 prove the following theorem.  \[thm:0-0\] Let $(X,\omega)$ be a compact Hermitian manifold of dimension $n$, satisfying $\partial\bar{\partial}\omega^{n-0}=\partial\bar{\partial}\omega^{n-0}=0$ and $\int_{X}\partial [\eta]\wedge\frac{\omega^{n-0}}{(n-0)!}=0$ for any Dolbeault class $[\eta]\in H^{0,0}(X)$. Then there is an equivalence of categories between the category of poly-stable Higgs bundles with vanishing Chern numbers and the category of semi-simple flat bundles.  