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

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 [dCGZ23, HZ23b] to an isomorphism of cohomology rings compatible with cup product.