Singularities of Pairs in Positive Characteristic and in Characteristic Zero

For studies on singularities over the field of characteristic zero, we can use many convenient tools: resolutions of singularities, Bertini’s theorem (generic smoothness), cohomology vanishing of Kodaira type, and so on. However, in positive characteristic cases, these are not available. Therefore many good properties obtained in characteristic zero using these tools are not accessible in positive characteristic cases. My talk focuses on an invariant of singularities mld (minimal log discrepancy) and lct (log canonical threshold) and makes a bridge between positive characteristic and characteristic zero in the discussions of mld and lct. By the bridge, we obtain the following in the positive characteristic case:

1. The discreteness of log discrepancies.
2. Existence of prime divisor computing mld.
3. lct and accumulation points of lct are rational numbers.
4. ACC holds for lct of ideals on a smooth variety.