ACC conjecture for log canonical thresholds of generalized pairs

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,DΓB,D\in\Gamma and M,NNef(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)dimX=d, B,DΓ, M,NNef(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.

Sources & referencesView supporting material

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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