The Miyaoka-Yau inequality for minimal Kähler klt spaces

In this paper, we obtain the generalized Bogomolov inequality for reflexive Higgs sheaves defined on the regular locus of compact Kähler klt spaces. As an application, we establish the Miyaoka-Yau inequality for all minimal Kähler klt spaces. Apart from providing a self-contained formulation and investigation of Higgs sheaves on complex normal spaces, the analytical part of our approach is the establishment of $L^p$-approximate critical Hermitian structures for Higgs orbi-bundles on Gauduchon orbifolds. This also leads to the semistability (resp. generically nefness) of torsion-free sheaves under symmetric, exterior powers and tensor products in the singular setting.