It is a classical fact that the Chern classes of any minimal smooth projective surface $X$ satisfy the so-called Bogomolov-Miyaoka-Yau inequality $3c_2(X)−c^2_1(X) \geq 0$ and it is known explicitely for which surfaces equality is attained. More generally, if $X$ is a minimal projective variety of dimension n, Miyaoka proved that $(3c_2(X) − c^2_1(X))H^{n−2} \geq 0$ for any ample divisor $H$ on $X$. In this talk I want to discuss the structure of those varieties $X$ attaining equality. In particular, we will see that abundance holds for such varieties. This is joint work with M. Iwai and S.-I. Matsumura.
