Stratifications on the Nilpotent Cone of the moduli space of Hitchin pairs

TL;DR

This paper investigates the asymptotic behavior of Higgs bundles in the nilpotent cone under a natural scaling action, providing explicit descriptions for regular nilpotent cases and complete solutions in rank three.

Contribution

It offers a detailed analysis of the limit behavior of Higgs bundles in the nilpotent cone, including explicit descriptions for regular nilpotent cases and a full solution in rank three.

Findings

Limit described via filtration of the vector bundle

Results apply to regular nilpotent Higgs fields

Complete characterization in rank three

Abstract

We consider the problem of finding the limit at infinity (corresponding to the downward Morse flow) of a Higgs bundle in the nilpotent cone under the natural $\mathbb{C}^*$-action on the moduli space. For general rank we provide an answer for Higgs bundles with regular nilpotent Higgs field, while in rank three we give the complete answer. Our results show that the limit can be described in terms of data defined by the Higgs field, via a filtration of the underlying vector bundle.