Projective geometry and the quaternionic Feix-Kaledin construction

TL;DR

This paper generalizes the Feix-Kaledin construction to produce quaternionic manifolds from complex manifolds with specific structures, extending known hypercomplex and hyperkaehler cases to broader geometric contexts.

Contribution

It introduces a new quaternionic construction based on complex manifolds with c-projective structures, unifying and extending previous hypercomplex and hyperkaehler constructions.

Findings

Constructs quaternionic manifolds from complex manifolds with (1,1) curvature

Extends hypercomplex and hyperkaehler constructions to new settings

Connects to self-dual conformal 4-manifolds and Einstein-Weyl geometry

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.