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C-projective symmetries of submanifolds in quaternionic geometry

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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-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.

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Acknowledgements

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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Authors and Affiliations

  1. Institute of Mathematics, Jagiellonian University, 30-348, Kraków, Poland

    Aleksandra Borówka

  2. Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlarska 2, 611 37, Brno, Czechia

    Henrik Winther

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  1. Aleksandra Borówka
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  2. Henrik Winther
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Correspondence to Henrik Winther.

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Borówka, A., Winther, H. C-projective symmetries of submanifolds in quaternionic geometry. Ann Glob Anal Geom 55, 395–416 (2019). https://doi.org/10.1007/s10455-018-9631-3

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  • DOI: https://doi.org/10.1007/s10455-018-9631-3

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