Very stable bundles and properness of the Hitchin map

TL;DR

This paper characterizes very stable vector bundles on a smooth projective curve as those for which the Hitchin map, restricted to Higgs fields, is proper, linking stability properties to geometric features of the Hitchin fibration.

Contribution

It establishes a precise criterion connecting the concept of very stability of vector bundles with the properness of the Hitchin map restricted to Higgs fields.

Findings

Very stable bundles correspond to proper Hitchin map restrictions.

Characterization of very stability via properness of the Hitchin map.

Provides a geometric criterion for stability properties.

Abstract

Let $X$ be a smooth complex projective curve of genus $g\geq 2$ and let $K$ be its canonical bundle. In this note we show that a stable vector bundle $E$ on $X$ is very stable, i.e. $E$ has no non-zero nilpotent Higgs field, if and only if the restriction of the Hitchin map to the vector space of Higgs fields $H^0(X, \mathrm{End}(E) \otimes K)$ is a proper map.