On the image of Hitchin morphism for algebraic surfaces: The case ${\rm GL}_n$

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-Ng\^o 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 proper birational morphisms, and apply the result to study $\mathscr{A}_X$ for ruled surfaces.