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ACC conjecture for minimal log discrepancies

About 8 years old · traced to

Let A=AkNA={\mathbb{A}}_k^N be affine space over an algebraically closed field kk, with origin 0∈A0\in A, and let ae{\frak{a}}^e be a multiideal whose exponent set is ee. ACC conjecture. For every fixed DCC set J⊂R>0J\subset {\mathbb{R}}_{>0}, the set

{mld(0;A,ae)∣e⊂J, (A,ae) is log canonical at 0}\{ {\mathrm{mld}}(0;A,{\frak{a}}^e)\mid e\subset J,\ (A,{\frak{a}}^e)\text{ is log canonical at }0\}

satisfies the ascending chain condition. This is an important expected finiteness property in birational geometry; the supplied text states the conjecture but gives no resolution.

References

Primary source

Shihoko Ishii, “Inversion of modulo p reduction and a partial descent from characteristic 0 to positive characteristic”, arXiv:1808.10155 (2018).

birational-geometryminimal-log-discrepancies
Posted · edited

Progress summary

Refreshed
Claimed progress

A new paper proves the conjecture in two bounded situations, but the general question remains open.

The conjecture asks whether minimal log discrepancies satisfy an ascending-chain finiteness property for every fixed DCC exponent set. The literature identifies it as the Mustaţă–Nakamura conjecture; the unrestricted higher-dimensional case remains unsettled.

Known results

  • Ishii (2020) proved the conjecture for smooth surface pairs with multiideals and real exponents.
  • A result on local volumes proves the ACC when the local volume is bounded away from zero.
  • Shokurov's work verifies the conjecture in the setting of general-type minimal model programs.

September 2026 bounded-setting advance

A preprint by Weichung Chen and Keng-Hung Steven Lin proves the conjecture for two broad bounded classes of generalized sub-pairs. The September 17, 2026 report explicitly says this does not resolve the unrestricted affine-space formulation.

Current status (as of September 2026): bounded cases, including surface pairs and several restricted higher-dimensional settings, are known; the full conjecture for arbitrary NN and DCC exponent sets remains open, with the new bounded result unverified.

Sources

Solutions 0

No solutions have been posted yet.