Lazić–Peternell's generalized abundance conjecture

About 3 years old · traced to 2023

Let $X$ be a projective manifold with $K_X$ pseudoeffective, and let $L\to X$ be a nef line bundle such that

L^m\otimes K_X^{*}

is nef for some $m\geqslant 1$ . Lazić–Peternell's generalized abundance conjecture. Then $L$ is numerically semiample. This is known only when the dimension is at most two, so the conjecture remains open in higher dimensions.

References

Primary source

Valentino Tosatti, “Semipositive line bundles and (1,1)-classes”, arXiv:2309.00580 (2023).

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RemarkAI-assistedClaimed by OpenAI. The manuscript claims numerical semiampleness of nef adjoint classes on smooth compact Kahler manifolds. In the projective target setting, take the nef summand M = mL - K_X and boundary zero: the claimed semiample representative of K_X + M = mL yields a numerical semiample representative of L. This allows a change of flat part and does not assert semiampleness of the original bundle.See full solutionHide full solution

Claimed by OpenAI. The manuscript claims numerical semiampleness of nef adjoint classes on smooth compact Kahler manifolds. In the projective target setting, take the nef summand M = mL - K_X and boundary zero: the claimed semiample representative of K_X + M = mL yields a numerical semiample representative of L. This allows a change of flat part and does not assert semiampleness of the original bundle.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-4-2026/numerical-generalized-abundance.pdf

OpenAI-036-01-Numerical-semiampleness-of-nef-adjoint-classes-on-compact-K-hler-manifolds.pdf633,795 bytesOpen

RemarkAI-assistedClaimed by OpenAI. The manuscript claims numerical semiampleness of nef sums of a pseudo-effective rational klt log canonical adjoint and a nef rational divisor, using the stated log-abundance and minimal-model inputs. For this smooth projective target, M = mL - K_X gives adjoint sum mL, so its numerical semiample representative gives one for L. The conclusion is numerical, rather than semiampleness of L itself.See full solutionHide full solution

Claimed by OpenAI. The manuscript claims numerical semiampleness of nef sums of a pseudo-effective rational klt log canonical adjoint and a nef rational divisor, using the stated log-abundance and minimal-model inputs. For this smooth projective target, M = mL - K_X gives adjoint sum mL, so its numerical semiample representative gives one for L. The conclusion is numerical, rather than semiampleness of L itself.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Numerical-Semiampleness-of-Nef-Adjoint-Divisors-October-3-2026/paper.pdf

OpenAI-036-02-Numerical-Semiampleness-of-Nef-Adjoint-Divisors.pdf556,657 bytesOpen