The central result is a family of three-dimensional normal Q-Gorenstein varieties in which minimal log discrepancy, a numerical label for the singularity at a point, stays exactly 1, while the Cartier index of the canonical divisor can be set to any positive integer. Here, Q-Gorenstein means that some positive multiple of the canonical divisor is Cartier; the Cartier index is the smallest multiplier with that property. In plain terms, the construction holds the paper's singularity value fixed and lets the divisor index grow as large as desired. That gives a counterexample to Shokurov's index conjecture for dimension 3 and target minimal log discrepancy 1.
The conjecture's promise
Shokurov's conjecture asks for a uniform ceiling. Once a dimension and a nonnegative target value for minimal log discrepancy are fixed, it predicts that a positive integer depending only on those two inputs should bound the Cartier index at every point with that value. The new construction keeps the dimension at 3 and the target at 1, but makes the index equal to every positive integer m. No single bound can therefore cover this family. The claim is about the unrestricted conjecture in this setting, not a conclusion that every singularity class behaves this way.
Building the examples
To make the examples, the authors begin with an affine cone over an elliptic curve, using an ample line bundle of degree m to polarize the curve. An order-m torsion translation gives a cyclic group of order m. The group then acts diagonally on the cone together with an affine-line factor, and the family member is their quotient. The same integer m thus appears in the geometric input and in the finite group action.
Following the calculation
The affine cone used in the construction is normal and Gorenstein. For the calculation, the argument uses a smooth affine-line bundle over the elliptic curve and a proper birational morphism from that space to the cone. The diagonal action is virtually free, and the paper invokes a quotient result stating that actions with this property preserve minimal log discrepancies. Together, these ingredients carry the singularity calculation from the factors to the quotient.
At the cone's vertex, the resolution calculation gives a minimal log discrepancy of 0. The affine-line factor contributes 1, and the quotient calculation gives the selected point on the resulting variety a minimal log discrepancy of 1. That is the fixed value required for the counterexample, even though the Cartier index changes with m.
Why the index is exact
Showing that the index is exactly m requires more than showing it divides m. Because the pre-quotient is Gorenstein and the quotient group has order m, the argument establishes that m times the canonical divisor on the quotient is Cartier, so the index divides m. It then tracks the canonical line bundle descended to F′ in the Picard group, the algebraic record of line bundles, and shows that this class has torsion order exactly m. The index must consequently be both a multiple of m and a divisor of m, forcing it to equal m for every member of the family.
A counterexample with limits
The counterexample has a clear boundary. The selected point is log canonical but not klt, so the result shows failure of the unrestricted conjecture in the non-klt regime. It does not show failure for klt varieties, does not address parameter pairs other than dimension 3 and target value 1, and does not establish that the authors' suggested revisions are universally valid.
The authors suggest assuming that the variety is klt or, in the log-canonical case, restricting to target value 0. They also state that the same virtually-free-action idea cannot produce a klt counterexample. The argument directly establishes the constructed family, not the universal validity of those proposed revisions. The work is stated over an algebraically closed field of characteristic zero. The supplied document is an arXiv version 1 dated 28 August 2026.
Paper data and sources
Original title: A non-klt counterexample to Shokurov's index conjecture
Authors: Yusuke Nakamura, Kohsuke Shibata
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text