Hyperkahler analogues of Kahler quotients

Details Hyperkahler analogues of Kahler quotients Hyperkahler analogues of Kahler quotients

2004-05-12

arxiv.org/abs/math/0405233v1

Ph. D. thesis, U.C. Berkeley, Spring 2004

Mathematics

Algebraic Geometry

86 pages, 10 figures

Scientific paper

Let X be a Kahler manifold that is presented as a Kahler quotient of C^n 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^*C^n 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.

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