Andrew Swann

HyperKähler and quaternionic Kähler geometry

 A quaternion-Hermitian manifold, of dimension at least 12, with closed fundamental 4-form is shown to be quaternionic Kahler. A similar result is proved for 8-manifolds. HyperKahler metrics are constructed on the fundamental quaternionic line bundle (with the zero-section removed) of a quaternionic Kahler manifold (indefinite if the scalar curvature is negative). This construction is compatible with the quaternionic Kahler and hyperKahier quotient constructions and allows quaternionic Kahler geometry to be subsumed into the theory of hyperKahler manifolds. It is shown that the hyperKahler metrics that arise admit a certain type of SU (2)- action, possess functions which are Kahler potentials for each of the complex structures simultaneously and determine quaternionic Kahler structures via a variant of the moment map construction. Quaternionic Kahler metrics are also constructed on the fundamental quaternionic line bundle and a twistor space analogy leads to a construction of hyperKahler metrics with circle actions on complex line bundles over Kahler-Einstein (complex) contact manifolds. Nilpotent orbits in a complex semi-simple Lie algebra, with the hyperKahler metrics defined by Kronheimer, are shown to give rise to quaternionic Kahler metrics and various examples of these metrics are identified. It is shown that any quaternionic Kahler manifold with positive scalar curvature and sufficiently large isometry group may be embedded in one of these manifolds. The twistor space structure of the projectivised nilpotent orbits is studied.

Cohomogeneity-one G2-structures

Abstract G 2 -manifolds with a cohomogeneity-one action of a compact Lie group G are studied. For G simple, all solutions with holonomy G 2 and weak holonomy G 2 are classified. The holonomy G 2 solutions are necessarily Ricci-flat and there is a one-parameter family with SU(3)-symmetry. The weak holonomy G 2 solutions are Einstein of positive scalar curvature and are uniquely determined by the simple symmetry group. During the proof the equations for G 2 -symplectic and G 2 -cosymplectic structures are studied and the topological types of the manifolds admitting such structures are determined. New examples of compact G 2 -cosymplectic manifolds and complete G 2 -symplectic structures are found.

The Einstein-Weyl equations in complex and quaternionic geometry*

Abstract Einstein–Weyl manifolds with compatible complex structures are shown to be given as torus bundles on Kahler–Einstein manifolds, extending known results on locally conformal Kahler manifolds. The Weyl structure is derived from a Ricci-flat metric constructed by Calabi on the canonical bundle of the Kahler–Einstein manifold. Similar questions are addressed when the Weyl geometry admits compatible hypercomplex or quaternionic structures.

Twisting Hermitian and hypercomplex geometries

A twist construction for manifolds with torus action is described generalising certain T-duality examples and constructions in hypercomplex geometry. It is applied to complex, SKT, hypercomplex and HKT manifolds to construct compact simply-connected examples. In particular, we find hypercomplex manifolds that admit no compatible HKT metric, and HKT manifolds whose Obata connection has holonomy contained in

Hypercomplex structures associated to quaternionic manifolds

SL(n,\mathbb H)

CLASSICAL NILPOTENT ORBITS AS HYPERKÄHLER QUOTIENTS

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Hyperkähler metrics of cohomogeneity one

Abstract If M is a quaternionic manifold and P is an S1-instanton over M, then Joyce constructed a hypercomplex manifold we call ℘(M) over M. These hypercomplex manifolds admit a U(2)-action of a special type permuting the complex structures. We show that up to double covers, all such hypercomplex manifolds arise in this way. Examples, including that of a hypercomplex structure on SU(3), show the necessity of including double covers of ℘(M) .

G2-STRUCTURES WITH TORSION FROM HALF-INTEGRABLE NILMANIFOLDS

We show that on an arbitrary nilpotent orbit in where is a direct sum of classical simple Lie algebras, there is a G-invariant hyperKahler structure obtainable as a hyperKaher quotient of the flat hyperKahler manifold ℝ4N≅ℍN. Coincidences between various low-dimensional simple Lie groups lead to some nilpotent orbits being described as hyperKahler quotients (in some cases in fact finite quotients) of other nilpotent orbits. For example, from the construction we are able to read off pairs of orbits in different classical Lie algebras such that there is a finite -equivariant surjection between the orbit closures. We include a table listing examples of hyperKahler quotients between small nilpotent orbits. The above-mentioned results have consequences in quaternionic Kahler geometry: it is known that nilpotent orbits in complex semisimple Lie algebras give rise to quaternionic Kahler manifolds. Our approach gives a more direct proof of this in the classical case as these manifolds turn out to be quaternionic Kahler quotients of quaternionic projective spaces. We find that many of these manifolds can also be constructed as quaternionic Kahler quotients of complex Grassmannians .

Multi-moment maps

Abstract Irreducible hyperkahler manifolds of dimension greater than four admitting a cohomogeneity-one action of a compact simple Lie group are classified via coadjoint orbits. It is shown that the only complete example is the Calabi metric on T ∗ C P (n) .

Hyperkähler metrics associated to compact Lie groups

Abstract The equations for a G 2 -structure with torsion on a product M 7 = N 6 × S 1 are studied in relation to the induced SU ( 3 ) -structure on N 6 . All solutions are found in the case when the Lee-form of the G 2 -structure is non-zero and N 6 is a six-dimensional nilmanifold with half-integrable SU ( 3 ) -structure. Special properties of the torsion of these solutions are discussed.