The non-abelian Hodge theory was developed by Corlette-Donaldson-Hitchin-Simpson, it bridges representations of fundamental group, flat bundles and Higgs bundles. In this talk, we introduce our recent works on semi-stable Higgs bundle and  the existence of Poisson metric. As application, we established a extended non-abelian Hodge correspondence on  the projectively flat bundles case and some non-K/"ahler case.  These works are joint with Changpeng Pan and  Chuanjing Zhang.

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