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Termination of generalized log canonical flips on compact Kähler fourfolds

OpenAI · October 7, 2026

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Abstract

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.

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

Christopher Hacon, Yi Li, Lingyao Xie, Fujiki Class C Varieties and a Kähler Criterion (arXiv:2608.20588, 20 Aug 2026). It shows that flips and divisorial contractions preserve the Kähler condition for strongly $\mathbb Q$-factorial generalized klt pairs with $B+\beta_X$ big. This relates to the theorem's hypothesis that every model $X_i$ in the sequence is compact Kähler.
Makoto Enokizono, Kenta Hashizume, On termination of minimal model program for log canonical pairs on complex analytic spaces (arXiv:2501.03531, v2 8 Dec 2025). It studies termination of lc MMPs in the complex analytic setting, which relates to the paper's ordinary lc case ($\mathbf M=0$) with projective flip diagrams on possibly nonprojective spaces.
Jingjun Han and Jihao Liu, On termination of flips and exceptionally non-canonical singularities (arXiv:2209.13122; Geometry & Topology 29, 2025). Its conditional deduction of four-dimensional flip termination from ACC for minimal log discrepancies of exceptionally non-canonical pairs provides a comparison for the manuscript’s ordinary projective specialization, without treating its generalized Kähler setting.

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Cite

@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}
}