ACC for minimal log discrepancies and termination of flips

May 14 to May 18, 2012

at the

American Institute of Mathematics, San Jose, California

organized by

Tommaso de Fernex and Christopher Hacon

Original Announcement

This workshop will be devoted to two closely connected conjectures in the minimal model program.

The minimal model program generalizes the classification of surfaces to higher dimensional varieties. After the recent proof of the existence of flips (due to Birkar-Cascini-Hacon-McKernan), one of the main remaining open problems in the field is the Termination of Flips Conjecture. This is an important conjecture with many applications to questions related to the minimal model program. As shown by Shokurov, termination of flips can be reduced to a question on minimal log discrepancies, an invariant that gives a sophisticated measure of singularities. Minimal log discrepancies are known to improve after each flip, and Shokurov conjectured that these invariants have no accumulation points from below, that is, that they satisfy the ascending chain condition (ACC). This conjecture, together with a conjecture on the semicontinuity of these invariants, is known to imply the termination of flips. Results on minimal log discrepancies are of independent interest as these invariants are important in the study of singularities.

The above conjectures constitute the main topics for the workshop:

- Termination of flips.
- ACC for minimal log discrepancies.

These topics are closely related to other central questions in this area of research such as the ACC for log canonical thresholds, the Borisov-Alexeev-Borisov Conjecture, and the Abundance Conjecture. In view of recent spectacular results in this area, we hope that this workshop will help to spur further progress on these conjectures.

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

Ideal-adic semi-continuity of minimal log discrepancies on surfaces
by  Masayuki Kawakita,  Michigan Math. J. 62 (2013), no. 2, 443-447  MR3079272
The index of a threefold canonical singularity
by  Masayuki Kawakita,  Amer. J. Math. 137 (2015), no. 1, 271-280  MR3318092
On log canonical rings
by  Osamu Fujino and Yoshinori Gongyo,  Higher dimensional algebraic geometry in honour of Professor Yujiro Kawamata's sixtieth birthday, 159–169, Adv. Stud. Pure Math., 74, Math. Soc. Japan, Tokyo, 2017  MR3791212
Singularities with the highest Mather minimal log discrepancy
by  Shihoko Ishii and Ana Reguera,  Math. Z. 275 (2013), no. 3-4, 1255-1274  MR3127058
On the log discrepancies in toric Mori contractions
by  Valery Alexeev and Alexander Borisov,  Proc. Amer. Math. Soc. 142 (2014), no. 11, 3687-3694  MR3251710