Result 066, Algebraic and complex geometry

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.

Proof

The bigger picture

Why it matters

Algebraic geometers use weighted hypersurfaces to modify a space's canonical class, a basic geometric invariant. This manuscript claims that, for certain Fano families, such modifications have uniformly bounded complexity while keeping singularities mild.

What changes?

The manuscript reports that, in fixed dimension d and for positive rational epsilon, every epsilon-log-canonical Fano contraction over an algebraically closed characteristic-zero field admits a klt complement near each closed base point. Such contractions have relative anticanonical positivity and singularities controlled by epsilon. A complement adjusts divisor weights so an integer multiple of the adjusted canonical class is trivial; klt means mild singularities. That integer, the index, is bounded only in terms of d and epsilon.

What does that help mathematicians do?

Over the complex numbers, the manuscript also reports bounded monotone klt complements for epsilon-log-canonical Fano-type pairs, meaning varieties equipped with weighted divisors. Here the negative adjusted canonical class must have nonnegative intersection with every curve, and weights must belong to a fixed finite rational set. Monotonicity means weights only increase. Researchers can thus obtain bounded-index triviality while retaining mild singularities without reducing existing boundary weights. This does not settle unrestricted coefficient settings.

Are there practical applications?

The immediate value is foundational: these complements provide controlled replacements for canonical-divisor data in the geometry of Fano families. The local guarantee applies around every base point, allowing researchers to study different parts of a family under one uniform index bound, rather than bounds tailored to individual locations.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Bounded klt complements for Fano contractions

September 25, 2026 93 pages

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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Cite (BibTeX) 
@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}
}

Uniform Cartier sections for Fano type contractions

September 25, 2026 106 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.