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Uniform Cartier sections for Fano type contractions
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Abstract
We prove the Cartier-divisor conjecture of Birkar and Shokurov for rational boundaries in characteristic zero and for real boundaries over ℂ. For an ϵ-lc Fano type contraction with positive-dimensional base and nef negative log canonical class, a Cartier divisor through any prescribed base point can be chosen with pullback log canonical threshold bounded below in terms of the dimension and ϵ alone.
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Lean scope notes for result 066 (lean/docs/066.md, 2026). The scope notes list this paper as the accompanying one; the formalization proves only that normalized chart-field retractions form a compact set in the pointwise topology on the fraction field, not the uniform Cartier-section theorem or the birational klt-complement theorem. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Sung Rak Choi, Chuyu Zhou, On existence of bounded relative-global complements for Fano fibrations (arXiv:2402.12031, 19 February 2024). Proves bounded relative-global complements for fixed-dimensional ε-lc Fano fibrations and klt complements over curves, next to the curve-base results (Birkar) that the paper surveys before Theorem 1.
- Florin Ambro, An explicit bound for complements on toric varieties (arXiv:2506.18509, June 2025). Gives an effective bound for klt complement indices on toric Fano varieties; the paper's toric discussion cites only Ambro's earlier arXiv:2212.13862, and its τ(d, ε) is not effective.
- Caucher Birkar, Boundedness of ε-log Canonical Complements on Surfaces (arXiv:math/0409254, September 2004). Proves Shokurov's ε-lc complement boundedness on surfaces, related to the stronger expectation from Shokurov's 2004 formulation cited in the introduction.
- Stefano Filipazzi and Joaquín Moraga, Strong (δ, n)-Complements for Semi-Stable Morphisms (Documenta Mathematica 25, 1953–1996, 2020). Theorem 1.4 gives local strong complements with a positive discrepancy bound under semistability, finite boundary coefficients and bounded Cartier index of the base canonical divisor, a restricted-setting comparison for bounded klt complements. [family-level]
Elsewhere
- Emergent Mind: family 066
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Uniform-Cartier-sections-for-Fano-type-contractions-September-25-2026,
author = {{OpenAI}},
title = {{Uniform Cartier sections for Fano type contractions}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Uniform-Cartier-sections-for-Fano-type-contractions-September-25-2026/Uniform-Cartier-sections-for-Fano-type-contractions-September-25-2026.pdf}{OAI:Uniform-Cartier-sections-for-Fano-type-contractions-September-25-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/066.md. The Lean statements have not been checked by people against the paper's claims.
Family 066: Bounded klt complements for Fano contractions
Proves the finite-rational-coefficient form of Shokurov’s bounded-klt-complement conjecture for ϵ-lc complex Fano-type pairs with nef anti-log-canonical divisor. For ϵ-lc Fano contractions over any algebraically closed characteristic-zero field, it gives klt complements near every base point, with index bounded only by dimension and positive rational ϵ.
Other papers in this family:
- Bounded klt complements for Fano contractions · September 25, 2026