Equivariant cohomology and rings of functions

TL;DR

This dissertation explores the relationship between cohomology rings of algebraic varieties and rings of functions on zero schemes, extending results to singular varieties, spherical varieties, and GKM spaces, with implications for equivariant cohomology and K-theory.

Contribution

It generalizes previous results to singular varieties, demonstrates the cohomology ring of spherical varieties as a ring of functions, and advances the understanding of the K-theory conjecture for GKM spaces.

Findings

Zero scheme is isomorphic to the spectrum of the equivariant cohomology ring under certain conditions.

Cohomology rings of spherical varieties can be realized as rings of functions on zero schemes.

Partial results on the K-theory conjecture for GKM spaces.

Abstract

This submission is a PhD dissertation. It constitutes the summary of the author's work concerning the relations between cohomology rings of algebraic varieties and rings of functions on zero schemes and fixed point schemes. It includes the results from the co-authored article arXiv:2212.11836. They are complemented by: an introduction to the theory of group actions on algebraic varieties, with particular focus on vector fields; a historical overview of the field; a few newer results of the author. The fundamental theorem from arXiv:2212.11836 says that if the principal nilpotent has a unique zero, then the zero scheme over the Kostant section is isomorphic to the spectrum of the equivariant cohomology ring, remembering the grading in terms of a $\mathbb{C}$ action. In this thesis, we also tackle the case of a singular variety. As long as it is embedded in a smooth variety with regular action, we are able to study its cohomology as well by means of the zero scheme. In largest generality, this allows us to see geometrically a subring of the cohomology ring. We also show that the cohomology ring of spherical varieties appears as the ring of functions on the zero scheme. Lastly, the K-theory conjecture is studied, with some results attained for GKM spaces.