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It is a classical fact that the Chern classes of any minimal smooth projective surfacesatisfy the so-called Bogomolov-Miyaoka-Yau inequality and it is known explicitely for which surfaces equality is attained. More generally, if is a minimal projective variety of dimension n, Miyaoka proved that for any ample divisor on . In this talk I want to discuss the structure of those varieties attaining equality. In particular, we will see that abundance holds for such varieties. This is joint work with M. Iwai and S.-I. Matsumura.