All the GIT quotients at once

Let G be an algebraic torus acting on a smooth variety V . We study the relationship between the various GIT quotients of V and the symplectic quotient of the cotangent bundle of V . Let G be a reductive algebraic group acting on a smooth variety V . The cotangent bundle T V admits a canonical algebraic symplectic structure, and the induced action of G on T V is hamiltonian, that is, it admits a natural moment map μ : T V → g (see Equation (1) for an explicit formula). Over the past ten years, a guiding principle has emerged that says that if X is an interesting variety which may be naturally presented as a GIT (geometric invariant theory) quotient of V by G, then the symplectic quotient μ(λ)/G of T V by G is also interesting. This mantra has been particularly fruitful on the level of cohomology, as we describe below. Over the complex numbers, a GIT quotient may often be interpreted as a Kähler quotient by the compact form of G, and an algebraic quotient as a hyperkähler quotient. For this reason, the symplectic quotient may be loosely thought of as a quaternionic or hyperkähler analogue of X. Let us review a few examples of this construction. Hypertoric varieties. These examples comprise the case where G is abelian and V is a linear representation of G. The geometry of toric varieties is deeply related to the combinatorics of polytopes; for example, Stanley [St] used the hard Lefschetz theorem for toric varieties to prove certain inequalities for the h-numbers of a simplicial polytope. The hyperkähler analogues of toric varieties, known as hypertoric varieties, interact in a similar way with the combinatorics of rational hyperplane arrangements. Introduced by Bielawski and Dancer [BD], hypertoric varieties were used by Hausel and Sturmfels [HS] to give a geometric interpretation of virtually every known property of the h-numbers of a rationally representable matroid. Webster and the author [PW] extended this line of research by studying the intersection cohomology groups of singular hypertoric varieties. Quiver varieties. A quiver is a directed graph, and a representation of a quiver is a vector space for each node along with a linear map for each edge. For any quiver, Nakajima [N1, N2, N3] defined a quiver variety to be the quaternionic analogue of the moduli space of framed representations. Examples include the Hilbert scheme of n points in the plane and the moduli space of instantons on an ALE space. He has shown that the cohomology and K-theory groups of quiver varieties carry actions of Kac-Moody algebras and their associated quantum algebras, and has exploited this fact to define canonical bases for highest weight representations. Crawley-Boevey and Van den Bergh [CBVdB] and Hausel [Ha] have used Betti numbers of quiver varieties to prove a long standing conjecture of Kac. Hyperpolygon spaces. Given an ordered n-tuple of positive real numbers, the associated polygon space is the moduli space of n-sided polygons in R with edges of the prescribed Partially supported by a National Science Foundation Postdoctoral Research Fellowship.