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Existence of a minimizer on a log Fano cone singularity

Let x∈(X,Δ,T)x\in (X,\Delta,\mathbb{T}) be a log Fano cone singularity over an uncountable algebraically closed field. For every T\mathbb{T}-invariant valuation ν0\nu_0 centered at xx with AX,Δ(ν0)<∞A_{X,\Delta}(\nu_0)<\infty, does there exist a valuation ν∗∈ValX,∋xT,∗\nu_*\in \mathrm{Val}^{\mathbb{T},*}_{X,\ni x} such that

δ(X,Δ;ν0):=inf⁡ν∈ValX,∋xT,∗AX,Δ(ν)S(ν0;ν)=AX,Δ(ν∗)S(ν0;ν∗)?\delta(X,\Delta;\nu_0):=\inf_{\nu\in \mathrm{Val}^{\mathbb{T},*}_{X,\ni x}}\frac{A_{X,\Delta}(\nu)}{S(\nu_0;\nu)}=\frac{A_{X,\Delta}(\nu_*)}{S(\nu_0;\nu_*)}?
References

Primary source

arXiv

Additional references

algebraic-geometryk-stabilityvaluation-theory
Posted

Progress summary

Refreshed
Claimed solved

A new generic-limit argument claims to prove that the desired minimum is attained for log Fano cone singularities, but the claim has not been independently verified.

The problem asks whether the valuation invariant of every log Fano cone singularity has a minimizer. The latest preprint claims attainment under precisely these hypotheses and indicates that the conclusion can fail without the log Fano cone structure.

Known results

  • Blum (2018) established existence of a normalized-volume minimizer for every klt singularity.
  • A 2019 proof using limits of Kollár-component valuations removed Blum's uncountability assumption.
  • For a K-semistable log Fano cone, the valuation induced by the Reeb vector field minimizes normalized volume.

August 2026 generic-limit proof

A newly reported preprint gives a generic-limit argument proving attainment in the log Fano cone setting and explicitly limits the conclusion to that framework. This is a claimed complete resolution, but the supplied sources contain no independent verification or discussion of possible gaps.

Current status (as of August 2026): Existence is claimed both in the broader klt setting and by the new log Fano cone argument, but the latest claim remains unverified in this report.

Sources

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