The Topology and Geometry of Hyperk\"ahler Quotients

TL;DR

This thesis explores the topology and geometry of hyperk"ahler quotients using Morse theory, providing new proofs, explicit computations, and models related to Nakajima and Hitchin systems.

Contribution

It introduces a Lojasiewicz inequality for non-compact quotients, proves hyperk"ahler Kirwan surjectivity, and develops explicit methods for Betti number computation and Poisson geometry analysis.

Findings

Proved a Lojasiewicz inequality for non-compact hyperk"ahler quotients.

Established hyperk"ahler Kirwan surjectivity for hypertoric varieties.

Developed an explicit inductive procedure for Betti number calculations.

Abstract

In this thesis we study the topology and geometry of hyperk\"ahler quotients, as well as some related non-compact K\"ahler 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 non-compact setting. We use this to deduce Kirwan surjectivity for an interesting class of non-compact quotients, and obtain a new proof of hyperk\"ahler 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 S^1 -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 hyperk\"ahler 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 hyperk\"ahler metric may be computed explicitly.