Positivity criteria for log canonical divisors and hyperbolicity

Abstract

Let X be a complex projective variety and D a reduced divisor on X. Under mild conditions on the singularities of (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 KX + D. Our geometric criterion for KX + D to be numerically effective yields also a geometric version of the cone theorem and a criterion for KX + 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 KX + D, a criterion verified under a canonical hyperbolicity assumption on (X, D). Without these conjectures, we verify this ampleness criterion with mild assumptions on D, being none in dimension two and D ≠ 0 in dimension three.