The ACC conjecture for lc thresholds of adjoint foliated structures

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Let (X,F,B,t)(X,\mathcal{F},B,t) be an adjoint foliated structure, with B=Bninv⁡+Binv⁡B=B^{\operatorname{ninv}}+B^{\operatorname{inv}}, and let DD be an R\mathbb{R}-divisor. The ACC conjecture for lc thresholds of adjoint foliated structures. For every positive integer dd, real number t∈[0,1]t\in[0,1], and DCC set Γ⊂[0,+∞)\Gamma\subset[0,+\infty), there is an ACC set Γ′\Gamma' such that, whenever dim⁡X=d\dim X=d, B,t,D∈ΓB,t,D\in\Gamma, one has

sup⁡{s∈[0,1]∣(X,F,B+s(Dninv⁡+(1−t)Dinv⁡),t) is lc}∈Γ′.\sup\{s\in[0,1]\mid (X,\mathcal{F},B+s(D^{\operatorname{ninv}}+(1-t)D^{\operatorname{inv}}),t)\text{ is lc}\}\in\Gamma'.

The paper lists this among three widely open ACC conjectures, including in dimension two.

References

Primary 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).

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