Stable Parabolic Higgs Bundles as Asymptotically Stable Decorated Swamps

TL;DR

This paper links the stability of parabolic Higgs bundles to asymptotic stability of decorated swamps, establishing a constant threshold beyond which stability conditions coincide, thus enabling a comprehensive understanding of their moduli spaces.

Contribution

It introduces the concept of asymptotic stability for decorated swamps and proves its equivalence to stability for large parameters, advancing the study of moduli spaces of parabolic Higgs bundles.

Findings

Existence of a stability threshold parameter.

Boundedness of semistable decorated swamps.

Recovery of classical stability as asymptotic stability.

Abstract

Parabolic Higgs bundles can be described in terms of decorated swamps, which we studied in a recent paper. This description induces a notion of stability of parabolic Higgs bundles depending on a parameter, and we construct their moduli space inside the moduli space of decorated swamps. We then introduce asymptotic stability of decorated swamps in order to study the behavior of the stability condition as one parameter approaches infinity. The main result is the existence of a constant, such that stability with respect to parameters greater than this constant is equivalent to asymptotic stability. This implies boundedness of all decorated swamps which are semistable with respect to some parameter. Finally, we recover the usual stability condition of parabolic Higgs bundles as asymptotic stability.