Result 056, Algebraic and complex geometry

Termination of projective and Kähler fourfold minimal model programs

Proves termination of every existing generalized log canonical flip sequence on globally Weil ℚ-factorial compact Kähler fourfolds, with rational boundary, fixed rational analytically nef b-data, and projective small flip diagrams with the prescribed ample signs. Also proves termination of arbitrary permitted minimal model programs for projective log canonical fourfolds with rational boundary in characteristic zero.

Proof

The bigger picture

Why it matters

A fourfold is a geometric space of dimension four; flips are controlled replacements of small parts that aim to simplify its geometry. These manuscripts claim that specified simplification processes cannot go on forever.

What changes?

For compact Kähler fourfolds, the manuscript claims that every existing generalized log canonical flip sequence terminates. Models must be normal, irreducible and globally Weil Q-factorial; boundaries have rational weights. The fixed rational nef b-divisor, auxiliary positivity data, must be represented by an analytically nef Q-Cartier divisor on a projective modification. Flips must be projective small diagrams with the prescribed opposite ample signs. Neither scaling nor pseudo-effectivity is assumed. This does not establish the existence of contractions or flips.

What does that help mathematicians do?

For projective log canonical fourfold pairs, spaces with weighted hypersurfaces and controlled singularities, the reported result covers every permitted minimal model program over an algebraically closed field of characteristic zero. It allows arbitrary negative extremal rays and mixed birational steps, without pseudo-effectivity or initial Q-factoriality. Thus researchers need not choose a special route through the permitted simplifications to secure termination: the claim rules out infinite runs for every such choice.

Are there practical applications?

The immediate value is foundational. Minimal model programs seek simpler representatives of geometric spaces while preserving their birational structure, meaning agreement away from exceptional subsets. The claimed finiteness removes endless iteration as an obstacle to this approach in the stated settings. For the Kähler flip theorem, constructing the required steps remains a separate question.

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.

5 manuscripts

Termination of generalized log canonical flips on compact Kähler fourfolds

October 5, 2026 46 pages

Every sequence of generalized log canonical flips on a normal irreducible globally Weil ℚ-factorial compact Kähler fourfold terminates, provided the flips are projective small diagrams with the stated opposite ample signs. The boundary is rational, and the nef b-divisor is fixed and represented by an analytically nef ℚ-Cartier divisor on a projective modification. No scaling rule or pseudo-effectivity assumption is required. The theorem concerns existing flip sequences; it does not assert the existence of all contractions or flips.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Termination-of-generalized-log-canonical-flips-on-compact-Kahler-fourfolds-October-5-2026,
  author = {{OpenAI}},
  title = {{Termination of generalized log canonical flips on compact K{\"a}hler fourfolds}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Termination-of-generalized-log-canonical-flips-on-compact-Kahler-fourfolds-October-5-2026/termination-generalized-lc-kahler-fourfolds.pdf}{OAI:Termination-of-generalized-log-canonical-flips-on-compact-Kahler-fourfolds-October-5-2026}},
  year = {2026}
}

Termination of generalized-canonical flips on compact Kähler fourfolds

October 5, 2026 22 pages

Every sequence of generalized-canonical flips on compact Kähler fourfolds with rational boundary and fixed rational analytically nef b-divisor terminates when the small morphisms are projective. We prove this for normal globally Weil ℚ-factorial models with boundary coefficients less than one, allowing exceptional log discrepancies equal to one. No scaling rule or pseudo-effectivity assumption is required. The theorem applies to existing projective small diagrams with the specified opposite ample signs.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds-October-5-2026,
  author = {{OpenAI}},
  title = {{Termination of generalized-canonical flips on compact K{\"a}hler fourfolds}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds-October-5-2026/termination-generalized-terminal-flips-compact-kahler-fourfolds.pdf}{OAI:Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds-October-5-2026}},
  year = {2026}
}

Termination for projective log canonical fourfolds with rational boundary

September 24, 2026 44 pages

We prove that every permitted minimal model program for a projective log canonical fourfold with rational boundary over an algebraically closed field of characteristic zero terminates. The result allows arbitrary negative extremal rays and mixed birational steps on the given models, without pseudo-effectivity or initial ℚ-factoriality.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Termination-for-projective-log-canonical-fourfolds-with-rational-boundary-September-24-2026,
  author = {{OpenAI}},
  title = {{Termination for projective log canonical fourfolds with rational boundary}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Termination-for-projective-log-canonical-fourfolds-with-rational-boundary-September-24-2026/paper.pdf}{OAI:Termination-for-projective-log-canonical-fourfolds-with-rational-boundary-September-24-2026}},
  year = {2026}
}

Finite ordinary minimal model programs on compact Kähler fourfolds

October 5, 2026 84 pages

We prove that every maximal ordinary negative-ray program starting from a compact Kähler klt fourfold pair with effective rational boundary in the global Weil-divisor ℚ-factorial category terminates. It ends at a nef model when the adjoint is pseudo-effective and at a projective Mori fibre space otherwise.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds-October-5-2026,
  author = {{OpenAI}},
  title = {{Finite ordinary minimal model programs on compact K{\"a}hler fourfolds}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds-October-5-2026/paper.pdf}{OAI:Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds-October-5-2026}},
  year = {2026}
}

Finite ordinary minimal model programs on compact Kähler fourfolds

September 24, 2026 35 pages

We prove that a globally Weil-ℚ-factorial compact Kähler klt fourfold pair with effective rational boundary and canonical rational line bundle admits a finite ordinary minimal model program starting on the given pair. It ends at a nef model when the adjoint class is pseudo-effective and at a Mori fibre space otherwise.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds-September-24-2026,
  author = {{OpenAI}},
  title = {{Finite ordinary minimal model programs on compact K{\"a}hler fourfolds}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds-September-24-2026/paper.pdf}{OAI:Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds-September-24-2026}},
  year = {2026}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 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.