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The reverse logarithmic Kodaira inequality and additivity

OpenAI · September 26, 2026

Lean

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Abstract

We prove the reverse logarithmic Kodaira inequality for a surjective connected-fiber morphism $f:(X,E)\to(Y,D)$ of smooth projective reduced simple-normal-crossing pairs, with $\mathop{\mathrm{Supp}}\nolimits (f^*D)\subseteq\mathop{\mathrm{Supp}}\nolimits E$, such that X and every boundary stratum are smooth over $Y\setminus\mathop{\mathrm{Supp}}\nolimits D$. Together with logarithmic subadditivity, the inequality gives additivity, including both negative-infinity cases. This resolves Popa's logarithmic additivity conjecture positively in the projective reduced-SNC, stratum-smooth setting.

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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 release's Lean formalization covers only the negative-fiber branch of the additivity theorem (fiber log Kodaira dimension $-\infty$ forces total $-\infty$) and vanishing of positive-degree logarithmic pluriforms; the finite-dimension and negative-base branches are not formalized.

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]

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:The-reverse-logarithmic-Kodaira-inequality-and-additivity-September-26-2026,
  author = {{OpenAI}},
  title = {{The reverse logarithmic Kodaira inequality and additivity}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-reverse-logarithmic-Kodaira-inequality-and-additivity-September-26-2026/paper.pdf}{OAI:The-reverse-logarithmic-Kodaira-inequality-and-additivity-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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