On minimal model program and Zariski decomposition of potential triples

In this paper, we investigate properties of potential triples $(X,\Delta,D)$ which consists of a pair $(X,\Delta)$ and a pseudoeffective $\mathbb{R}$-Cartier divisor $D$. In particular, we show that if $D$ admits a birational Zariski decomposition, then one can associate a generalized pair structure to the potential triple $(X,\Delta,D)$. Moreover, we can run the generalized MMP on $(K_X+\Delta+D)$ as special cases. As an application, we also show that for a pklt pair $(X,\Delta)$, if $-(K_X+\Delta)$ admits a birational Zariski decomposition with $\mathrm{NQC}$ positive part, then there exists a $-(K_X+\Delta)$-minimal model.