Implosions and hypertoric geometry

Hyperkahler manifolds occupy a special position at the intersection of Riemannian, symplectic and algebraic geometry. A hyperkahler structure involves a Riemannian metric, as well as a triple of complex structures satisfying the quaternionic relations. Moreover we require that the metric is Kahler with respect to each complex structure, so we have a triple (in fact a whole two-sphere) of symplectic forms. Of course, there is no Darboux theorem in hyperkahler geometry because the metric contains local information. However, many of the constructions and results of symplectic geometry, especially those related to moment maps, do have analogues in the hyperkahler world. The prototype is the hyperkahler quotient construction, and more recent examples include hypertoric varieties and cutting. In this article we shall explore a hyperkahler analogue of Guillemin, Jeffrey and Sjamaar's construction of symplectic implosion. This may be viewed as an abelianisation procedure: given a symplectic manifold M with a Hamiltonian action of a compact group K, the implosion Mimpl is a new symplectic space with an action of the maximal torus T of K, such that the symplectic reductions of Mimpl by T agree with the reductions of M by K. However the implosion is usually not smooth but is a singular space with a stratified symplectic structure. The implosion of the cotangent bundle T*K acts as a universal object here; implosions of general Hamiltonian K-manifolds may be defined using the symplectic implosion (T*K) impl . This space also has an algebrogeometric description as the geometric invariant theory quotient of KC by a maximal unipotent subgroup N .