A survey of hypertoric geometry and topology

Hypertoric varieties are quaternionic analogues of toric varieties, important for their interaction with the combinatorics of matroids as well as for their prominent place in the rapidly expanding field of algebraic symplectic and hyperkähler geometry. The aim of this survey is to give clear definitions and statements of known results, serving both as a reference and as a point of entry to this beautiful subject. Given a linear representation of a reductive complex algebraic group G, there are two natural quotient constructions. First, one can take a geometric invariant theory (GIT) quotient, which may also be interpreted as a Kähler quotient by a maximal compact subgroup of G. Examples of this sort include toric varieties (when G is abelian), moduli spaces of spacial polygons, and, more generally, moduli spaces of semistable representations of quivers. A second construction involves taking an algebraic symplectic quotient of the cotangent bundle of V , which may also be interpreted as a hyperkähler quotient. The analogous examples of the second type are hypertoric varieties, hyperpolygon spaces, and Nakajima quiver varieties. The subject of this survey will be hypertoric varieties, which are by definition the varieties obtained from the second construction when G is abelian. Just as the geometry and topology of toric varieties is deeply connected to the combinatorics of polytopes, hypertoric varieties interact richly with the combinatorics of hyperplane arrangements and matroids. Furthermore, just as in the toric case, the flow of information goes in both directions. On one hand, Betti numbers of hypertoric varieties have a combinatorial interpretation, and the geometry of the varieties can be used to prove combinatorial results. Many purely algebraic constructions involving matroids acquire geometric meaning via hypertoric varieties, and this has led to geometric proofs of special cases of the g-theorem for matroids [HSt, 7.4] and the Kook-Reiner-Stanton convolution formula [PW, 5.4]. Future plans include a geometric interpretation of the Tutte polynomial and of the phenomenon of Gale duality of matroids [BLP]. On the other hand, hypertoric varieties are important to geometers with no interest in combinatorics simply because they are among the most explicitly understood examples of algebraic symplectic or hyperkähler varieties, which are becoming increasingly prevalent in many areas of mathematics. For example, Nakajima’s quiver varieties include resolutions of Slodowy slices and Hilbert schemes of points on ALE spaces, both of which play major roles in modern representation theory. Moduli spaces of Higgs bundles are currently receiving a lot of attention in string theory, and character varieties of fundamental groups of surfaces and 3-manifolds have become an important tool in low-dimensional topology. Hypertoric Supported by a National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship.