ACC conjecture for log canonical thresholds

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Let I⊆[0,1]I\subseteq[0,1] and J⊆R>0J\subseteq\mathbb{R}_{>0} be sets satisfying the descending chain condition. For a log pair (X,Δ)(X,\Delta) of dimension nn over an arbitrary field, with coefficients of Δ\Delta in II, and an effective R\mathbb{R}-Cartier divisor MM whose coefficients lie in JJ, define

lct⁡(X,Δ;M)=sup⁡{t∈R∣(X,Δ+tM) is log canonical}\operatorname{lct}(X,\Delta;M)=\sup\{t\in\mathbb{R}\mid (X,\Delta+tM)\text{ is log canonical}\}

and let LCT⁡n(I,J)\operatorname{LCT}_n(I,J) be the set of all such thresholds. ACC conjecture for log canonical thresholds. The set LCT⁡n(I,J)\operatorname{LCT}_n(I,J) satisfies the ascending chain condition.

This conjecture concerns the discreteness of log canonical thresholds under DCC conditions on the coefficient sets. Its status is not resolved by the supplied context.

References

Primary source

Omprokash Das, “On the boundedness of anti-canonical volumes of singular Fano 3-folds in characteristic p>5”, arXiv:1808.02102 (2019).

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