The hypertoric intersection cohomology ring

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. T. Braden’s research was supported in part by NSF grant DMS-0201823. N. Proudfoot’s research was supported in part by an NSF Postdoctoral Research Fellowship and NSF grant DMS-0738335.     

From moment graphs to intersection cohomology

We describe a method of computing equivariant and ordinary intersection cohomology of certain varieties with actions of algebraic tori, in terms of structure of the zero- and one-dimensional orbits. The class of varieties to which our formula applies includes Schubert varieties in flag varieties and affine flag varieties. We also prove a monotonicity result on local intersection cohomology stalks. Received: 9 November 2000 / Published online: 24 September 2001     

Sheaves on moment graphs and a localization of Verma flags

To any moment graph G we assign a subcategory V of the category of sheaves on G together with an exact structure. We show that in the case that the graph is associated to a non-critical block of the equivariant category O over a symmetrizable Kac-Moody algebra, V is equivalent (as an exact category) to the subcategory of modules that admit a Verma flag. The projective modules correspond under this equivalence to the intersection cohomology sheaves on the graph, and hence, by a theorem of Braden and MacPherson, to the equivariant intersection cohomologies of Schubert varieties associated to Kac-Moody groups.     

Equivariant cohomology of K-contact manifolds

We investigate the equivariant cohomology of the natural torus action on a K-contact manifold and its relation to the topology of the Reeb flow. Using the contact moment map, we show that the equivariant cohomology of this action is Cohen–Macaulay, the natural substitute of equivariant formality for torus actions without fixed points. As a consequence, generic components of the contact moment map are perfect Morse-Bott functions for the basic cohomology of the orbit foliation ${{\mathcal F}}$ of the Reeb flow. Assuming that the closed Reeb orbits are isolated, we show that the basic cohomology of ${{\mathcal F}}$ vanishes in odd degrees, and that its dimension equals the number of closed Reeb orbits. We characterize K-contact manifolds with minimal number of closed Reeb orbits as real cohomology spheres. We also prove a GKM-type theorem for K-contact manifolds which allows to calculate the equivariant cohomology algebra under the nonisolated GKM condition.     

Symplectic Cohomology and q-Intersection Numbers

Given a symplectic cohomology class of degree 1, we define the notion of an ??equivariant" Lagrangian submanifold (this roughly corresponds to equivariant coherent sheaves under mirror symmetry). The Floer cohomology of equivariant Lagrangian submanifolds has a natural endomorphism, which induces an ${\mathbb{R}}$ -grading by generalized eigenspaces. Taking Euler characteristics with respect to the induced grading yields a deformation of the intersection number. Dehn twists act naturally on equivariant Lagrangians. Cotangent bundles and Lefschetz fibrations give fully computable examples. A key step in computations is to impose the ??dilation" condition stipulating that the BV operator applied to the symplectic cohomology class gives the identity.     

Mod 3 cohomology algebras of finite H-spaces

Let X be a 3 local, finite, simply connected H-space with associative homology ring . Some known examples are the Lie group , Harper's H-space X(3) and any odd dimensional sphere . We prove the cohomology algebra is isomorphic to the cohomology algebra of a finite product of and odd dimensional spheres. Received: 15 May 2001; in final form: 22 May 2001 / Published online: 28 February 2002     

The universality of equivariant complex bordism

We show that if A is an abelian compact Lie group, all A-equivariant complex vector bundles are orientable over a complex orientable equivariant cohomology theory. In the process, we calculate the complex orientable homology and cohomology of all complex Grassmannians. Received: 14 February 2000; in final form: 4 August 2000 / Published online: 19 October 2001     

The Witten genus and equivariant elliptic cohomology

We construct a Thom class in complex equivariant elliptic cohomology extending the equivariant Witten genus. This gives a new proof of the rigidity of the Witten genus, which exhibits a close relationship to recent work on non-equivariant orientations of elliptic spectra. Received: 24 May 2001; in final form: 4 October 2001 / Published online: 28 February 2002     

Group-valued equivariant localization

We prove a localization formula for group-valued equivariant de Rham cohomology of a compact G-manifold. This formula is a non-trivial generalization of the localization formula of Berline-Vergne and Atiyah-Bott for the usual equivariant de Rham cohomology. We derive from this result a Duistermaat-Heckman formula for group valued moment maps. As an application, we prove part of Witten’s conjectures about intersection pairings on moduli spaces of flat connections on 2-manifolds. Oblatum 24-VI-1999 & 29-X-1999?Published online: 21 February 2000     

A cohomological characterization of Alexander schemes

Vistoli defined Alexander schemes in [19], which behave like smooth varieties from the viewpoint of intersection theory with Q-coefficients. In this paper, we will affirmatively answer Vistoli’s conjecture that Alexander property is Zariski local. The main tool is the abelian category of bivariant sheaves, and we will spend most of our time for proving basic properties of this category. We show that a scheme is Alexander if and only if all the first cohomology groups of bivariant sheaves vanish, which is an analogy of Serre’s theorem, which says that a scheme is affine if and only if all the first cohomology groups of quasi-coherent sheaves vanish. Serre’s theorem implies that the union of affine closed subschemes is again affine. Mimicking the proof line by line, we will prove that the union of Alexander open subschemes is again Alexander. Oblatum 1-XII-1997 & 14-XII-1998 / Published online: 10 May 1999     

