Projective geometry and the quaternionic Feix–Kaledin construction

Borówka, Aleksandra; Calderbank, David

doi:10.1090/tran/7719

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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