The Hitchin morphism for certain surfaces fibered over a curve

Abstract

The Chen-Ngô Conjecture predicts that the Hitchin morphism from the moduli stack of $G$-Higgs bundles on a smooth projective variety surjects onto the space of spectral data. The conjecture is known to hold for the group $G{L}_n$ and any surface, and for the group $G{L}_2$ and any smooth projective variety. We prove the Chen-Ngô Conjecture for any reductive group when the variety is a ruled surface or (a blowup of) a nonisotrivial elliptic fibration with reduced fibers. Furthermore, if the group is a classical group, i.e. $G\in \{S{L}_n,S{O}_n,S{p}_{2n}\}$, then we prove the Hitchin morphism restricted to the Dolbeault moduli space of semiharmonic $G$-Higgs bundles surjects onto the space of spectral data.

final version; to appear in PAMQ