Cohomology of the moduli of Higgs bundles on a curve via positive characteristic

Abstract

For a curve of genus $g$ and any two degrees coprime to the rank, we construct a family of ring isomorphisms parameterized by the complex Lie group $\mathrm{GSp}(2g),$ between the cohomology of the moduli spaces of stable Higgs bundles which preserve the perverse filtrations. As a consequence, we prove two structural results concerning the cohomology of Higgs moduli which are predicted by the P = W Conjecture in Non-Abelian Hodge Theory: (1) Galois conjugation for character varieties preserves the perverse filtrations for the corresponding Higgs moduli spaces. (2) The restriction of the Hodge–Tate decomposition for a character variety to each piece of the perverse filtration for the corresponding Higgs moduli space also gives a decomposition. Our proof uses reduction to positive characteristic and relies on the non-abelian Hodge correspondence in characteristic $p$ between Dolbeault and de Rham moduli spaces.