Abstract

Minimal log discrepancies (mld's) are related not only to termination of log flips [Shokurov, Algebr. Geom. Metody 246: 328–351, (2004)] 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 ≦ d imply the following: the ACC conjecture for a-lc thresholds, in particular, for canonical and log canonical (lc) thresholds in dimension ≦ d; the ACC conjecture for lc thresholds in dimension ≦ d + 1; and termination of log flips in dimension ≦ d + 1 for effective lc pairs. In particular, when d = 3 we can drop the assumptions on LMMP and boundedness of canonical Mori-Fano varieties.