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

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