We study singular hyperkähler quotients of the cotangent bundle of a complex semisimple Lie group as stratified spaces whose strata are hyperkähler. We focus on one particular case where the stratification satisfies the frontier condition and the partial order on the set of strata can be described explicitly by Lie theoretic data.

When a compact Lie group acts freely and in a Hamiltonian way on a symplectic manifold, the Marsden-Weinstein theorem says that the reduced space is a smooth symplectic manifold. If we drop the freeness assumption, the reduced space might be singular, but Sjamaar-Lerman (1991) showed that it can still be partitioned into smooth symplectic manifolds which "fit together nicely" in the sense that they form a stratification. In this paper, we prove a hyperkähler analogue of this statement, using the hyperkähler quotient construction. We also show that singular hyperkähler quotients are complex spaces which are locally biholomorphic to affine complex-symplectic GIT quotients with biholomorphisms that are compatible with natural holomorphic Poisson brackets on both sides.

Abstract. The equations of motion of a charged particle in the field of Yang's SU(2) monopole in 5-dimensional Euclidean space are derived by applying the Kaluza-Klein formalism to the principal bundle R 8 \ {0} → R 5 \ {0} obtained by radially extending the Hopf fibration S 7 → S 4 , and solved by elementary methods. The main result is that for every particle trajectory r : I → R 5 \ {0}, there is a 4-dimensional cone with vertex at the origin on which r is a geodesic. We give an explicit expression of the cone for any initial conditions.

Hyperkähler quotients by non-free actions are typically singular, but are nevertheless partitioned into smooth hyperkähler manifolds. We show that these partitions are topological stratifications, in a strong sense. We also endow the quotients with global Poisson structures which recover the hyperkähler structures on the strata. Finally, we give a local model which shows that these quotients are locally isomorphic to linear complex-symplectic reductions in the GIT sense. These results can be thought of as the hyperkähler analogues of Sjamaar–Lerman’s theorems for singular symplectic reduction. They are based on a local normal form for the underlying complex-Hamiltonian manifold, which may be of independent interest.

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