Bounded total Cartier indices in families
Are the total Cartier indices bounded for a bounded family of varieties with rational singularities?
References
Primary source
Progress summary
A 2026 paper proves that total Cartier indices are uniformly bounded for rational singularities in bounded families, settling the problem.
Han--Jiang Problem 4.5 asked whether the boundedness theorem for varieties of klt type remains true for varieties with rational singularities. The answer is affirmative: for a projective family over a finite-type scheme, the total Cartier index of every normal, projective, pure-dimensional fiber with rational singularities divides one fixed positive integer.
Known results
- Han--Jiang, 2025: bounded total Cartier indices for bounded families of projective varieties of klt type.
- Han--Jiang, 2025: the rational-singularity extension was posed as Problem 4.5.
- Han--Jiang, 2025: arbitrary fibers cannot satisfy such a bound, since an elliptic cone can have infinite total Cartier index.
May 2026 proof
The paper proves the rational-singularity case using local-to-global boundedness, Brieskorn’s surface class-group computation, link cohomology, Kollár’s exponential-sequence argument, and semialgebraic torsion bounds. It credits Rethlas with originating the proof structure, while stating that substantial correction, verification, and rewriting were done by hand.
Current status (as of May 2026): Han--Jiang Problem 4.5 is settled by the arXiv paper, subject only to the ordinary status of a preprint rather than peer review.
Sources
Solutions 0
No solutions have been posted yet.