Algebraic and complex geometry · Result 033

Iitaka subadditivity, variation, and logarithmic additivity

5 papersLean

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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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 5

Orbifold and logarithmic Iitaka subadditivity

OpenAI · September 26, 2026

We prove Campana's orbifold Iitaka subadditivity conjecture for rational simple normal crossing boundaries on compact manifolds in Fujiki class \(\mathcal C\), including coefficient one. Ordinary and logarithmic subadditivity follow.

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@misc{OAI:Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026,
  author = {{OpenAI}},
  title = {{Orbifold and logarithmic Iitaka subadditivity}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/paper.pdf}{OAI:Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026}},
  year = {2026}
}

Logarithmic Kodaira dimension and whole-fiber variation

OpenAI · September 26, 2026

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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@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}
}

The reverse logarithmic Kodaira inequality and additivity

OpenAI · September 26, 2026

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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@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}
}

Projective Hodge lines and ordinary Iitaka subadditivity

OpenAI · September 27, 2026

We prove the ordinary Iitaka subadditivity conjecture for surjective projective morphisms with connected fibers between smooth connected projective varieties over algebraically closed fields of characteristic zero. If F is the geometric generic fiber of \(f:X\to Z\), then \(\kappa(X)\geq\kappa(F)+\kappa(Z)\).

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@misc{OAI:Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026,
  author = {{OpenAI}},
  title = {{Projective Hodge lines and ordinary Iitaka subadditivity}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/paper.pdf}{OAI:Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026}},
  year = {2026}
}

B-semiampleness for compact log-smooth Kähler fibrations

OpenAI · September 10, 2026

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 M_S 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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@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}
}

What was formalized in Lean

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

The reverse logarithmic Kodaira inequality and additivity

Scope

The paper studies logarithmic Kodaira additivity for connected-fiber morphisms of smooth projective reduced simple-normal-crossing pairs that are smooth on all boundary strata away from the base boundary. The linked formalization proves the negative-fiber branch: for a very general base point, if the logarithmic Kodaira dimension of the fiber is \(-\infty\), then the total logarithmic Kodaira dimension equals the sum of the base and fiber dimensions and is \(-\infty\). Every positive-degree logarithmic pluriform section on the total space then vanishes.

The finite-dimension and negative-base branches of the paper's additivity theorem are outside this selected statement.

Comparator links

Result	Comparator statement
Logarithmic Kodaira additivity in the negative-fiber branch	LogKodairaFiberNegative.lean

Scope note on GitHub →