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ARC SPACES AND EQUIVARIANT COHOMOLOGY DAVE ANDERSON AND ALAN STAPLEDON
 

Summary: ARC SPACES AND EQUIVARIANT COHOMOLOGY
DAVE ANDERSON AND ALAN STAPLEDON
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
JX, 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 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
Z-bases for the equivariant cohomology rings of a smooth toric variety (with re-
spect to the dense torus) and a partial flag variety (with respect to the general
linear group).
Contents

  

Source: Anderson, Dave - Department of Mathematics, University of Washington at Seattle

 

Collections: Mathematics