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Existence of the minimal model program for log canonical generalized pairs

Zhengyu Hu, Jihao Liu

Abstract

We introduce linearly decomposable (LD) generalized pairs, which serve as a workable substitute for rational decompositions in the non-NQC setting. Using LD generalized pairs, together with a refinement of special termination and Kollár-type gluing theory, we prove the existence of flips for log canonical generalized pairs without assuming the klt condition, the NQC condition, or $\mathbb Q$-factoriality. Together with the cone and contraction theorems, this yields the existence of the minimal model program for arbitrary log canonical generalized pairs.

Existence of the minimal model program for log canonical generalized pairs

Abstract

We introduce linearly decomposable (LD) generalized pairs, which serve as a workable substitute for rational decompositions in the non-NQC setting. Using LD generalized pairs, together with a refinement of special termination and Kollár-type gluing theory, we prove the existence of flips for log canonical generalized pairs without assuming the klt condition, the NQC condition, or -factoriality. Together with the cone and contraction theorems, this yields the existence of the minimal model program for arbitrary log canonical generalized pairs.
Paper Structure (38 sections, 68 theorems, 175 equations, 1 table)

This paper contains 38 sections, 68 theorems, 175 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Special notation
  4. Generalized pairs
  5. Relative Nakayama-Zariski decomposition
  6. Perturb generalized pair to non-lc pair
  7. Iitaka dimensions
  8. Linearly decomposable generalized pair
  9. Definition of LD generalized pair
  10. Lc strata of LD generalized pair
  11. Nef and semi-ampleness of LD generalized pairs
  12. LD generalized pair under standard transformations
  13. LD property under ample polarization
  14. Models
  15. Basic properties of models
  16. ...and 23 more sections

Key Result

Theorem 1.1

Let $(X,B,{\bf{M}})/U$ be a log canonical generalized pair and let $f: X\rightarrow Z$ be a $(K_X+B+{\bf{M}}_X)$-flipping contraction$/U$. Then the flip $f^+: X^+\rightarrow Z$ of $f$ exists.

Theorems & Definitions (155)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Remark 1.7
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 145 more