The non-abelian Hodge correspondence on some non-Kahler manifolds

The non-abelian Hodge correspondence was established by Corlette (1988), Donaldson (1987), Hitchin (1987) and Simpson (1988, 1992). It states that on a compact Kahler manifold (X, omega), there is a one-to-one correspondence between the moduli space of semi-simple flat complex vector bundles and the moduli space of poly-stable Higgs bundles with vanishing Chern numbers. In this paper, we extend this correspondence to the projectively flat bundles over some non-Kahler manifold cases. Firstly, we prove an existence theorem of Poisson metrics on simple projectively flat bundles over compact Hermitian manifolds. As its application, we obtain a vanishing theorem of characteristic classes of projectively flat bundles. Secondly, on compact Hermitian manifolds which satisfy Gauduchon and astheno-Kaller conditions, we combine the continuity method and the heat flow method to prove that every semi-stable Higgs bundle with Delta(E, (partial derivative) over bar (E)) . [omega(n-2)] = 0 must be an extension of stable Higgs bundles. Using the above results, over some compact non-Kahler manifolds (M, omega), we establish an equivalence of categories between the category of semi-stable (poly-stable) Higgs bundles (E, (partial derivative) over bar (E), phi) with Delta(E, (partial derivative) over bar (E)) . [omega(n-2)] = 0 and the category of (semi-simple) projectively flat bundles (E, D) with root-1F(D) = alpha circle times Id(E) for some real (1,1)-form alpha.