Nonabelian Hodge theory for Fujiki class $\mathcal{C}$ manifolds

The nonabelian Hodge correspondence (Corlette-Simpson correspondence), between the polystable Higgs bundles with vanishing Chern classes on a compact K"ahler manifold $X$ and the completely reducible flat connections on $X$, is extended to the Fujiki class $\mathcal{C}$ manifolds.