Orateur
Description
The classical Corlette--Simpson (CS) correspondence relates local systems on complex varieties to Higgs bundles; it is highly transcendental in nature. Its characteristic $p$ counterpart surprisingly turns out to be purely algebraic: Bezrukavnikov identified de Rham local systems on a smooth variety $X$ over $\mathbb{F}_p$ with Higgs bundles twisted by a natural $\mathbb{G}_m$-gerbe on the cotangent bundle $T^*X$. By trivializing the gerbe over suitable loci in $T^*X$ using additional choices, Ogus--Vologodsky then recovered an honest CS correspondence (i.e., with untwisted Higgs bundles). In this talk, I'll explain that this story has an exact analog for a smooth rigid space $X$ over a perfectoid $p$-adic field: (generalized) local systems identify with Higgs bundles twisted by a natural $\mathbb{G}_m$-gerbe on $T^*X$, and honest CS correspondes (as studied by many authors in the last 2 decades) can be recovered by trivializing the gerbe over suitable loci in $T^*X$.
This is joint work in progress with Mingjia Zhang, and is inspired by recent work of Heuer.
