Projective geometry and the quaternionic Feix-Kaledin construction

Aleksandra Borowka, David M. J. Calderbank

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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 L → 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.

Original languageEnglish
Pages (from-to)4729-4760
Number of pages32
JournalTransactions of the American Mathematical Society
Volume372
Issue number7
Early online date4 Jan 2019
DOIs
Publication statusPublished - 1 Oct 2019

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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