Three instability stratifications of the stack of Higgs bundles on a smooth projective curve

TL;DR

This paper explores three different ways to classify the instability of Higgs bundles on a smooth projective curve, providing explicit criteria and connections to geometric invariant theory and stratification techniques.

Contribution

It introduces and compares three instability stratifications of the stack of twisted Higgs bundles, including a refined stratification, and relates them to GIT and Bialynicki-Birula stratifications.

Findings

Explicit criteria for semistability of low-rank Higgs bundles with unstable underlying bundles.

Comparison and filtration of the Higgs bundle stack using HN and bHN stratifications.

Connection of refined bHN stratification to Bialynicki-Birula stratification.

Abstract

We study three instability stratifications of the stack of twisted Higgs bundle of a fixed rank and degree on a smooth complex projective curve. The first is the Harder-Narasimhan (HN) stratification, defined by the instability type of the Higgs bundle. The second is the bundle Harder-Narasimhan (bHN) stratification, defined by the instability type of the underlying bundle. While an unstable HN stratum fibres over the stack parametrising Higgs bundles which are isomorphic to their graded, this is not true for Higgs bundles of unstable bHN type. Obtaining such a fibration requires refining the bHN stratification; this is the third instability stratification. After introducing these three stratifications, we establish comparison results. In particular we obtain explicit criteria for determining semistability of a Higgs bundle of low rank with unstable underlying bundle. Then we show how the HN and bHN stratifications can be used to filter the stack of Higgs bundles by global quotient stacks in two different ways. Finally we use these filtrations to relate the HN and bHN stratifications to GIT instability stratifications, and the refined bHN stratification to a Bialynicki-Birula stratification.