Topology of $\mathbb{G}_m$-actions and applications to the moduli of Higgs bundles

TL;DR

This paper studies the topology of varieties with $ ext{G}_m$-actions and applies these insights to Higgs bundle moduli, notably enhancing the cohomological Nonabelian Hodge Theorem in positive characteristic.

Contribution

It introduces new topological methods for varieties with $ ext{G}_m$-actions and extends the cohomological Nonabelian Hodge Theorem to positive characteristic.

Findings

Established isomorphism of cohomology rings in positive characteristic

Developed techniques for analyzing $ ext{G}_m$-actions on varieties and stacks

Enhanced understanding of Higgs bundle moduli topology

Abstract

We explain some results concerning the topology of varieties and stacks equipped with an action of the multiplicative group $\mathbb{G}_m$. We apply these techniques to the moduli of Higgs bundles. Our main application is to upgrade the cohomological Nonabelian Hodge Theorem in positive characteristic to an isomorphism of cohomology rings compatible with cup product.