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Termination for projective log canonical fourfolds with rational boundary

OpenAI · September 24, 2026

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Abstract

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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Possibly relevant literature (found by the agent, not checked)

Kenta Hashizume, Full termination of MMP for klt pairs with big boundaries (arXiv:2609.33679, v1 27 Sep 2026, v2 13 Oct 2026). It proves in arbitrary dimension that every MMP terminates for projective klt pairs with $K_X+\Delta$ or $\Delta$ big. This bears on the paper's comparison with the big-boundary klt fourfold case of Han–Liu–Zhuang, and it appeared three days after this paper's date.
Jingjun Han and Jihao Liu, On termination of flips and exceptionally non-canonical singularities (arXiv:2209.13122; Geometry & Topology 29, 2025). Its reduction of four-dimensional flip termination to ACC for minimal log discrepancies of exceptionally non-canonical pairs offers a conditional comparison with the manuscript’s unrestricted rational-boundary termination claim.
Vladimir Lazić and Zhixin Xie, Nakayama-Zariski decomposition and the termination of flips (arXiv:2305.01752; Épijournal de Géométrie Algébrique 9, 2025). Its conditional passage from one terminating sequence to all sequences for pseudo-effective projective pairs relates directly to the manuscript’s claim of termination for arbitrary permitted choices.

Elsewhere

VibeMathed: Problem record
Emergent Mind: family 056

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Cite

@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}
}