2004/05/12 by Nicholas J. Proudfoot
Mathematics · #math.AG #math.DG #math.SG #msc:53C26 #msc:53D20 #msc:14D20 #msc:52C35
published as Ph. D. thesis, U.C. Berkeley, Spring 2004 · 86 pages, 10 figures
arxiv created 2004/05/12 · arxiv updated 2009/12/01
Let X be a Kahler manifold that is presented as a Kahler quotient of Cn by the linear action of a compact group G. We define the hyperkahler analogue M of X as a hyperkahler quotient of the cotangent bundle T^*Cn by the induced G-action. Special instances of this construction include hypertoric varieties and quiver varieties. Our aim is to provide a unified treatment of these two previously studied examples, with specific attention to the geometry and topology of the circle action on M that descends from the scalar action on the fibers of the cotangent bundle. We provide a detailed study of this action in the cases where M is a hypertoric variety or a hyperpolygon space. Most of this document consists of material from the papers math.DG/0207012, math.AG/0308218, and math.SG/0310141. Sections 2.2 and 3.5 contain previously unannounced results.