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Logarithmic Kodaira dimension and whole-fiber variation

OpenAI · September 26, 2026

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Abstract

We prove the logarithmic Iitaka–Viehweg inequality for projective surjective morphisms with connected fibers between smooth complex quasi-projective varieties whose base has nonnegative logarithmic Kodaira dimension. The variation measures the birational field of definition of the whole geometric generic fiber. This resolves Popa's logarithmic variation conjecture positively.

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work in progress

Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Formalization

  • OpenAI, Family 033 Lean scope note (OpenAI math release, 2026-10-06). The family scope note covers only the reverse-additivity companion’s negative-fiber branch, not this paper’s main theorem. [family-level]

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

Digests and expositions

  • AI Proved What?, Do the forms add up? (Episode 033, October 2026). AI-assisted family exposition outlining subadditivity, variation, additivity, and the limited Lean scope. [AI-generated per source] [family-level]

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Elsewhere

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Cite

@misc{OAI:Logarithmic-Kodaira-dimension-and-whole-fiber-variation-September-26-2026,
  author = {{OpenAI}},
  title = {{Logarithmic Kodaira dimension and whole-fiber variation}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Logarithmic-Kodaira-dimension-and-whole-fiber-variation-September-26-2026/paper.pdf}{OAI:Logarithmic-Kodaira-dimension-and-whole-fiber-variation-September-26-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/033.md. The Lean statements have not been checked by people against the paper's claims.

Family 033: Iitaka subadditivity, variation, and logarithmic additivity

Proves Campana's orbifold Iitaka subadditivity conjecture for smooth Fujiki-class-$\mathcal C$ manifolds with rational simple-normal-crossing boundaries. For projective fibrations $f:U\to V$ of smooth complex quasi-projective varieties with connected fibers, general fiber F, and $\bar\kappa(V)\ge0$, proves Popa's inequality $\bar\kappa(U)\ge\kappa(F)+\max\{\bar\kappa(V),\mathop{\mathrm{Var}}\nolimits (f)\}$, where variation measures the whole geometric generic fiber.

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