Shokurov; Mld’s vs thresholds and flips

Minimal log discrepancies (mld’s) are related not only to termina-tion of log flips [22], and thus to the existence of log flips [11] but also to the ascending chain condition (acc) of some global invariants and in-variants 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; termination of log flips in dimension ≤ d + 1 for effective pairs; and existence of pl flips in dimension ≤ d + 2. This also gives new proofs of some well-known and new results in the field in low dimensions: the acc conjecture holds for a-lc thresholds of sur