Boundedness in general type MMP

TL;DR

This paper proves that in the minimal model program for general type varieties, the singularities' minimal log discrepancies are finitely many and the fibers of contractions are bounded, using local volume analysis.

Contribution

It establishes finiteness of minimal log discrepancies and boundedness of fibers in the MMP for general type, advancing understanding of singularity behavior.

Findings

Minimal log discrepancies take finitely many values.

Fibers of extremal contractions and flips are bounded.

Local volume analysis is key to the proof.

Abstract

We show that in any sequence of a general type MMP, the minimal log discrepancy of singularities takes at most finitely many values, and the fibers of all the extremal contractions and flips belong to a bounded family. A key ingredient in the proof is an analysis of the behavior of local volumes in the MMP.