[HG_AG] Anticanonical minimal models and Zariski decomposition
ABSTRACT
Let $(X,\Delta)$ be an lc pair. The minimal model program is a sequence of birational maps, which makes the divisor $K_{X}+\Delta$ closer to a nef divisor. Now, we know that we can run a minimal model program. However, we do not know whether the sequence terminates in finite steps or not. If the sequence stops, then we obtain either a minimal model or a mori fiber space. Besides the termination, Birkar and Hu showed that if a pair $(X,\Delta)$ is lc and $K_{X}+\Delta$ admits a birational Zariski decomposition, then $(X,\Delta)$ has a minimal model. Analogously, we prove that if a pair $(X,\Delta)$ is pklt and $-(K_{X}+\Delta)$ admits a birational Zariski decomposition, then $(X,\Delta)$ has an anticanonical minimal model.