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We prove Bogomolov's inequality on a normal projective variety in positivecharacteristic and we use it to show some new restriction theorems and a newboundedness result. Then we redefine Higgs sheaves on normal varieties and weprove restriction theorems and Bogomolov type inequalities for semistablelogarithmic Higgs sheaves on some normal varieties in an arbitrarycharacteristic.

Bogomolov's inequality and Higgs sheaves on normal varieties in positive characteristic