Abundance for threefolds in positive characteristic when $\nu =2$

Abstract

In this paper, we prove the abundance conjecture for threefolds over a perfect field $k$ of characteristic $p>3$ in the case of numerical dimension equals to $2$. More precisely, we prove that if $(X,B)$ be a projective lc threefold pair over $k$ such that $K_X+B$ is nef and $\nu (K_X+B)=2$, then $K_X+B$ is semiample.

Final version, to appear in Compositio Mathematica