On minimal model program and Zariski decomposition of potential triples

Abstract

In this paper, we investigate properties of potential triples $(X,\hat{I}”,D)$ which consists of a pair $(X,\hat{I}”)$ 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 $(X,\hat{I}”,D)$. Moreover, we can run the generalized MMP on $(K_X+\hat{I}”+D)$ as special cases. As an application, we also show that for a pklt pair $(X,\hat{I}”)$, if $-(K_X+\hat{I}”)$ admits a birational Zariski decomposition with $\mathrm{NQC}$ positive part, then there exists a $-(K_X+\hat{I}”)$-minimal model.