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

TL;DR

This paper proves that for smooth projective surfaces, the spectral data space $ ext{A}_X$ coincides with the image of the Hitchin morphism for vector bundles, and explores its invariance under birational transformations.

Contribution

It confirms Chen-Ng extsuperscript{o}'s conjecture for surfaces and studies the spectral data space's invariance and behavior on ruled surfaces.

Findings

$ ext{A}_X$ equals the Hitchin morphism's image for surfaces.

$ ext{A}_X$ is invariant under proper birational morphisms.

Application to spectral data on ruled surfaces.

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-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.