On very stablity of principal $G-$bundles

TL;DR

This paper extends the concept of very stability from vector bundles to principal G-bundles over smooth projective curves, establishing a criterion involving the properness of the Hitchin map and exploring stability relations for SL_2-bundles.

Contribution

It generalizes the characterization of very stability via the Hitchin map to principal G-bundles for any semisimple group, and examines stability relations specifically for SL_2-bundles.

Findings

Very stability of principal G-bundles characterized by Hitchin map properness.

Established equivalence between very stability and Hitchin map properties for principal G-bundles.

Explored the relationship between very stability and other stability notions for SL_2-bundles.

Abstract

Let $X$ be a smooth irreducible projective curve. Recently, Pauly and Pe\'on-Nieto shows that a vector bundle over $X$ is very stable if and only if the Hitchin map on the vector space of Higgs field on that vector bundle is proper. In this notes, we generalize this result to principal $G-$bundles for any semisimple linear algebraic group $G$. We also study the relation between very stability and other stability conditions in the case of $\text{SL}_2-$bundles.