On the Image of Hitchin Morphism for Algebraic Surfaces: The Case GLn

Abstract

Abstract The Hitchin morphism is a map from the moduli space of Higgs bundles $\mathscr {M}_{X}$ to the Hitchin base $\mathscr {B}_{X}$, where $X$ is a smooth projective variety. When $X$ has dimension at least two, this morphism is not surjective in general. Recently, Chen-Ngô introduced a closed subscheme $\mathscr {A}_{X}$ of $\mathscr {B}_{X}$, which is called the space of spectral data. They proved that the Hitchin morphism factors through $\mathscr {A}_{X}$ and conjectured that $\mathscr {A}_{X}$ is the image of the Hitchin morphism. We prove that when $X$ is a smooth projective surface, this conjecture is true for vector bundles. Moreover, we show that $\mathscr {A}_{X}$, for any dimension, is invariant under any proper birational morphism and apply the result to study $\mathscr {A}_{X}$ for ruled surfaces.