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Let X be a Kahler manifold that is presented as a Kahler quotient of C\textasciicircumn 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\textasciicircum*C\textasciicircumn 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.

@article{proudfoot_hyperkahler_2004, title = {Hyperkahler analogues of {Kahler} quotients}, url = {http://arxiv.org/abs/math/0405233}, abstract = {Let X be a Kahler manifold that is presented as a Kahler quotient of C{\textasciicircum}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{\textasciicircum}*C{\textasciicircum}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.}, language = {en}, urldate = {2019-04-15}, journal = {arXiv:math/0405233}, author = {Proudfoot, Nicholas J.}, month = may, year = {2004}, note = {arXiv: math/0405233}, keywords = {14D20, 52C35, 53C26, 53D20, Mathematics - Algebraic Geometry, Mathematics - Differential Geometry, Mathematics - Symplectic Geometry} }

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