ACC conjecture for log canonical thresholds of generalized pairs

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Let dd be a positive integer and let Γ⊂[0,+∞)\Gamma\subset [0,+\infty) be a DCC set. For a projective generalized pair (X,B,M)(X,B,\mathbf{M}) of dimension dd, with B,D∈ΓB,D\in\Gamma and M,N∈Nef(X,Γ)\mathbf{M},\mathbf{N}\in\mathrm{Nef}(X,\Gamma), let lct⁡(X,B,M;D,N)\operatorname{lct}(X,B,\mathbf{M};D,\mathbf{N}) denote its log canonical threshold.

ACC conjecture for log canonical thresholds. The set

{lct⁡(X,B,M;D,N)∣dim⁡X=d, B,D∈Γ, M,N∈Nef(X,Γ)}\{\operatorname{lct}(X,B,\mathbf{M};D,\mathbf{N})\mid \dim X=d,\ B,D\in\Gamma,\ \mathbf{M},\mathbf{N}\in\mathrm{Nef}(X,\Gamma)\}

satisfies the ascending chain condition.

This conjecture asserts uniform ACC for log canonical thresholds in fixed dimension and with coefficients drawn from a DCC set, including nef parts of generalized pairs. The supplied text gives no resolution status or further evidence, so the conjecture is recorded as open.

References

Primary source

Tianle Yang, Zelin Ye and Zhiyao Zhang, “Existence of minimal models for threefold generalized pairs in positive characteristic”, arXiv:2408.12269 (2026).

Additional references

19 papers in this index state this conjecture (1999–2024). The statement above is taken from the most recent of them; the others are arXiv:2305.06493, arXiv:2209.13122, arXiv:2207.04610, arXiv:2202.11346, arXiv:2112.09501, arXiv:1904.09642, arXiv:1903.04338, arXiv:1803.02539, arXiv:1501.06248, arXiv:1405.5649, arXiv:1104.4981, arXiv:0912.3186, and 6 more.

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