ACC conjecture for generalized pairs

Let dZ>0d\in\mathbb{Z}_{>0} and let I[0,+)I\subset[0,+\infty) be a subset satisfying the DCC. Define

Agen(d,I):={mldx(X,Δ+M) | (X,Δ+M)/Z is a generalized pair, dimX=d, ΔI, MI, xXcl},A_{\mathrm{gen}}(d,I):=\left\{\operatorname{mld}_x(X,\Delta+M)\ \middle|\ (X,\Delta+M)/Z\text{ is a generalized pair},\ \dim X=d,\ \Delta\in I,\ M\in I,\ x\in |X|_{\rm cl}\right\},

where MIM\in I has the meaning specified in the source's definition of generalized pairs. ACC conjecture for generalized pairs. The set Agen(d,I)A_{\mathrm{gen}}(d,I) satisfies the ACC. This extends the minimal-log-discrepancy ACC question from log pairs to generalized pairs; the paper establishes partial results but does not resolve it in full generality.

Sources & referencesView supporting material

Primary source

Weichung Chen, Yoshinori Gongyo and Yusuke Nakamura, “On generalized minimal log discrepancy”, arXiv:2112.09501 (2024).

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