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In this paper, we first prove a complete version of theDonaldson-Uhlenbeck-Yau theorem over normal varieties, including normal Kaehlervarieties and projective normal varieties with multiple polarizations. Inparticular, this gives the polystability of reflexive sheaves under symmetricand exterior powers and tensor products. As a consequence of the singularDonaldson-Uhlenbeck-Yau theorem, the complete Hitchin-Kobayashi correspondenceover normal varieties smooth in codimension two is built by showing that anadmissible Hermitian-Yang- Mills connection defines a polystable reflexivesheaf. Furthermore, it is shown that the Hermitian-Yang-Mills connection givesa lower bound for the discriminants of any Kaehler resolutions, which gives aBogomolov-Gieseker inequality over normal varieties and a characterization ofthe equality using projectively flat connections. We discuss typical casesincluding normal surfaces and varieties smooth in codimension two where wecould simplify the Bogomolov-Gieseker inequality and endow it with topologicalmeanings. We also prove the Bogomolov-Gieseker inequality for semistablereflexive sheaves and characterize the class of semistable sheaves that satisfythe Bogomolov-Gieseker equality. Finally, as another application, we give a newcriteria for when a normal Kaehler variety with trivial first Chern class is afinite quotient of torus.

Admissible Hermitian-Yang-Mills connections over normal varieties