Uniform Weighted Bound for Divisorial Discrepancies

Establish whether, for every integer n≥2 and every finite set Γ⊆[0,1], there exists a finite constant C_{n,Γ} uniformly bounding the local volume multiplied by the sum of the reciprocals of the log discrepancies of all prime divisors over an n-dimensional klt germ whose log discrepancy is less than the minimal log discrepancy plus one, when the boundary coefficients belong to Γ.

Background

The proposed bound concerns the number and distribution of prime divisors with discrepancies in a bounded interval above the minimal log discrepancy. The weighting by reciprocal discrepancy emphasizes divisors with small log discrepancy, while multiplication by the local volume makes the expression compatible with the local invariants studied throughout the paper. The paper asks whether this quantity is uniformly bounded when the dimension and a finite coefficient set are fixed; no such bound is established.

References

Fix an integer $n\ge2$ and a finite set $\Gamma\subset[0,1]$. Does there exist a constant $C_{n,\Gamma}<+\infty$ such that every $n$-dimensional klt germ $x\in(X,\Delta)$ with $\operatorname{Coeff}(\Delta)\subseteq\Gamma$ satisfies

(x,X,\Delta) \sum_{\substack{ E\text{ a prime divisor over }X\ni x\ A_{X,\Delta}(E)<x(X,\Delta)+1} \frac{1}{A{X,\Delta}(E)} \le C_{n,\Gamma}.

— A Sharp inequality between local volumes and minimal log discrepancies  (2608.16726 - Han, 17 Aug 2026) in Question 3, Section 5.3 (Questions)