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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Projective geometry and the quaternionic Feix–Kaledin construction
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by Aleksandra W. Borówka and David M. J. Calderbank PDF
Trans. Amer. Math. Soc. 372 (2019), 4729-4760 Request permission

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
  • MR Author ID: 622381
  • Email: D.M.J.Calderbank@bath.ac.uk
  • 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.
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 4729-4760
  • MSC (2010): Primary 53A20, 53C26; Secondary 32L25, 53B10, 53C28
  • DOI: https://doi.org/10.1090/tran/7719
  • MathSciNet review: 4009396