ACC conjecture for minimal log discrepancies of exceptional non-canonical pairs

Let dd be a positive integer and let Γ[0,1]\Gamma\subset[0,1] be a set of real numbers. Define

eMLDd(Γ):={mld(X,B)(X,B) is enc, dimX=d, BΓ}.\operatorname{eMLD}_{d}(\Gamma):=\{\operatorname{mld}(X,B)\mid (X,B)\text{ is enc},\ \dim X=d,\ B\in\Gamma\}.

ACC conjecture for mlds of enc pairs. If Γ\Gamma satisfies the DCC, then eMLDd(Γ)\operatorname{eMLD}_{d}(\Gamma) satisfies the ACC. If Γ\Gamma is finite, then eMLDd(Γ)\operatorname{eMLD}_{d}(\Gamma) is discrete away from 00; equivalently, it satisfies the ACC.

This is an ACC and discreteness prediction for minimal log discrepancies in fixed dimension and with prescribed boundary coefficients. The source gives no resolution status, so it remains open.

Sources & referencesView supporting material

Primary source

Jingjun Han and Jihao Liu, “On termination of flips and exceptionally non-canonical singularities”, arXiv:2209.13122 (2025).

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