Existence of the minimal model program for log canonical generalized pairs

TL;DR

This paper proves the existence of the minimal model program for log canonical generalized pairs by introducing linearly decomposable generalized pairs and refining key theoretical tools, removing several traditional assumptions.

Contribution

It introduces LD generalized pairs and extends the MMP to all log canonical generalized pairs without requiring klt, NQC, or Q-factoriality assumptions.

Findings

Proves existence of flips for log canonical generalized pairs.

Establishes the minimal model program for arbitrary log canonical generalized pairs.

Refines special termination and Kollár-type gluing theory.

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 Koll\'ar-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.