Marsden-Weinstein Reductions for Kähler, Hyperkähler and Quaternionic Kähler Manifolds

If a Lie group G acts on a symplectic manifold (M, ω) and preserves the symplectic form ω, then in some cases there may exist a moment map ([C]) Φ from M to the dual of the Lie algebra. When the action is (locally) free, the preimage of a point in the dual of the Lie algebra modulo the isotropy group of this point will still be a symplectic manifold (orbifold). This process is called the MarsdenWeinstein reduction (MW reduction) and the reduced manifold is called MarsdenWeinstein quotient (MW quotient) ([MW]). It has been generalized to hyperkähler and quaternionic Kähler maniflods by Hitchin et al. ([HKLR]) and Galicki and Lawson ([GL]), respectively. In this term paper, we will show how MarsdenWeinstein reduction works in Kähler, hyperkähler and quaternionic Kähler cases and then give some examples to see how MW reduction gives a new approach to get manifolds or orbifolds in each case.