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McKernan's ACC conjecture for interpolated lc thresholds

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Let dd be a positive integer. An ACC set is a subset of the real numbers satisfying the ascending chain condition. For a Q\mathbb Q-Gorenstein foliation F\mathcal{F} on a Q\mathbb Q-Gorenstein variety XX, define the interpolated log canonical threshold by

lct⁡(X;F):=sup⁡{t≥0∣(X,F,t) is log canonical}.\operatorname{lct}(X;\mathcal{F}):=\sup\{t\geq 0\mid (X,\mathcal{F},t)\text{ is log canonical}\}.

McKernan's ACC conjecture. There exists an ACC set Γ⊂[0,1]\Gamma\subset [0,1] depending only on dd such that, for every such F\mathcal{F} on XX, the threshold lct⁡(X;F)\operatorname{lct}(X;\mathcal{F}) belongs to Γ\Gamma. This is the foliated analogue of Shokurov's ACC conjecture for log canonical thresholds. The classical counterpart was proved by Hacon, McKernan, and Xu, while the foliated statement is presented here as the uniformity principle underlying the parameter in the birational boundedness theorem and remains unresolved in the stated generality.

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Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. McKernan's ACC conjecture for interpolated lc thresholds

    Given a Q\mathbb{Q}-factorial normal variety XX of dimension dd with a foliation F\mathcal{F}, define its interpolated lc threshold by

    tˉ=sup⁡{t∈[0,1]∣(X,F,t) is lc}.\bar t=\sup\{t\in[0,1]\mid (X,\mathcal{F},t)\text{ is lc}\}.

    McKernan's ACC conjecture. For every positive integer dd and every DCC set Γ⊂[0,+∞)\Gamma\subset[0,+\infty), there is an ACC set Γ′=Γ′(d,Γ)\Gamma'=\Gamma'(d,\Gamma) such that, for every foliated triple (X,F,B)(X,\mathcal{F},B) with KF+BK_{\mathcal{F}}+B and KXK_X R\mathbb{R}-Cartier, dim⁡X=d\dim X=d, and B∈ΓB\in\Gamma,

    sup⁡{t∈[0,1]∣(X,F,tB,t) is lc}∈Γ′.\sup\{t\in[0,1]\mid (X,\mathcal{F},tB,t)\text{ is lc}\}\in\Gamma'.

    This conjecture is widely open in dimension at least three, even for algebraically integrable foliations; McKernan sketched a proof in dimension two.

    source: Paolo Cascini, Jingjun Han, Jihao Liu, Fanjun Meng, Calum Spicer, Roberto Svaldi and Lingyao Xie, “Minimal model program for algebraically integrable adjoint foliated structures”, arXiv:2408.14258 (2024).

Sources & referencesView supporting material

Primary source

Paolo Cascini, Jihao Liu, Calum Spicer and Roberto Svaldi, “Birational boundedness of stable families”, arXiv:2604.24106 (2026).

accbirational-geometryfoliated-geometryfoliationslog-canonical-thresholds

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