Algebraic and complex geometry · Result 034

Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity

14 papers

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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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 14

Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity

OpenAI · October 6, 2026

Assume logarithmic Iitaka subadditivity for surjective morphisms with connected fibers between smooth projective complex varieties with compatible reduced simple normal crossing boundaries. We prove log abundance for normal irreducible compact Kähler spaces in every dimension: for a log canonical pair \((X,\Delta)\) with effective rational boundary and \(K_X+\Delta\) ℚ-Cartier, analytic nefness of \(K_X+\Delta\) implies semiampleness.

This version supplies base-presentation and multiplier-ideal arguments for the induction, including the adjoint pullback and contraction-descent steps; see the previous version.

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@misc{OAI:Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026,
  author = {{OpenAI}},
  title = {{Log abundance for compact K{\"a}hler spaces under logarithmic Iitaka subadditivity}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/main.pdf}{OAI:Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026}},
  year = {2026}
}

Uniform indices for semi-log-canonical log Calabi–Yau pairs

OpenAI · October 5, 2026

We prove a uniform index theorem for connected projective semi-log-canonical log Calabi–Yau pairs in every fixed dimension at least four over an algebraically closed field of characteristic zero. For boundary coefficients in a fixed finite rational set, a single multiple of the log canonical divisor is Cartier and linearly trivial. The multiple depends only on the dimension and coefficient set, not on the number of irreducible components. Together with the established theorem in dimensions at most three, this resolves the finite-rational-coefficient semi-log-canonical index conjecture.

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@misc{OAI:Uniform-indices-for-semi-log-canonical-log-Calabi-Yau-pairs-October-5-2026,
  author = {{OpenAI}},
  title = {{Uniform indices for semi-log-canonical log Calabi--Yau pairs}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Uniform-indices-for-semi-log-canonical-log-Calabi-Yau-pairs-October-5-2026/uniform-slc-index.pdf}{OAI:Uniform-indices-for-semi-log-canonical-log-Calabi-Yau-pairs-October-5-2026}},
  year = {2026}
}

Conditional good minimal models for compact Kähler fourfolds

OpenAI · October 6, 2026

Assuming orbifold Iitaka subadditivity and abundance for nef fourfold adjoints of nonnegative Kodaira dimension, we prove the existence of good minimal models for globally strongly ℚ-factorial compact Kähler klt fourfold pairs with effective rational boundary and analytically pseudo-effective actual ℚ-Cartier adjoint. The ordinary fourfold minimal model program supplies the nef endpoint. We prove nonvanishing by fibration arguments and, in algebraic dimension zero, by singular metrics, holomorphic foliations, and extension from a reduced boundary. The projective abundance argument used in the proof is included in full.

This version clarifies the independent ordinary-MMP input and states the remaining Iitaka-subadditivity and effective-abundance hypotheses explicitly; see the previous version.

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@misc{OAI:Conditional-good-minimal-models-for-compact-Kahler-fourfolds-October-6-2026,
  author = {{OpenAI}},
  title = {{Conditional good minimal models for compact K\"ahler fourfolds}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Conditional-good-minimal-models-for-compact-Kahler-fourfolds-October-6-2026/paper.pdf}{OAI:Conditional-good-minimal-models-for-compact-Kahler-fourfolds-October-6-2026}},
  year = {2026}
}

Log abundance in characteristic zero

OpenAI · September 24, 2026

We prove the rational-boundary log abundance conjecture in every dimension over algebraically closed fields of characteristic zero: every nef ℚ-Cartier log canonical divisor on a projective log canonical pair with effective rational boundary is semiample. Over ℂ, the proof also establishes canonical nonvanishing for smooth projective varieties in every dimension.

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@misc{OAI:Log-abundance-in-characteristic-zero-September-24-2026,
  author = {{OpenAI}},
  title = {{Log abundance in characteristic zero}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Log-abundance-in-characteristic-zero-September-24-2026/paper.pdf}{OAI:Log-abundance-in-characteristic-zero-September-24-2026}},
  year = {2026}
}

Minimal metrics and interior injectivity for nef adjoints

OpenAI · September 27, 2026

Let \((H,\Theta)\) be a projective complex klt pair with effective rational boundary and nef ℚ-Cartier adjoint. On any projective log resolution, minimal semipositive metrics on the pulled-back adjoint exist and have zero Lelong numbers everywhere. On smooth projective complex varieties, we also prove an H¹-injectivity theorem for rational interior boundaries with simple-normal-crossing support when both endpoint bundles carry zero-Lelong semipositive metrics.

