The Hitchin morphism for certain surfaces fibered over a curve
Matthew Huynh
TL;DR
This work extends the Chen–Ngô Hitchin-morphism program to higher-dimensional fibrations by proving surjectivity of the Hitchin morphism onto the Chen–Ngô spectral base for fibered surfaces in broad cases, including ruled surfaces and blowups of nonisotrivial elliptic fibrations with reduced fibers. It develops and leverages a refined spectral theory via companion sections and Cohen–Macaulay spectral surfaces to realize Higgs bundles with prescribed spectral data, and establishes pullback machinery along fibrations and morphisms to transfer Hitchin-data between bases. For classical groups, the Dolbeault moduli space on fibered surfaces maps surjectively onto the curve Hitchin base, cementing a strong link between spectral data on the base curve and Higgs bundles on the total space. The results bridge general G-Higgs theory with explicit constructions through spectral covers, CM-spectral covers, and pullbacks, enabling a robust verification of Chen–Ngô predictions in new geometric settings with notable implications for the geometry of moduli stacks and spectral data on surfaces.
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 $GL_n$ and any surface, and for the group $GL_2$ 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. $G \in \{SL_n,SO_n,Sp_{2n}\}$, then we prove the Hitchin morphism restricted to the Dolbeault moduli space of semiharmonic $G$-Higgs bundles surjects onto the space of spectral data.