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

Relative denominators and effective systems for log Calabi-Yau fibrations

OpenAI · September 27, 2026

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Abstract

We prove uniform denominator and effective-system bounds for log Calabi-Yau fibrations from projective log canonical complex pairs of dimension at most four onto positive-dimensional bases, with boundary coefficients in a fixed finite rational set. The bounds give a uniform trivializing degree and a uniform b-Cartier multiple of the moduli b-divisor. When the base divisor is big, a uniform complete rounded adjoint system has section ratios generating the full base function field.

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Digests and expositions

  • AI Proved What?, Always semiample? (Episode 034, October 2026; formalization correction October 15, 2026). Explains the family’s abundance claims, proof strategy, and dependencies, and reports a supporting Lean development recovering the base function field from adjoint section ratios, rather than a formalization of this paper's main theorems. [AI-generated per source] [family-level]

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Cite

@misc{OAI:Relative-denominators-and-effective-systems-for-log-Calabi-Yau-fibrations-September-27-2026,
  author = {{OpenAI}},
  title = {{Relative denominators and effective systems for log Calabi-Yau fibrations}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Relative-denominators-and-effective-systems-for-log-Calabi-Yau-fibrations-September-27-2026/paper.pdf}{OAI:Relative-denominators-and-effective-systems-for-log-Calabi-Yau-fibrations-September-27-2026}},
  year = {2026}
}

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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