Hyperkähler Analogues of Kähler Quotients

Hyperkähler Analogues of Kähler Quotients by Nicholas James Proudfoot Doctor of Philosophy in Mathematics University of California, Berkeley Professor Allen Knutson, Chair Let X be a Kähler manifold that is presented as a Kähler quotient of Cn by the linear action of a compact group G. We define the hyperkähler analogue M of X as a hyperkähler quotient of the cotangent bundle T ∗Cn by the inducedG-action. Special instances of this construction include hypertoric varieties [BD, K1, HS, HP1] and quiver varieties [N1, N2, N3]. One of our aims is to provide a unified treatment of these two previously studied examples. The hyperkähler analogue M is noncompact, but this noncompactness is often “controlled” by an action of C× descending from the scalar action on the fibers of T ∗Cn. Specifically, we are interested in the case where the moment map for the action of the circle S1 ⊆ C× is proper. In such cases, we define the core of M, a reducible, compact subvariety onto which M admits a circle-equivariant deformation retraction. One of the components of the core is isomorphic to the original Kähler manifold X. When X is a moduli space of polygons in R3, we interpret each of the other core components of M as related polygonal moduli spaces. Using the circle action with proper moment map, we define an integration theory on the circle-equivariant cohomology of M, motivated by the well-known localization theorem of [AB] and [BV]. This allows us to prove a hyperkähler analogue of Martin’s theorem [Ma], which describes the cohomology ring of an arbitrary Kähler quotient in terms of the cohomology of the quotient by a maximal torus. This theorem, along with a direct analysis of the equivariant cohomology ring of a hypertoric variety, gives us a method for computing the equivariant cohomology ring of many hyperkähler analogues, including all