Semi-stable Higgs Sheaves

This paper is a summary about the existence of approximate Hermitian-Einstein structure on Higgs sheaves.As an important structure over compact kahler manifolds,approximate Hermitian-Einstein structure is an important studied object of differential geometry,Higgs sheaves was first introduced by Hitchin in the 1980s,which has some well structure and wide applications.In the first chapter,we will introduce some basic definitions such as Higgs sheaves,approximate Hermitian-Einstein structure,semi-stability and related properties.In the second chapter,we will introduce the long time existence of the heat flow equation to prepare for the proof of the theorem in the next chapter.In the third chapter,the proof in the reference[48]of the existence of approximate Hermitian-Einstein structure over semi-table reflexive Higgs sheaf over compact Kahler manifold is mainly introduced.