A remark on generalized abundance for surfaces

Abstract

Let $(X,\Delta )$ be a projective klt pair of dimension $2$ and let $L$ be a nef $\mathbb{Q}$-divisor on $X$ such that $K_X+\Delta +L$ is nef. As a complement to the Generalized Abundance Conjecture by Lazić and Peternell, we prove that if $K_X+\Delta$ and $L$ are not proportional modulo numerical equivalence, then $K_X+\Delta +L$ is semiample. An example due to Lazić shows that this is no longer true in any dimension $n\ge 3$.

Final version, to appear in European Journal of Mathematics