A cohomological nonabelian Hodge theorem in positive characteristic

By using specialization in cohomology techniques, under suitable natural coprimality conditions, we prove a cohomological Simpson correspondence between the moduli space of Higgs bundles and that of connections on a curve over an algebraically closed ground field of positive characteristic p > 0. We also prove a new p-multiplicative periodicity concerning the cohomology rings of Dolbeault moduli spaces of degrees differing by a factor of p. By coupling this p-periodicity in characteristic p with lifting/ specialization techniques in mixed characteristic, we find, in arbitrary characteristic, cohomology ring isomorphisms between the cohomology rings of Dolbeault moduli spaces for different degrees coprime to the rank. Interestingly, this last result is proved by passing from positive characteristic to characteristic zero and back.