematics Subject Classification

Primary 53C26. Secondary 22F50, 53A15.

Received 23 January 2020

Accepted 13 March 2020

Published 30 September 2021