# Twistor constructions of quaternionic manifolds and asymptotically hyperbolic Einstein--Weyl spaces

### Reference:

Borowka, A., 2012. Twistor constructions of quaternionic manifolds and asymptotically hyperbolic Einstein--Weyl spaces. Thesis (Doctor of Philosophy (PhD)). University of Bath.

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### Abstract

Let \$S\$ be a \$2n\$-dimensional complex manifold equipped with a line bundle with a real-analytic complex connection such that its curvature is of type \$(1,1)\$, and with a real analytic h-projective structure such that its h-projective curvature is of type \$(1,1)\$. For \$n=1\$ we assume that \$S\$ is equipped with a real-analytic 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 h-projective 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 Feix--Kaledin 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 quaternion-K\"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 self-dual conformal \$4\$-manifold and from Jones--Tod correspondence we know that the quotient \$B\$ of \$M\$ by an \$S^1\$ action is an asymptotically hyperbolic Einstein--Weyl manifold. Using a result of LeBrun \cite{Le}, we prove that \$B\$ is an asymptotically hyperbolic Einstein--Weyl manifold. We also give a natural construction of a minitwistor space \$T\$ of an asymptotically hyperbolic Einstein--Weyl manifold directly from \$S\$, such that \$T\$ is the Jones--Tod quotient of \$Z\$. As a consequence, we deduce that the Einstein--Weyl manifold constructed using \$T\$ is equipped with a distinguished Gauduchon gauge.

### Details

 Item Type Thesis (Doctor of Philosophy (PhD)) Creators Borowka, A. Uncontrolled Keywords quaternionic manifolds,einstein--weyl manifolds,h-projective structures Departments Faculty of Science > Mathematical Sciences Status Unpublished ID Code 40580

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