Variations on cohomology rings and zero schemes

TL;DR

This paper generalizes the relationship between equivariant cohomology rings and zero schemes for various classes of varieties, including smooth, singular, and GKM spaces, extending previous results and providing new insights.

Contribution

It proves that equivariant cohomology rings can be realized as functions on zero schemes under certain conditions for a broad class of varieties, including spherical and singular varieties.

Findings

Equivariant cohomology rings correspond to functions on zero schemes for smooth varieties.

Partial realization of equivariant cohomology for singular varieties via Chern classes.

Equivariant K-theory relates to fixed-point schemes in GKM spaces.

Abstract

We extend the theorem of Hausel and the author from arXiv:2212.11836 that relates equivariant cohomology rings and algebras of functions on zero schemes. This paper combines three separate results. We prove that for a reductive group G acting on a smooth projective variety one can see the equivariant cohomology ring as the ring of functions on the zero scheme over the Kostant section, provided that some transversality condition is satisfied. In particular, we show that the conclusion holds for spherical varieties. We then show a version for singular varieties, e.g. discriminant varieties, where in general we only recover a part of equivariant cohomology ring, generated by Chern classes. We also show that an analogous result, connecting equivariant K-theory to the ring of functions on the fixed-point scheme, holds for GKM spaces. This is a concise version of some results from the PhD thesis arXiv:2407.14659, which contains a broader introduction to the topic.