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INTRODUCTION WE DEVELOP here a generalization to singular spaces of the Poincare-Lefschetz theory of intersections of homology cycles on manifolds, as announced in [6]. Poincart, in his 1895 paper which founded modern algebraic topology ([18], p. 218; corrected in [19]), studied the intersection of an i-cycle V and a j-cycle W in a compact oriented… (More)

(1.1) This paper concerns three aspects of the action of a compact group K on a space X . The ®rst is concrete and the others are rather abstract. (1) Equivariantly formal spaces. These have the property that their cohomology may be computed from the structure of the zero and one dimensional orbits of the action of a maximal torus in K. (2) Koszul duality.… (More)

Suppose that a d-dimensional convex polytope P ⊂ R is rational, i.e. its vertices are all rational points. Then P gives rise to a polynomial g(P ) = 1 + g1(P )q + g2(P )q 2 + · · · with non-negative coefficients as follows. Let XP be the associated toric variety (see §6 – our variety XP is d + 1-dimensional and affine). The coefficient gi is the rank of the… (More)

- TOM BRADEN, Robert Macpherson
- 2008

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 zeroand 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… (More)

Assuming a certain “purity” conjecture, we derive a formula for the (complex) cohomology groups of the affine Springer fiber corresponding to any unramified regular semisimple element. We use this calculation to present a complex analog of the fundamental lemma for function fields. We show that the “kappa” orbital integral that arises in the fundamental… (More)

The affine Springer fiber corresponding to a regular integral equivalued semisimple element admits a paving by vector bundles over Hessenberg varieties and hence its homology is “pure”.

We define an ‘enriched’ notion of Chow groups for algebraic varieties, agreeing with the conventional notion for complete varieties, but enjoying a functorial push-forward for arbitrary maps. This tool allows us to glue intersectiontheoretic information across elements of a stratification of a variety; we illustrate this operation by giving a direct… (More)

If an algebraic torus T acts on a complex projective algebraic variety X then the affine scheme Spec H∗ T (X;C) associated to the equivariant cohomology is often an arrangement of linear subspaces of the vector space H 2 (X;C). In many situations the ordinary cohomology ring of X can be described in terms of this arrangement.

In this largely expository note we give some homological properties of algebraic maps of complex algebraic varieties which are rather surprising from the topological point of view. These include a generalisation to higher dimension of the invariant cycle theorem for maps to curves. These properties are all corollaries of a recent deep theorem of Deligne,… (More)