Some Studies Of Higgs Bundles And Flat Bundles Over Non-k?hler Manifolds

In this thesis,we study Higgs bundles and flat bundles over non-K?hler mani-folds.The goal of this paper is to generalize Corlette-Donaldson-Hitchin-Simpson's correspondence between the moduli space of semi-stable(polystable)Higgs bundles with vanishing Chern numbers and the moduli space of(semi-simple)flat bundles to some non-K?hler manifolds.We always assume the base Hermitian manifold(X,?)satisfying(?)?^n-1?0 and(?)?^n-2=0.In the first part,we study the existence of special metrics on flat bundles over non-K?hler manifolds.Under the additional condition ?_X(?)[?]??^n-1?0 for(?)[?]?H^0,1(X),we prove that there must exists Hermitian harmonic metric on semi-simple flat bundles.Using this metric,we can construct a Higgs structure on this flat bundle.Therefore we get the one-to-one correspondence between moduli space of poly-stable Higgs bundles with vanishing Chern numbers and moduli space of semi-simple flat bundles.In the second part,we study semi-stable Higgs bundles with vanishing Chern num-bers over non-K?hler manifolds.We first study the blow-up analysis of Yang-Mills-Higgs flow,we prove the monotonicity inequality and ?-regularity along the flow.Us-ing these technique of analysis,we prove that,for any semi-stable Higgs bundle with vanishing Chern numbers there is a filtration of Higgs sub-bundles such that all the successive quotients are stable Higgs bundles with vanishing Chern numbers.In the third part,we study the correspondence between semi-stable Higgs bundles and flat bundles.Suppose the base manifold satisfying the above conditions and the rank of the bundle equals 2,we prove that there is a one-to-one correspondence between moduli space of semi-stable Higgs bundles with vanishing Chern numbers and moduli space of flat bundles.