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B-semiampleness for compact log-smooth Kähler fibrations

OpenAI · September 10, 2026

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Abstract

We prove the compact log-smooth Kähler case of b-semiampleness. Let $f:Y\to X$ be a surjective holomorphic map with connected fibers between smooth compact connected Kähler manifolds, and let Δ be an effective rational divisor with simple normal crossing support and coefficients in $[0,1]$, with $K_Y+\Delta\sim_{\mathbb Q}f^*L$ for $L\in\mathop{\mathrm{Pic}}\nolimits (X)_{\mathbb Q}$. There is a smooth compact Kähler modification $S\to X$ for which the threshold-moduli line satisfies $M_{S_1}=\nu^*M_S$ in $\mathop{\mathrm{Pic}}\nolimits (S_1)_{\mathbb Q}$ for every smooth compact Kähler modification $\nu:S_1\to S$, and some positive multiple of MS is represented by a holomorphic line bundle generated by global sections. Horizontal components of coefficient one are allowed; neither projectivity nor a Campana orbifold Iitaka hypothesis is assumed.

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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]

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:B-semiampleness-for-compact-log-smooth-Kahler-fibrations-September-10-2026,
  author = {{OpenAI}},
  title = {{B-semiampleness for compact log-smooth K{\"a}hler fibrations}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/B-semiampleness-for-compact-log-smooth-Kahler-fibrations-September-10-2026/paper.pdf}{OAI:B-semiampleness-for-compact-log-smooth-Kahler-fibrations-September-10-2026}},
  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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