Cohomology of tilting modules over quantum groups and t-structures on derived categories of coherent sheaves

The paper is concerned with cohomology of the small quantum group at a root of unity, and of its upper triangular subalgebra, with coefficients in a tilting module. It turns out to be related to irreducible objects in the heart of a certain t-structure on the derived category of equivariant coherent sheaves on the Springer resolution, and to equivariant coherent IC sheaves on the nil-cone. The support of the cohomology is described in terms of cells in affine Weyl groups. The basis in the Grothendieck group provided by the cohomology modules is shown to coincide with the Kazhdan-Lusztig basis, as predicted by J. Humphreys and V. Ostrik. The proof is based on the results of [ABG ], [AB] and [B], which allow us to reduce the question to purity of IC sheaves on affine flag varieties. To the memory of my father     

Hyperbolic localization of intersection cohomology

For a normal variety X defined over an algebraically closed field with an action of the multiplicative group T = G_m, we consider the "hyperbolic localization" functor D^b(X) → D^b(X^T), which localizes using closed supports in the directions flowing into the fixed points, and compact supports in the directions flowing out. We show that the hyperbolic localization of the intersection cohomology sheaf is a direct sum of intersection cohomology sheaves.     

The equivariant Serre spectral sequence as an application of a spectral sequence of Spanier

E.H. Spanier (1992) has constructed, for a cohomology theory defined on a triangulated space and locally constant on each open simplex, a spectral sequence whose E₂-term consists of certain simplicial cohomology groups, converging to the cohomology of the space. In this paper we study a closed G-fibration ƒ: Y → X, where G is a finite group. We show that if the base-G-spaceX is equivariantly triangulated and Y is paracompact, then Spanier's spectral sequence yields an equivariant Serre spectral sequence for ƒ. The main point here is to identify the equivariant singular cohomology groups of X with appropriate simplicial cohomology groups of the orbit space X/G.     

Equivariant evaluation on free loop spaces

There is a natural evaluation map on the free loop space which sends a loop to its values at the kth roots of unity. This map is equivariant with respect to the action of the cyclic group on k elements . We study the induced map in -equivariant cohomology with mod two coefficients in the cases where for . Received: 17 February 2000; in final form: 28 February 2001 / Published online: 18 January 2002     

Reconstructing the orbit type stratification of a torus action from its equivariant cohomology

We investigate what information on the orbit type stratification of a torus action on a compact space is contained in its rational equivariant cohomology algebra. Regarding the (labelled) poset structure of the stratification, we show that equivariant cohomology encodes the subposet of ramified elements. For equivariantly formal actions, we also examine what cohomological information of the stratification is encoded. In the smooth setting, we show that under certain conditions—which in particular hold for a compact orientable manifold with discrete fixed point set—the equivariant cohomologies of the strata are encoded in the equivariant cohomology of the manifold.

     

An effective algorithm for the cohomology ring of symplectic reductions

Let G be a compact torus acting on a compact symplectic manifold M in a Hamiltonian fashion, and T a subtorus of G. We prove that the kernel of is generated by a small number of classes satisfying very explicit restriction properties. Our main tool is the equivariant Kirwan map, a natural map from the G-equivariant cohomology of M to the G/T-equivariant cohomology of the symplectic reduction of M by T . We show this map is surjective. This is an equivariant version of the well-known result that the (nonequivariant) Kirwan map is surjective. We also compute the kernel of the equivariant Kirwan map, generalizing the result due to Tolman and Weitsman [TW] in the case T = G and allowing us to apply their methods inductively. This result is new even in the case that dim T = 1. We close with a worked example: the cohomology ring of the product of two , quotiented by the diagonal 2-torus action. Submitted: September 2001, Revised: December 2001, Revised: February 2002.     

Mixed Hodge structures and equivariant sheaves on the projective plane

We describe an equivalence of categories between the category of mixed Hodge structures and a category of equivariant vector bundles on a toric model of the complex projective plane which verify some semistability condition. We then apply this correspondence to define an invariant which generalizes the notion of R ‐split mixed Hodge structure and give calculations for the first group of cohomology of possibly non smooth or non‐complete curves of genus 0 and 1. Finally, we describe some extension groups of mixed Hodge structures in terms of equivariant extensions of coherent sheaves. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

On topological invariants of stratified maps with non-Witt target

The Cappell-Shaneson decomposition theorem for self-dual sheaves asserts that on a space with only even-codimensional strata any self-dual sheaf is cobordant to an orthogonal sum of twisted intersection chain sheaves associated to the various strata. In sharp contrast to this result, we prove that on a space with only odd-codimensional strata (not necessarily Witt), any self-dual sheaf is cobordant to an intersection chain sheaf associated to the top stratum: the strata of odd codimension do not contribute terms. As a consequence, we obtain formulae for the pushforward of characteristic classes under a stratified map whose target need not satisfy the Witt space condition. To prove these results, we introduce a new category of superperverse sheaves, which we show to be abelian. Finally, we apply the results to the study of desingularization of non-Witt spaces and exhibit a singular space which admits a PL resolution in the sense of M. Kato, but no resolution by a stratified map.