On minimal model program and Zariski decomposition of potential triples

TL;DR

The paper develops a framework to study potential triples $(X,\Delta,D)$ by leveraging birational Zariski decompositions of $D$ to attach generalized-pair structures and run the MMP on $K_X+\Delta+D$. The authors prove that when $D$ admits a birational Zariski decomposition, the potentially non-klt locus $\operatorname{pNklt}(X,\Delta,D)$ becomes Zariski closed and, for $\mathbb{Q}$-factorial plc triples, the $(K_X+\Delta+D)$-MMP can be executed. As applications, they obtain existence results for minimal models in pklt situations with an NQC positive part and derive MMP conclusions under suitable base-locus conditions, tying their theory to established results in the literature. The work broadens the MMP toolkit beyond standard pairs to potential triples, clarifying when good minimal models exist and how base loci influence the process.

Abstract

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

Theorems & Definitions (25)

• Theorem 1.1
• Corollary 1.2: cf. Jan
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Definition 2.4
• Lemma 2.5: cf. CJK
• proof
• Proposition 2.6: CJK