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

Abstract

The non-abelian Hodge correspondence was established by Corlette (1988), Donaldson (1987), Hitchin (1987) and Simpson (1988, 1992). It states that on a compact Kähler manifold (X,ω), 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-Kähler 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-Kähler conditions, we combine the continuity method and the heat flow method to prove that every semi-stable Higgs bundle with $\Delta \left( {E,{{\bar \partial }_E}} \right) \cdot \left[ {{\omega ^{n - 2}}} \right] = 0$ must be an extension of stable Higgs bundles. Using the above results, over some compact non-Kähler manifolds (M,ω), we establish an equivalence of categories between the category of semi-stable (poly-stable) Higgs bundles $\left( {E,{{\bar \partial }_E},\phi } \right)$ with $\Delta \left( {E,{{\bar \partial }_E}} \right) \cdot \left[ {{\omega ^{n - 2}}} \right] = 0$ and the category of (semi-simple) projectively flat bundles (E, D) with $\sqrt { - 1} {F_D} = \alpha \otimes {\rm{I}}{{\rm{d}}_E}$ for some real (1,1)-form α).