Non-Abelian Hodge Theory And Some Specializations

This dissertation mainly works on non-Abelian Hodge theory and some specializa-tions,which contains two parts.The first part is the geometry of non-Abelian Hodge theory,which consists of the second,third and the fourth chapter.Chapter 2 is an introduction to non-Abelian Hodge theory,which can be thought as the background,and the basis of this dissertation.In this chapter,we try to give a self-contained introduction to this theory within limited pages,at last we give some estimates that related to 1-flat bundles,as an application,we will also give some examples.Chapter 3 is the first core chapter of this part,which contains the most contents.The first main result of this chapter is the construction of a dynamical system of two parameters on the moduli space of Higgs bundles via non-Abelian Hodge theory,this generalizes the most classical dynamical system on this moduli space,namely the C*-action.By calculation,we show that the fixed points of this dynamical system are ex-actly those points fixed by the C*-action,that is,all C-VHS in the moduli space of Higgs bundles.This property provides much convenience on the deep study of the limiting be-haviour of such dynamical system,we introduce some limits for such dynamical system,and we also find some special points in moduli space such that these limits exist and co-incide.The second main result of this chapter is the proof of a conjecture by Simpson on the stratification of the moduli space of flat bundles(weak form)via the theory of moduli space of holomorphic chains,we show that in the Bialynicki-Birula stratifica-tion of the moduli space of flat bundles given C*-action,the oper stratum is the unique closed stratum of minimal dimension.Chapter 4 is the second core chapter of this part.In this chapter,we generalize Deligne’s construction of Hitchin twistor space associated to the moduli space of Higgs bundles over a compact Riemann surface X.In Deligne’s interpretation,Hitchin twistor space can be obtained via gluing the Hodge moduli space over X and the Hodge moduli space over X,this gluing map is given by non-Abelian Hodge correspondence,the re-sulting twistor space is called the Deligne-Hitchin twistor space.In our generalization,the gluing map can be induced by any non-trivial element of the outer automorphism group Out(π1(x))of the fundamental group of the underlying smooth surface X,namely gluing the two Hodge moduli spaces associated to the two Riemann surfaces induced by the element.So we can obtain a new twistor space,such that the Deligne-Hitchin twistor space is exactly the new twistor space by taking an element which induces the orientation reversing map.We construct a kind of holomorphic sections for this new twistor space,that is,the de Rham sections.We also calculate the normal bundle of de Rham sections,and we found that de Rham sections in Deligne-Hitchin twistor space also have weight 1 property,so they are ample rational curves.We also show Torelli theorem for the new twistor space.The second part of this dissertation is the study of some specializations of non-Abelian Hodge correspondence,which contains the fifth and sixth chapters.Chapter 5 mainly deals with a fundamental proof of a conjecture related to quiver representations proposed by Reineke in 2003.We show that for representations of quiv-ers of An-type,there exists a weight system such that the stable representations with respect to this weight system are precisely these indecomposable ones.In chapter 6,we build the Kobayashi-Hitchin correspondence for quiver bundles over generalized Kahler manifolds,that is,a quiver bundle over a generalized Kahler manifold is(α,σ,τ)-polystable if and only if it admits an(α,σ,τ)-Hermitian-Einstein metric.