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A counterexample to Liu's conjecture on Fano indices in dimension four

A mix of human-written and AI-generated textHuman understanding: some partsmath.AG — Algebraic Geometry

Contributed by Deep Bhattacharjee ↗, Ushashi Bhattacharya ↗

Version 1 / Oct 22, 2026 / CC BY 4.0

Abstract

We prove that Liu's conjecture on Fano indices fails in dimension four: 13891389 is the index of a terminal Calabi--Yau fourfold and of a four-dimensional log canonical singularity, but it is not the Fano index of any canonical weighted projective space of dimension four.

Provenance statement

Deep Bhattacharjee chose the problem and directed the work. Claude (Anthropic), through Claude Code, carried out the searches and computations, wrote the checking programs and the Lean 4 proofs and drafted the paper. Refutes Liu's Conjecture 4.2 in dimension four. The proofs are written out in the paper; the programs and their outputs are in the release. Both authors approved the final version. Code and data: release v2.0.0, https://github.com/creelie/fuzzy-guacamole/releases/tag/v2.0.0.
AI provenance filesAI_PROVENANCE_LiuFanoIndexConjecture.md · 5 KB

Tools used

Anthropic
Claude CodeVersion Opus 5.5 Ultracode

References

  1. D. Bhattacharjee and U. Bhattacharya, A counterexample to Liu's conjecture on Fano indices in dimension four: Lean formalization and programs , Zenodo, 2026, doi: (https://doi.org/\zenodoID).link
  2. The mathlib Community, The Lean mathematical library , in: Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs (CPP 2020), ACM, 2020, pp. 367--381, doi:10.1145/3372885.3373824 (https://doi.org/10.1145/3372885.3373824).DOI
  3. L. Esser, B. Totaro and C. Wang, Calabi--Yau varieties of large index , Algebr. Geom., to appear; arXiv:2209.04597.arXiv
  4. A. R. Iano-Fletcher, Working with weighted complete intersections , in: Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser. 281, Cambridge Univ. Press, Cambridge, 2000, pp. 101--174, doi:10.1017/CBO9780511758942.005 (https://doi.org/10.1017/CBO9780511758942.005).DOI
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  6. J. Kollár and S. Mori, Birational geometry of algebraic varieties , Cambridge Tracts in Math. 134, Cambridge Univ. Press, Cambridge, 1998, doi:10.1017/CBO9780511662560 (https://doi.org/10.1017/CBO9780511662560).DOI
  7. H. Liu, On Fano indices of weighted projective spaces , arXiv:2608.03434v3, 2026.arXiv
  8. H. Liu, Fano indices of four-dimensional canonical weighted projective spaces (data and code), Zenodo, doi:10.5281/zenodo.21806548 (https://doi.org/10.5281/zenodo.21806548).DOI
  9. L. de Moura and S. Ullrich, The Lean 4 theorem prover and programming language , in: Automated Deduction -- CADE 28, Lecture Notes in Comput. Sci. 12699, Springer, 2021, pp. 625--635, doi:10.1007/978-3-030-79876-5_37 (https://doi.org/10.1007/978-3-030-79876-5_37).DOI

Version history

  1. v1Submitted by Deep BhattacharjeeInitial depositCurrentOct 22, 2026