On the Abundance Problem for $3$-folds in characteristic $p>5$

Abstract

In this article we prove two cases of the abundance conjecture for $3$-folds in characteristic $p>5$: $(i)$ $(X, \Delta)$ is KLT and $\kappa(X, K_X+\Delta)=1$, and $(ii)$ $(X, 0)$ is KLT, $K_X\equiv 0$ and $X$ is not uniruled.Comment: With an Appendix by Christopher Hacon. Some reference updated and minor changes. This is the final versio