Abstract Let X be a complex projective variety and D a reduced divisor on X. Under mild conditions on the singularities of ( X , D ) $(X,D)$ , which includes the case of smooth X with simple normal crossing D, and by running the minimal model program, we obtain by induction on dimension via adjunction geometric criteria guaranteeing various positivity conditions for K X + D $K_{X}+D$ . Our geometric criterion for K X + D $K_{X}+D$ to be numerically effective yields also a geometric version of the cone theorem and a criterion for K X + D $K_{X}+D$ to be pseudo-effective with mild hypothesis on D. We also obtain, assuming the abundance conjecture and the existence of rational curves on Calabi–Yau manifolds, an optimal geometric sharpening of the Nakai–Moishezon criterion for the ampleness of a divisor of the form K X + D $K_{X}+D$ , a criterion verified under a canonical hyperbolicity assumption on ( X , D ) $(X,D)$ . Without these conjectures, we verify this ampleness criterion with mild assumptions on D, being none in dimension two and D ≠ 0 $D\neq 0$ in dimension three.