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@misc{OAI:Minimal-metrics-and-interior-injectivity-for-nef-adjoints-September-27-2026,
  author = {{OpenAI}},
  title = {{Minimal metrics and interior injectivity for nef adjoints}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Minimal-metrics-and-interior-injectivity-for-nef-adjoints-September-27-2026/paper.pdf}{OAI:Minimal-metrics-and-interior-injectivity-for-nef-adjoints-September-27-2026}},
  year = {2026}
}

Fourfold nonvanishing by minimal metrics and moving jets

OpenAI · September 27, 2026

We prove canonical nonvanishing for smooth connected projective complex fourfolds: if K_X is pseudo-effective, then \(H^0(X,mK_X)\ne0\) for some positive integer m.

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@misc{OAI:Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets-September-27-2026,
  author = {{OpenAI}},
  title = {{Fourfold nonvanishing by minimal metrics and moving jets}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets-September-27-2026/paper.pdf}{OAI:Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets-September-27-2026}},
  year = {2026}
}

Lifting sections from the reduced support of an adjoint

OpenAI · September 27, 2026

For a projective ℚ-factorial dlt pair with effective rational boundary over an algebraically closed field of characteristic zero, we prove that the adjoint has positive Iitaka dimension whenever a nonzero effective Cartier multiple is supported on the coefficient-one boundary and restricts to a semiample line bundle on its whole reduced support. This gives log abundance after nonvanishing in dimension at most four over ℂ.

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@misc{OAI:Lifting-sections-from-the-reduced-support-of-an-adjoint-September-27-2026,
  author = {{OpenAI}},
  title = {{Lifting sections from the reduced support of an adjoint}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Lifting-sections-from-the-reduced-support-of-an-adjoint-September-27-2026/paper.pdf}{OAI:Lifting-sections-from-the-reduced-support-of-an-adjoint-September-27-2026}},
  year = {2026}
}

Schnell fiber spaces and good canonical models

OpenAI · September 24, 2026

We prove the Campana–Peternell inequality \(\kappa(X)\geq\kappa(D)\) for a smooth connected projective complex variety X and an effective Cartier divisor D whenever \(m_0K_X-D\) is pseudo-effective for some positive integer m₀. For an algebraic fiber space \(f\colon X\to Y\) between smooth connected projective complex varieties, the hypothesis that \(m_0K_X-f^*H\) is pseudo-effective with H ample Cartier gives \(\kappa(X)=\kappa(F)+\dim Y\) for a very general smooth fiber F, as well as nonzero sections of \(mK_X-f^*H\) for all sufficiently large divisible m.

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@misc{OAI:Schnell-fiber-spaces-and-good-canonical-models-September-24-2026,
  author = {{OpenAI}},
  title = {{Schnell fiber spaces and good canonical models}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Schnell-fiber-spaces-and-good-canonical-models-September-24-2026/paper.pdf}{OAI:Schnell-fiber-spaces-and-good-canonical-models-September-24-2026}},
  year = {2026}
}

Uniform log Iitaka fibrations and bounded moduli denominators

OpenAI · October 4, 2026

For normal projective log canonical pairs over algebraically closed fields of characteristic zero, of fixed dimension d ≥ 5 and with effective boundary coefficients in a fixed finite rational set, we prove that one complete rounded pluricanonical system generates the full Iitaka field whenever the ℚ-Cartier log canonical divisor has nonnegative Kodaira dimension. Its degree depends only on the dimension and coefficient set. For the paper's normalized canonical bundle formulae over ℂ, we also bound the Cartier denominators of the moduli divisors on smooth projective determining models in terms of the dimension and coefficient set.

