arXiv subjects
Publications and source records attributed to Nicholas J. Proudfoot.
Etingof and Schedler formulated a conjecture about the degree zero Poisson homology of an affine cone that admits a projective symplectic resolution. We strengthen this conjecture in general and prove the strengthened version for hypertoric varieties. We also formulate an analogous conjecture for the degree zero Hochschild homology of a quantization of such a variety.
We present a functorial computation of the equivariant intersection cohomology of a hypertoric variety, and endow it with a natural ring structure. When the hyperplane arrangement associated with the hypertoric variety is unimodular, we show that this ring structure is induced by a ring structure on the equivariant intersection cohomology sheaf in the equivariant derived category. The computation is given in terms of a localization functor which takes equivariant sheaves on a sufficiently nice stratified space to sheaves on a poset.
Given a sufficiently nice collection of sheaves on an algebraic variety V, Bondal explained how to build a quiver Q along with an ideal of relations in the path algebra of Q such that the derived category of representations of Q subject to these relations is equivalent to the derived category of coherent sheaves on V. We consider the case in which these sheaves are all locally free and study the moduli spaces of semistable representations of our quiver with relations for various stability conditions. We show that V can often be recovered as a connected component of such a moduli space and we describe the line bundle induced by a GIT construction of the moduli space in terms of the input data. In certain special cases, we interpret our results in the language of topological string theory.
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 hyperkahler 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 matroid M represented by a linear subspace L in n-space (equivalently by an arrangement of n hyperplanes in L), we define a graded ring R(L) which degenerates to the Stanley-Reisner ring of the broken circuit complex for any choice of ordering of the ground set. In particular, R(L) is Cohen-Macaulay, and may be used to compute the h-vector of the broken circuit complex of M. We give a geometric interpretation of Spec R(L), as well as a stratification indexed by the flats of M.
We study the Z/2-equivariant K-theory of the complement of the complexification of a real hyperplane arrangement. We compute the rational K and KO rings, and give two different combinatorial descriptions of the subring of the integral KO ring generated by line bundles.
Let T 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.
For physicists: We show that the quiver gauge theory derived from a Calabi-Yau cone via an exceptional collection of line bundles on the base has the original cone as a component of its classical moduli space. For mathematicians: We use data from the derived category of sheaves on a Fano surface to construct a quiver, and show that its moduli space of representations has a component which is isomorphic to the anticanonical cone over the surface.
Given a hyperplane arrangement A in a real vector space, we introduce a real algebraic prevariety Z(A), and exhibit the complement of the complexification of A as the total space of an affine bundle with fibers modeled on the dual of the ambient vector space.
This is an expository paper in which we define projective GIT quotients and introduce toric varieties from this perspective. It is intended primarily for readers who are learning either invariant theory or toric geometry for the first time.
Given a real arrangement $A$, the complement $M(A)$ of the complexification of $A$ admits an action of $\mathbb{Z}_2$ by complex conjugation. We define the equivariant Orlik-Solomon algebra of $A$ to be the $\mathbb{Z}_2$-equivariant cohomology ring of $M(A)$ with coefficients in $\mathbb{Z}_2$. We give a combinatorial presentation of this ring, and interpret it as a deformation of the ordinary Orlik-Solomon algebra into the Varchenko-Gel'fand ring of locally constant $\mattbb{Z}_2$-valued functions on the complement $C(A)$ of $A$ in $\mathbb{R}^n$. We also show that the $\mathbb{Z}_2$-equivariant homotopy type of $M(A)$ is determined by the oriented matroid of $A$. As an application, we give two examples of pairs of arrangements $A$ and $A'$ such that $M(A)$ and $M(A')$ have the same nonequivariant homotopy type, but are distinguished by the equivariant Orlik-Solomon algebra.
Let X be a Kahler manifold that is presented as a Kahler quotient of C^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^*C^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.
We consider a manifold X obtained by a Kahler reduction of C^n, and we define its hyperkahler analogue M as a hyperkahler reduction of T^*C^n = H^n by the same group. In the case where the group is abelian and X is a smooth toric variety, M is a toric hyperkahler manifold, as defined by Bielawski-Dancer, and further studied by Konno and Hausel-Sturmfels. The manifold M carries a natural action of S^1, induced by the scalar action of S^1 on the fibers of T^*C^n. In this paper we study this action, computing its fixed points and its equivariant cohomology. As an application, we use the associated Z/2 action on the real locus of M to compute a deformation of the Orlik-Solomon algebra of a smooth, generic, real hyperplane arrangement, depending nontrivially on the affine structure of the arrangement. This deformation is given by the Z/2-equivariant cohomology of the complement of the complexification, where Z/2 acts by complex conjugation.
Given an n-tuple of positive real numbers, Konno defines an algebraic variety called a hyperpolygon space, a hyperkahler analogue of the Kahler variety parametrizing spacial polygons with fixed edge lengths. The ordinary polygon space can be interpreted as the moduli space of stable representations of a certain quiver with fixed dimension vector; from this point of view, the hyperpolygon space is the hyperkahler quiver variety defined by Nakajima. A quiver variety admits a natural action of the nonzero complex numbers, and the union of the precompact orbits is called the core. We study the components of the core of the hyperpolygon space, interpreting each one as a moduli space of pairs of spatial polygons with certain properties. Konno gives a presentation of the cohomology ring of the hyperpolygon space; we extend this result by computing the circle-equivariant cohomology ring, as well as the ordinary and equivariant cohomology rings of the core components.
Let \A be a complex hyperplane arrangement, and let $X$ be a modular element of arbitrary rank in the intersection lattice of \A. We show that projection along $X$ restricts to a fiber bundle projection of the complement of \A to the complement of the localization $\A_X$ of \A at $X$. The fiber is the decone of a realization of the complete principal truncation of the underlying matroid of \A along the flat corresponding to $X$. This result gives a topological realization of results of Stanley, Brylawsky, and Terao on modular factorization. We show that (generalized) parallel connection of matroids corresponds to pullback of fiber bundles, clarifying the notion that all examples of diffeomorphisms of complements of inequivalent arrangements result from the triviality of the restriction of the Hopf bundle to the complement of a hyperplane. The modular fibration result also yields a new method for identifying $K(π,1)$ arrangements of rank greater than three. We identify a new families of $K(π,1)$ arrangements, providing more evidence for the conjecture that factored arrangements of arbitrary rank are $K(π,1)$.