On termination of flips and exceptionally noncanonical singularities

Han, Jingjun¹ ; Liu, Jihao²

¹ Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, China
² Department of Mathematics, Peking University, Beijing, China

Geometry & topology, Tome 29 (2025) no. 1, pp. 399-441.

Résumé

We systematically introduce and study a new type of singularity, namely, exceptionally noncanonical (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.

Détail

DOI : 10.2140/gt.2025.29.399

Keywords: minimal model program, minimal log discrepancy, termination of flips

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