Anticanonical minimal models and Zariski decomposition

Abstract

Birkar and Hu showed that if a pair $(X,\hat{I}”)$ is lc and $K_X+\hat{I}”$ admits a birational Zariski decomposition, then $(X,\hat{I}”)$ has a minimal model. Analogously, we prove that if a pair $(X,\hat{I}”)$ is pklt and $-(K_X+\hat{I}”)$ admits a birational Zariski decomposition, then $(X,\hat{I}”)$ has an anticanonical minimal

In previous version, there was a mistake when we apply the theorem of Birkar and Hu. So I revised section 4 and added Lemma 4.3