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Arc Spaces and Equivariant Cohomology

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Abstract

We present a new geometric interpretation of equivariant cohomology in which one replaces a smooth, complex G-variety X by its associated arc space J ∞ X, with its induced G-action. This not only allows us to obtain geometric classes in equivariant cohomology of arbitrarily high degree, but also provides more flexibility for equivariantly deforming classes and geometrically interpreting multiplication in the equivariant cohomology ring. Under appropriate hypotheses, we obtain explicit bijections between \( \mathbb{Z} \)-bases for the equivariant cohomology rings of smooth varieties related by an equivariant, proper birational map. We also show that self-intersection classes can be represented as classes of contact loci, under certain restrictions on singularities of subvarieties.

We give several applications. Motivated by the relation between self-intersection and contact loci, we define higher-order equivariant multiplicities, generalizing the equivariant multiplicities of Brion and Rossmann; these are shown to be local singularity invariants, and computed in some cases. We also present geometric \( \mathbb{Z} \)-bases for the equivariant cohomology rings of a smooth toric variety (with respect to the dense torus) and a partial flag variety (with respect to the general linear group).

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Author notes

  1. Dave Anderson*

    Present address: Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, 22460, Brasil

  2. Alan Stapledon**

    Present address: School of Mathematics and Statistics, University of Sydney, Sydney, NSW, 2006, Australia

Authors and Affiliations

  1. Department of Mathematics, University of Washington, Seattle, WA, 98195, USA

    Dave Anderson*

  2. Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada

    Alan Stapledon**

Authors
  1. Dave Anderson*
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  2. Alan Stapledon**
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Correspondence to Dave Anderson*.

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*Partially supported by NSF Grants DMS-0502170 and DMS-0902967, and by a Clay Mathematics Institute Liftoff Fellowship.

**Funded in part with assistance from the Australian Research Council project DP0559325 at Sydney University, Chief Investigator Professor G. I. Lehrer.

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Anderson*, D., Stapledon**, A. Arc Spaces and Equivariant Cohomology. Transformation Groups 18, 931–969 (2013). https://doi.org/10.1007/s00031-013-9239-4

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