C-projective symmetries of submanifolds in quaternionic geometry

TL;DR

This paper explores the relationship between c-projective and quaternionic geometries, demonstrating how symmetries in one setting extend to the other and analyzing specific hyperkähler structures.

Contribution

It establishes the connection between submaximally symmetric c-projective and quaternionic models, and provides conditions for symmetry extension between these geometries.

Findings

Submaximal c-projective model is a submanifold of a submaximal quaternionic model.

Conditions are given for extending c-projective symmetries to quaternionic symmetries.

All quaternionic symmetries of certain hyperkähler structures can be obtained via this construction.

Abstract

The generalized Feix--Kaledin construction shows that c-projective $2n$-manifolds with curvature of type $(1,1)$ are precisely the submanifolds of quaternionic $4n$-manifolds which are fixed points set of a special type of quaternionic $S^1$ action $v$. In this paper, we consider this construction in the presence of infinitesimal symmetries of the two geometries. First, we prove that the submaximally symmetric c-projective model with type $(1,1)$ curvature is a submanifold of a submaximally symmetric quaternionic model, and show how this fits into the construction. We give conditions for when the c-projective symmetries extend from the fixed points set of $v$ to quaternionic symmetries, and we study the quaternionic symmetries of the Calabi-- and Eguchi-Hanson hyperk\"ahler structures, showing that in some cases all quaternionic symmetries are obtained in this way.