On termination of flips and fundamental groups

TL;DR

This paper introduces a boundedness conjecture for the regional fundamental group of klt singularities and shows its implications for the termination of flips with scaling, providing proofs in specific singularity cases.

Contribution

It proposes a new boundedness conjecture and demonstrates its implications for the termination of flips, with proofs in particular classes of singularities.

Findings

Boundedness conjecture proven for toric, quotient, isolated 3-fold, and exceptional singularities.

Boundedness conjecture, Zariski closedness, and minimal log discrepancy bounds imply flip termination.

Establishes connections between fundamental groups and minimal model program termination.

Abstract

In this article, we propose a boundedness conjecture for the regional fundamental group of klt singularities. We prove that this boundedness conjecture, the Zariski closedness of the diminished base locus of $K_X$, and an upper bound for minimal log discrepancies, imply the termination of flips with scaling. We prove the boundedness conjecture in the case of toric singularities, quotient singularites, isolated $3$-fold singularities, and exceptional singularities.