On minimal model program and Zariski decomposition of potential triples

TL;DR

This paper explores the properties of potential triples involving pairs and pseudoeffective divisors, establishing links to Zariski decompositions and generalized minimal model programs, with applications to pklt pairs and minimal models.

Contribution

It introduces a method to associate generalized pair structures to potential triples with Zariski decompositions and demonstrates running the generalized MMP in this context.

Findings

If $D$ admits a birational Zariski decomposition, a generalized pair structure can be associated.

The generalized MMP can be run on $(K_X+ riangle+D)$ in these cases.

Existence of a $-(K_X+ riangle)$-minimal model under certain Zariski decomposition conditions.

Abstract

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.