Arc spaces and equivariant cohomology

TL;DR

This paper introduces a geometric approach to equivariant cohomology using arc spaces, enabling new classes, deformations, and explicit bases for smooth varieties, with applications to singularity invariants and specific varieties.

Contribution

It provides a novel geometric interpretation of equivariant cohomology via arc spaces, allowing for high-degree classes, explicit bases, and new invariants, extending prior algebraic methods.

Findings

Geometric classes in high degree are obtained via arc spaces.

Explicit bijections between cohomology rings of related varieties are established.

Higher-order equivariant multiplicities are defined and computed in some cases.

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 $J_{\infty} X$, 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).