Arithmetic and topology of hypertoric varieties

A hypertoric variety is a quaternionic analogue of a toric variety. Just as the topology of toric varieties is closely related to the combinatorics of polytopes, the topology of hypertoric varieties interacts richly with the combinatorics of hyperplane arrangements and matroids. Using finite field methods, we obtain combinatorial descriptions of the Betti numbers of hypertoric varieties, both for ordinary cohomology in the smooth case and intersection cohomology in the singular case. We also introduce a conjectural ring structure on the intersection cohomology of a hypertoric variety. Let T k be an algebraic torus acting linearly and effectively on an affine space A, and let α be a character of T . Then α defines a linearization of the T k action to the trivial bundle on A, and the corresponding GIT quotient X = An/αT k is a toric variety. A hypertoric variety is a symplectic quotient M = T ∗An//(α,0)T k = μ−1(0)/αT , where μ : T A → (t) is the algebraic moment map for the T k action on T A. Over the complex numbers, this construction may be interpreted as a hyperkähler quotient [BD], and M is the hyperkähler analogue of X in the sense of [Pr]. In this paper, however, we will focus on the algebro-geometric construction, which lets us work over arbitrary fields. The data of T k acting on A, along with the character α, can be conveniently encoded by an arrangement A of cooriented hyperplanes in an affine space of dimension d = n− k. The topology of the corresponding toric variety X(A) is deeply related to the combinatorics of the polytope cut out by A [S1, S2]. The hypertoric variety M(A) is sensitive to a different side of the combinatorial data. As a topological space, the complex variety M(A)C does not depend on the coorientations of the hyperplanes [HP], and hence has little relationship to the polytope that controls X(A). Instead, the topology of M(A) interacts richly with the combinatorics of the matroid associated to A [Ha], which we describe below. Let ∆ be a simplicial complex of dimension d− 1 on the ground set {1, . . . , n}. The f vector of ∆ is the (d+1)-tuple (f0, . . . , fd), where fi is the number of faces of ∆ of cardinality i (and therefore of dimension i − 1). The h-vector (h0, . . . , hd) and h-polynomial h∆(q) of ∆ are defined by the equations