Equality in the Bogomolov--Miyaoka--Yau inequality in the non-general type case.

We classify all minimal models X of dimension n, Kodaira dimension n-1 and with vanishing Chern number $c_1^{n-2}c_2(X)=0$. This solves a problem of Koll\'ar. Completing previous work of Koll\'ar and Grassi, we also show that there is a universal constant $\epsilon>0$ such that any minimal threefold satisfies either $c_1c_2=0$ or $-c_1c_2>\epsilon$. This settles completely a conjecture of Koll\'ar.