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$P$-trivial MMP, Zariski decompositions and minimal models for generalised pairs

Zhengyu Hu

TL;DR

The paper tackles extending minimal model theory to generalised pairs in the non-NQC setting by introducing a $P$-trivial MMP framework, where steps preserve a nef divisor $P$. By coupling this approach with large-$ ho$ MMPs on $K_X+B+M+ ho P$ and Nakayama-Zariski decompositions with nef positive parts, the authors establish existence of minimal models in several regimes, including g-klt and g-lc pairs, and in dimension $3$. They also prove a special termination result for $P$-trivial MMP and connect these methods to Zariski decompositions and weak minimal models, offering a robust pathway toward minimal models beyond the NQC hypothesis. Collectively, these results advance the birational geometry of generalized pairs and inform broader conjectures such as BAB and the role of Zariski decompositions in minimal model theory.

Abstract

We develop a theory of $P$-trivial MMP whose each step is $P$-trivial for a given nef divisor $P$. As an application, we prove that, given a projective generalised klt pair $(X,B+M)$ with data $M'$ being just a nef $\mathbb{R}$-divisor, if $K_X+B+M$ birationally has a Nakayama-Zariski decomposition with nef positive part, and either if $M'$ or the positive part is log numerically effective, then it has a minimal model. Furthermore, we prove this for generalised lc pairs in dimension $3$. This is a generalisation of the main theorem of [Birkar-Hu14]. We also prove some related results.

$P$-trivial MMP, Zariski decompositions and minimal models for generalised pairs

TL;DR

The paper tackles extending minimal model theory to generalised pairs in the non-NQC setting by introducing a -trivial MMP framework, where steps preserve a nef divisor . By coupling this approach with large- MMPs on and Nakayama-Zariski decompositions with nef positive parts, the authors establish existence of minimal models in several regimes, including g-klt and g-lc pairs, and in dimension . They also prove a special termination result for -trivial MMP and connect these methods to Zariski decompositions and weak minimal models, offering a robust pathway toward minimal models beyond the NQC hypothesis. Collectively, these results advance the birational geometry of generalized pairs and inform broader conjectures such as BAB and the role of Zariski decompositions in minimal model theory.

Abstract

We develop a theory of -trivial MMP whose each step is -trivial for a given nef divisor . As an application, we prove that, given a projective generalised klt pair with data being just a nef -divisor, if birationally has a Nakayama-Zariski decomposition with nef positive part, and either if or the positive part is log numerically effective, then it has a minimal model. Furthermore, we prove this for generalised lc pairs in dimension . This is a generalisation of the main theorem of [Birkar-Hu14]. We also prove some related results.
Paper Structure (10 sections, 54 theorems, 37 equations)

This paper contains 10 sections, 54 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Minimal model theory for non-NQC g-pairs
  4. Non-NQC g-pairs
  5. Termination with scaling versus the existence of minimal models
  6. MMP on $K_X+B+M+\alpha P$ for $\alpha \gg 0$
  7. $P$-trivial MMP and its applications
  8. $P$-trivial MMP
  9. A special termination on $P$-trivial MMP
  10. Zariski decompositions

Key Result

Theorem 1.3

Let $(X/Z,B+M)$ be a $\mathbb Q$-factorial g-dlt pair and $P$ be a nef$/Z$ divisor. Write $N=K_X+B+M-P$. Suppose one of the followings holds: Then, there is a number $\alpha >0$ such that any MMP on $K_X+B+M + a P$ is $P$-trivial for $a \ge \alpha$.

Theorems & Definitions (112)

  • Conjecture 1.1
  • Example 1.2
  • Theorem 1.3: =Lemma \ref{['lem-P-trivial']}+Corollary \ref{['cor-klt']}
  • Theorem 1.4: =Corollary \ref{["cor-klt'"]}
  • Theorem 1.5: =Corollary \ref{["cor-3-dim'"]}
  • Theorem 1.6: =Corollary \ref{['cor-dlt-abund']}
  • Definition 1.7: $P$-trivial MMP, see Definition \ref{['defn-P-mmp']}
  • Theorem 1.8: A special termination, =Theorem \ref{['thm-sp-term']}
  • Definition 1.9: Degenerations, =Definition \ref{['defn-degen']}
  • Conjecture 1.10
  • ...and 102 more