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Bounded klt complements for Fano contractions
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Abstract
For fixed dimension d and positive rational ϵ, we prove that every ϵ-log-canonical Fano contraction over an algebraically closed field of characteristic zero admits, near each closed base point, a klt complement of index bounded only by d and ϵ. Over ℂ, we also obtain monotone klt complements for Fano type pairs with nef anti-log-canonical divisor and coefficients in a fixed finite rational set. This proves the finite-rational-coefficient form of Shokurov's bounded-klt-complement conjecture.
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Notes
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-23) and not checked by a person. May be wrong or incomplete. Report an error.
Comments and discussion
- Marcus Bintz (@spurious-bows.bsky.social), post on Bluesky (12 October 2026). Pairs this paper with the release's Minimal models and Mori fibre spaces for generalized log canonical Q-pairs, equating its Proposition 14.3 with that paper's Lemma 8.3 and matching Lemma 11.6 through Proposition 14.5 with Lemma 6.3 through Section 8.3 "broadly".
Formalization
- OpenAI math release, Lean scope notes for result 066 (lean/docs/066.md, 2026). The release's Lean development covers only a supporting compactness statement, in the pointwise topology on the fraction field, for normalized chart-field retractions tied to the Cartier-sections companion; bounded klt complements themselves are outside the formalized statement. [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 one-dimensional bases, a special case of the setting of Theorem 1.1.
- Caucher Birkar, Boundedness of ε-log Canonical Complements on Surfaces (arXiv:math/0409254, September 2004). Treats bounded ε-lc complements in dimension two, the stronger form that the paper explicitly does not assert (citing Chen, Remark 1.2(1)).
- Florin Ambro, An explicit bound for complements on toric varieties (arXiv:2506.18509, June 2025). Gives an effective bound on the index of klt complements for toric Fano varieties, whereas the paper's index N(d, ε) is not made effective.
- 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
- VibeMathed: Problem record
- Emergent Mind: family 066
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Bounded-klt-complements-for-Fano-contractions-September-25-2026,
author = {{OpenAI}},
title = {{Bounded klt complements for Fano contractions}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Bounded-klt-complements-for-Fano-contractions-September-25-2026/Bounded-klt-complements-for-Fano-contractions-September-25-2026.pdf}{OAI:Bounded-klt-complements-for-Fano-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:
- Uniform Cartier sections for Fano type contractions · September 25, 2026