ACC for log canonical thresholds. (English) Zbl 1320.14023

A partially ordered set is said to satisfies the ascending chain condition, ACC, (resp. the descending chain condition, DCC), if it does not contained infinite increasing (decreasing) sequences.
The minimal model conjecture would follow from the proof that there is no infinite sequence of flips. V. V. Shokurov [in: Algebraic geometry. Methods, relations, and applications. Collected papers. Dedicated to the memory of Andrei Nikolaevich Tyurin. Moscow: Maik Nauka/Interperiodica. 315–336 (2004; Zbl 1107.14012)] introduced an invariant of singularities, the minimal log discrepancy (mld): a rational number associated to every singularity, with bigger numbers corresponding to milder singularities. Since flips improve the singularities, minimal log discrepancies increase under flips. Therefore, the ACC condition for mld would imply that any sequence of flips terminates.
Shokurov conjectured also the ACC conjecture for an other invariant of singularities, the log canonical threshold, that is a bit simpler and more studied that the minimal log discrepancy. Given a log canonical pair (X, \(\Delta\)) and \(M \geq 0\), an \(\mathbb{R}\)-Cartier divisor, the log canonical threshold of \(M\) with respect to \((X, \Delta)\) is \[ \text{lct}(X, \Delta; M) = \text{sup}\{ t \in \mathbb{R} | (X, \Delta + tM) \text{ is log canonical}\} \]
Fixing a positive integer \(n\), let \(\mathcal{I}_n(I)\) denote the set of log canonical pairs \((X, \Delta)\), where \(X\) is a variety of dimension \(n\) and the coefficients of \(\Delta\) belong to set \(I \subseteq [0, 1]\), then set \[ \text{LCT}_n(I, J) = \{\text{lct}(X, \Delta; M) | (X, \Delta) \in \mathcal{I}_n(I)\} \] where the coefficients of \(M\) belong to a subset \(J\) of the positive real numbers.
The main theorem of the paper under review states that if \(I\) and \(J\) satisfy the DCC, then \(\text{LCT}_n(I, J)\) satisfies the ACC.
This powerful result has important applications in the minimal model program. For examples, after the work of C. Birkar [Duke Math. J. 136, No. 1, 173–180 (2007; Zbl 1109.14018)], one important consequence is termination of flips for klt pairs (X, \(\Delta\)) in dimension \(n - 1\) implies termination in dimenison \(n\), for \(K_X + \Delta\) numerically equivalent to an effective divisor.


14E30 Minimal model program (Mori theory, extremal rays)
14C20 Divisors, linear systems, invertible sheaves
Full Text: DOI arXiv


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