Projective geometry and the quaternionic Feix-Kaledin construction

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 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 hyperkaehler metrics on cotangent bundles, obtained independently by B. Feix and D. 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 a construction presented by the first author in a previous paper. The extension also applies to quaternionic Kaehler manifolds with circle actions, as studied by A. Haydys and N. Hitchin.

28 pages, (v2) added material on Swann bundles, quaternionic Kaehler metrics and the Haydys-Hitchin correspondence, (v3) refereed version, restructured content, to appear in TAMS