C-projective symmetries of submanifolds in quaternionic geometry
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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-point set of a special type of quaternionic circle action. 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-point set of the circle action to quaternionic symmetries, and we study the quaternionic symmetries of the Calabi and Eguchi–Hanson hyperkähler structures, showing that in some cases all quaternionic symmetries are obtained in this way.
Keywordsc-projective structure Quaternionic structure Symmetries Submaximally symmetric spaces Calabi metric
Mathematics Subject Classification58D19 53B15 53A20 53C28
We would like to thank David Calderbank, Boris Kruglikov and Lenka Zalabova for helpful discussions and comments. This work was partially supported by the Simons Foundation Grant 346300 and the Polish Government MNiSW 2015-2019 matching fund, and by the Grant P201/12/G028 of the Grant Agency of the Czech Republic.
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