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@misc{OAI:Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators-October-4-2026,
  author = {{OpenAI}},
  title = {{Uniform log Iitaka fibrations and bounded moduli denominators}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators-October-4-2026/uniform-log-iitaka.pdf}{OAI:Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators-October-4-2026}},
  year = {2026}
}

Uniform Pluricanonical Iitaka Fibrations

OpenAI · October 3, 2026

We prove the effective Iitaka fibration conjecture in characteristic zero. For each dimension, one pluricanonical degree defines the Iitaka fibration of every smooth integral projective variety of that dimension and nonnegative Kodaira dimension over an algebraically closed field. The associated sections generate the full Iitaka function field.

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@misc{OAI:Uniform-Pluricanonical-Iitaka-Fibrations-October-3-2026,
  author = {{OpenAI}},
  title = {{Uniform Pluricanonical Iitaka Fibrations}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Uniform-Pluricanonical-Iitaka-Fibrations-October-3-2026/paper.pdf}{OAI:Uniform-Pluricanonical-Iitaka-Fibrations-October-3-2026}},
  year = {2026}
}

Relative denominators and effective systems for log Calabi-Yau fibrations

OpenAI · September 27, 2026

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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@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}
}

Arithmetic Stein-degree bounds for log Calabi–Yau pairs

OpenAI · September 25, 2026

Fix d ≥ 1 and t > 0. Let \((X,B)\) be an ordinary projective log canonical ℚ-pair of dimension d over a characteristic-zero field k, with X normal and integral, \(H^0(X,\mathcal O_X)=k\), B effective, and \(K_X+B\sim_{\mathbb Q}0\). We prove that every prime component S of B with coefficient at least t satisfies \([k_S:k]\leq N(d,t)\), where k_S is the relative algebraic closure of k in \(k(S)\). This also bounds the Stein degree of S over k, proving the contraction-to-a-point formulation of Birkar's Stein-degree conjecture for ordinary ℚ-pairs.

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@misc{OAI:Arithmetic-Stein-degree-bounds-for-log-Calabi-Yau-pairs-September-25-2026,
  author = {{OpenAI}},
  title = {{Arithmetic Stein-degree bounds for log Calabi--Yau pairs}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Arithmetic-Stein-degree-bounds-for-log-Calabi-Yau-pairs-September-25-2026/paper.pdf}{OAI:Arithmetic-Stein-degree-bounds-for-log-Calabi-Yau-pairs-September-25-2026}},
  year = {2026}
}

Uniform effective log Iitaka fibrations for fourfolds

OpenAI · September 26, 2026

We prove the finite-rational-coefficient case of the effective log Iitaka conjecture in dimension four. For normal projective log canonical complex fourfolds with boundary coefficients in a fixed finite rational set and pseudo-effective rational Cartier adjoint, one uniform degree makes the complete rounded reflexive system nonempty and its section ratios generate the full Iitaka field. We also prove a uniform canonical index bound for projective klt complex fourfolds with rationally trivial canonical divisor.

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@misc{OAI:Uniform-effective-log-Iitaka-fibrations-for-fourfolds-September-26-2026,
  author = {{OpenAI}},
  title = {{Uniform effective log Iitaka fibrations for fourfolds}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Uniform-effective-log-Iitaka-fibrations-for-fourfolds-September-26-2026/paper.pdf}{OAI:Uniform-effective-log-Iitaka-fibrations-for-fourfolds-September-26-2026}},
  year = {2026}
}

Abundance after nonvanishing for compact Kähler fourfolds

OpenAI · September 27, 2026

We prove semiampleness of the actual ℚ-Cartier adjoint \(K_X+\Delta\) of a normal connected compact Kähler klt fourfold whenever it is analytically nef and some positive Cartier multiple has a nonzero section. The boundary is effective and rational; the fourfold need not be projective or ℚ-factorial. In Iitaka dimension zero, a positive Cartier multiple is the trivial holomorphic line bundle.

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@misc{OAI:Abundance-after-nonvanishing-for-compact-Kahler-fourfolds-September-27-2026,
  author = {{OpenAI}},
  title = {{Abundance after nonvanishing for compact K\"ahler fourfolds}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Abundance-after-nonvanishing-for-compact-Kahler-fourfolds-September-27-2026/paper.pdf}{OAI:Abundance-after-nonvanishing-for-compact-Kahler-fourfolds-September-27-2026}},
  year = {2026}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.