The Topology and Geometry of Hyperkahler Quotients

In this thesis we study the topology and geometry of hyperkahler quotients, as well as some related non-compact Kahler quotients, from the point of view of Hamiltonian group actions. The main technical tool we employ is Morse theory with moment maps. We prove a Lojasiewicz inequality which permits the use of Morse theory in the noncompact setting. We use this to deduce Kirwan surjectivity for an interesting class of non-compact quotients, and obtain a new proof of hyperkahler Kirwan surjectivity for hypertoric varieties. We then turn our attention to quiver varieties, obtaining an explicit inductive procedure to compute the Betti numbers of the fixed-point sets of the natural S1-action on these varieties. To study the kernel of the Kirwan map, we adapt the Jeffrey- Kirwan residue formula to our setting. The residue formula may be used to compute intersection pairings on certain compact subvarieties, and in good cases these provide a complete description of the kernel of the hyperkahler Kirwan map. We illustrate this technique with several detailed examples. Finally, we investigate the Poisson geometry of a certain family of Nakajima varieties. We construct an explicit Lagrangian fibration on these varieties by embedding them into Hitchin systems. This construction provides an interesting class of toy models of Hitchin systems for which the hyperkahler metric may be computed explicitly.