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Browse › Algebraic and complex geometry › 034 Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity

Conditional good minimal models for compact Kähler fourfolds

OpenAI · October 6, 2026

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Abstract

Assuming orbifold Iitaka subadditivity and abundance for nef fourfold adjoints of nonnegative Kodaira dimension, we prove the existence of good minimal models for globally strongly ℚ-factorial compact Kähler klt fourfold pairs with effective rational boundary and analytically pseudo-effective actual ℚ-Cartier adjoint. The ordinary fourfold minimal model program supplies the nef endpoint. We prove nonvanishing by fibration arguments and, in algebraic dimension zero, by singular metrics, holomorphic foliations, and extension from a reduced boundary. The projective abundance argument used in the proof is included in full.

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Digests and expositions

  • AI Proved What?, Always semiample? (Episode 034, October 2026). Explains the family’s abundance claims, proof strategy, and dependencies. [AI-generated per source] [family-level]

Elsewhere

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Cite

@misc{OAI:Conditional-good-minimal-models-for-compact-Kahler-fourfolds-October-6-2026,
  author = {{OpenAI}},
  title = {{Conditional good minimal models for compact K\"ahler fourfolds}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Conditional-good-minimal-models-for-compact-Kahler-fourfolds-October-6-2026/paper.pdf}{OAI:Conditional-good-minimal-models-for-compact-Kahler-fourfolds-October-6-2026}},
  year = {2026}
}

Versions

This version clarifies the independent ordinary-MMP input and states the remaining Iitaka-subadditivity and effective-abundance hypotheses explicitly; see the previous version.

Family 034: Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity

Using logarithmic Iitaka subadditivity, proves log abundance in every dimension for normal compact Kähler log canonical pairs with effective rational boundary: an analytically nef ℚ-Cartier adjoint is semiample. It also proves projective log abundance over every algebraically closed field of characteristic zero and the effective Iitaka fibration conjecture for smooth projective varieties of nonnegative Kodaira dimension in that setting. A further result resolves the finite-rational-coefficient index conjecture for connected projective semi-log-canonical log Calabi–Yau pairs over such fields in each fixed dimension and for each fixed finite set of rational boundary coefficients, with a uniform index independent of the number of components.

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