Title:

Twistor constructions of quaternionic manifolds and asymptotically hyperbolic EinsteinWeyl spaces

Let $S$ be a $2n$dimensional complex manifold equipped with a line bundle with a realanalytic complex connection such that its curvature is of type $(1,1)$, and with a real analytic hprojective structure such that its hprojective curvature is of type $(1,1)$. For $n=1$ we assume that $S$ is equipped with a realanalytic M\"obius structure. Using the structure on $S$, we construct a twistor space of a quaternionic $4n$manifold $M$. We show that $M$ can be identified locally with a neighbourhood of the zero section of the twisted (by a unitary line bundle) tangent bundle of $S$ and that $M$ admits a quaternionic $S^1$ action given by unit scalar multiplication in the fibres. We show that $S$ is a totally complex submanifold of $M$ and that a choice of a connection $D$ in the hprojective class on $S$ gives extensions of a complex structure from $S$ to $M$. For any such extension, using $D$, we construct a hyperplane distribution on $Z$ which corresponds to the unique quaternionic connection on $M$ preserving the extended complex structure. We show that, in a special case, the construction gives the FeixKaledin construction of hypercomplex manifolds, which includes the construction of hyperk\"ahler metrics on cotangent bundles. We also give an example in which the construction gives the quaternionK\"ahler manifold $\mathbb{HP}^n$ which is not hyperk\"ahler. We show that the same construction and results can be obtained for $n=1$. By convention, in this case, $M$ is a selfdual conformal $4$manifold and from JonesTod correspondence we know that the quotient $B$ of $M$ by an $S^1$ action is an asymptotically hyperbolic EinsteinWeyl manifold. Using a result of LeBrun \cite{Le}, we prove that $B$ is an asymptotically hyperbolic EinsteinWeyl manifold. We also give a natural construction of a minitwistor space $T$ of an asymptotically hyperbolic EinsteinWeyl manifold directly from $S$, such that $T$ is the JonesTod quotient of $Z$. As a consequence, we deduce that the EinsteinWeyl manifold constructed using $T$ is equipped with a distinguished Gauduchon gauge.
