Non-vanishing implies numerical dimension one abundance

Abstract

We show that the non-vanishing conjecture implies the abundance conjecture when $\hat{I}½\le 1$. We also prove the abundance conjecture in dimension $\le 5$ when $\hat{I}º\ge 0$ and $\hat{I}½\le 1$ unconditionally.

24 pages. A part of the proof is refined by applying a result of Lazic-Peternell and now the log canonical case of Theorem 1.6 also holds