The Limit of the Yang-Mills-Higgs Flow for twisted Higgs pairs

TL;DR

This paper studies the long-term behavior of the Yang-Mills-Higgs flow for twisted Higgs pairs on Kähler manifolds, showing convergence to a reflexive sheaf related to the initial bundle's filtration.

Contribution

It establishes the convergence of the Yang-Mills-Higgs flow to a specific reflexive sheaf, linking the flow's limit to the Harder-Narasimhan--Eshadri filtration.

Findings

Flow converges outside a codimension 4 subset

Limit is isomorphic to the double dual of the graded sheaves

Connects flow limits to the initial bundle's filtration

Abstract

In this paper, we consider the Yang-Mills-Higgs flow for twisted Higgs pairs over K\"ahler manifolds. We prove that this flow converges to a reflexive twisted Higgs sheaf outside a closed subset of codimension $4$, and the limiting twisted Higgs sheaf is isomorphic to the double dual of the graded twisted Higgs sheaves associated to the Harder-Narasimhan--eshadri filtration of the initial twisted Higgs bundle.