Algebraic and complex geometry · Result 056

Termination of projective and Kähler fourfold minimal model programs

5 papers

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.

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Papers 5

Termination of generalized log canonical flips on compact Kähler fourfolds

OpenAI · October 7, 2026

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.

This version clarifies the independent relative MMP and birational-comparison inputs used by the proof and updates citations to revised companion papers; see the previous version.

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@misc{OAI:Termination-of-generalized-log-canonical-flips-on-compact-Kahler-fourfolds-October-7-2026,
  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-7-2026/termination-generalized-lc-kahler-fourfolds.pdf}{OAI:Termination-of-generalized-log-canonical-flips-on-compact-Kahler-fourfolds-October-7-2026}},
  year = {2026}
}

Termination of generalized-canonical flips on compact Kähler fourfolds

OpenAI · October 6, 2026

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.

This version clarifies that only independent negativity, codimension-two smoothness, and cycle-comparison results are used from the companion fourfold paper; see the previous version.

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@misc{OAI:Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds-October-6-2026,
  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-6-2026/termination-generalized-terminal-flips-compact-kahler-fourfolds.pdf}{OAI:Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds-October-6-2026}},
  year = {2026}
}

Termination for projective log canonical fourfolds with rational boundary

OpenAI · September 24, 2026

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.

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@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

OpenAI · October 6, 2026

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.

This version adds carrier-positivity and multiplier-ideal arguments and revises the contraction induction for ordinary negative-ray programs; see the previous version.

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@misc{OAI:Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds-October-6-2026,
  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-6-2026/paper.pdf}{OAI:Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds-October-6-2026}},
  year = {2026}
}

Finite ordinary minimal model programs on compact Kähler fourfolds

OpenAI · September 24, 2026

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.

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@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}
}

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