Mld's vs thresholds and flips

TL;DR

This paper establishes deep connections between key conjectures in the Log Minimal Model Program, showing how certain boundedness and chain conditions imply broader termination and existence results across various dimensions.

Contribution

It proves that the LMMP, the acc conjecture for mld's, and boundedness of Mori-Fano varieties imply several other important conjectures and results in the LMMP, providing new proofs in low dimensions.

Findings

Acc conjecture holds for $a$-lc thresholds of surfaces.

Acc conjecture holds for lc thresholds of 3-folds.

Termination of 3-fold log flips for effective pairs.

Abstract

Minimal log discrepancies (mld's) are related not only to termination of log flips, and thus to the existence of log flips but also to the ascending chain condition (acc) of some global invariants and invariants of singularities in the Log Minimal Model Program (LMMP). In this paper, we draw clear links between several central conjectures in the LMMP. More precisely, our main result states that the LMMP, the acc conjecture for mld's and the boundedness of canonical Mori-Fano varieties in dimension $\le d$ imply the following: the acc conjecture for $a$-lc thresholds, in particular, for canonical and log canonical (lc) thresholds in dimension $\le d$; the acc conjecture for lc thresholds in dimension $\le d+1$; termination of log flips in dimension $\le d+1$ for effective pairs; and existence of pl flips in dimension $\le d+2$. This also gives new proofs of some well-known and new results in the field in low dimensions: the acc conjecture holds for $a$-lc thresholds of surfaces; the acc conjecture holds for lc thresholds of 3-folds; termination of 3-fold log flips holds for effective pairs; and the existence of 4-fold pl flips holds.