The ideal-adic semi-continuity conjecture for minimal log discrepancies

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Let XX be a klt variety, let x∈Xx \in X be a closed point, and let mx\mathfrak{m}_x denote the maximal ideal corresponding to xx. Ideal-adic semi-continuity conjecture. For every given r∈R>0r \in \mathbb{R}_{>0}, there exists ℓ∈Z≥0\ell \in \mathbb{Z}_{\ge 0} such that, for any ideal sheaves a\mathfrak{a} and b\mathfrak{b} on XX satisfying

a+mxℓ=b+mxℓ,\mathfrak{a}+\mathfrak{m}_x^\ell=\mathfrak{b}+\mathfrak{m}_x^\ell,

one has

mld⁡x(X,ar)=mld⁡x(X,br).\operatorname{mld}_x(X,\mathfrak{a}^r)=\operatorname{mld}_x(X,\mathfrak{b}^r).

This conjecture asserts that the minimal log discrepancy at xx is determined by a sufficiently high ideal-adic jet. The source attributes it to Mustață and Nakamura and presents it in a paper that gives a counterexample to a related conjectural property; the supplied text does not state that this conjecture itself is resolved.

References

Primary source

Yusuke Nakamura and Kohsuke Shibata, “A counterexample to the PIA conjecture for minimal log discrepancies”, arXiv:2404.06164 (2026).

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