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

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.