On termination of flips and exceptionally non-canonical singularities
Jingjun Han, Jihao Liu
TL;DR
The paper introduces exceptionally non-canonical (enc) singularities and uses them to study the termination of flips in the minimal model program, linking this question to the ACC for minimal log discrepancies (mlds) of enc pairs and to terminal flip termination. The authors develop an inductive framework based on lower mld bounds, complements, index‑one covers, and decomposable $\ ext{R}$‑complements to prove ACC results for enc mlds in dimension $3$, and derive termination in dimension $3$ and, under enc ACC, in dimension $4$. They further show that termination in higher dimensions can be reduced to ACC/LSC hypotheses for terminal and enc pairs, and they treat canonical/terminal threefolds, cyclic quotient enc singularities, and enc cDV quotients with detailed classifications. The work connects enc singularities to local K‑stability, Calabi–Yau and mirror symmetry phenomena, and conjectural boundedness theories, providing new pathways toward the flip termination problem in higher dimensions and suggesting tighter control over singularities that arise in MMP sequences.
Abstract
We systematically introduce and study a new type of singularities, namely, exceptionally non-canonical (enc) singularities. This class of singularities plays an important role in the study of many questions in birational geometry, and has tight connections with local K-stability theory, Calabi-Yau varieties, and mirror symmetry. We reduce the termination of flips to the termination of terminal flips and the ACC conjecture for minimal log discrepancies (mlds) of enc pairs. As a consequence, the ACC conjecture for mlds of enc pairs implies the termination of flips in dimension $4$. We show that, in any fixed dimension, the termination of flips follows from the lower-semicontinuity for mlds of terminal pairs, and the ACC for mlds of terminal and enc pairs. Moreover, in dimension $3$, we give a rough classification of enc singularities, and prove the ACC for mlds of enc pairs. These two results provide a second proof of the termination of flips in dimension $3$ which does not rely on any difficulty function. Finally, we propose and prove the special cases of several conjectures on enc singularities and local K-stability theory. We also discuss the relationship between enc singularities, exceptional Fano varieties, and Calabi-Yau varieties with small mlds or large indices via mirror symmetry.