Optimal destabilization conjecture for K-unstable log Fano pairs

Let (X,D)(X,D) be a log Fano pair. For a divisor EE over XX, let AX,D(E)A_{X,D}(E) denote its log discrepancy and let SX,D(E)S_{X,D}(E) denote its expected vanishing order. The stability threshold is

δ(X,D):=infEAX,D(E)SX,D(E).\delta(X,D):=\inf_E \frac{A_{X,D}(E)}{S_{X,D}(E)}.

Optimal destabilization conjecture. If (X,D)(X,D) is a K-unstable log Fano pair, then there exists a divisor EE over XX computing the infimum:

δ(X,D)=AX,D(E)SX,D(E).\delta(X,D)=\frac{A_{X,D}(E)}{S_{X,D}(E)}.

This asserts that the stability threshold of every K-unstable log Fano pair is attained by a divisorial valuation, providing an optimal destabilizing divisor and implying that the corresponding normalized Futaki infimum is a minimum. The supplied context does not state whether the conjecture has been resolved.

Sources & referencesView supporting material

Primary source

Harold Blum, Daniel Halpern-Leistner, Yuchen Liu and Chenyang Xu, “On properness of K-moduli spaces and optimal degenerations of Fano varieties”, arXiv:2011.01895 (2021).

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