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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Projective geometry and the quaternionic Feix-Kaledin construction


Authors: Aleksandra W. Borówka and David M. J. Calderbank
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 53A20, 53C26; Secondary 32L25, 53B10, 53C28
DOI: https://doi.org/10.1090/tran/7719
Published electronically: January 4, 2019
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Abstract: Starting from a complex manifold $ S$ with a real-analytic c-projective structure whose curvature has type $ (1,1)$, and a complex line bundle $ \mathscr {L}\to S$ with a connection whose curvature has type $ (1,1)$, we construct the twistor space $ Z$ of a quaternionic manifold $ M$ with a quaternionic circle action which contains $ S$ as a totally complex submanifold fixed by the action. This extends a construction of hypercomplex manifolds, including hyperkähler metrics on cotangent bundles, obtained independently by Feix and Kaledin.

When $ S$ is a Riemann surface, $ M$ is a self-dual conformal $ 4$-manifold and the quotient of $ M$ by the circle action is an Einstein-Weyl manifold with an asymptotically hyperbolic end, and our construction coincides with the construction presented by Borówka. The extension also applies to quaternionic Kähler manifolds with circle actions, as studied by Haydys and Hitchin.


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Additional Information

Aleksandra W. Borówka
Affiliation: Institute of Mathematics, Jagiellonian University, ulica Professor Stanislawa Lojasiewicza 6, 30-348 Kraków, Poland
Email: Aleksandra.Borowka@uj.edu.pl

David M. J. Calderbank
Affiliation: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
Email: D.M.J.Calderbank@bath.ac.uk

DOI: https://doi.org/10.1090/tran/7719
Received by editor(s): January 31, 2018
Received by editor(s) in revised form: September 26, 2018
Published electronically: January 4, 2019
Additional Notes: We thank the Eduard Cech Institute and the Czech Grant Agency, grant no. P201/12/G028, for their hospitality and financial support.
Article copyright: © Copyright 2019 American Mathematical Society