Semistability of the tangent sheaf of singular varieties

The main goal of this paper is to prove the polystability of the logarithmic tangent sheaf T X (-log D) of a log canonical pair (X, D) whose canonical bundle K X + D is ample, generalizing in a significant way a theorem of Enoki.We apply this result and the techniques involved in its proof to get a Bogomolov-type inequality for Chern numbers for log canonical pairs, a version of the semistability theorem for stable varieties (the higher-dimensional analogue of Deligne-Mumford's stable curves) and finally the generic semipositivity of the sheaf of logarithmic differentials of a log canonical pair with pseudo-effective log canonical class, in the spirit of Miyaoka's theorem.