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Numerical Semiampleness of Nef Adjoint Divisors

OpenAI · October 3, 2026

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Abstract

We prove the Generalised Abundance Conjecture: if $(X,B)$ is a projective klt ℚ-pair over an algebraically closed field of characteristic zero, $K_X+B$ is pseudo-effective, M is a nef ℚ-Cartier divisor on X, and $K_X+B+M$ is nef, then $K_X+B+M$ is numerically equivalent to a semiample ℚ-Cartier divisor on X.

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  • Vladimir Lazić and Thomas Peternell, On Generalised Abundance, I (arXiv:1808.00438; Publications of the Research Institute for Mathematical Sciences 56, 2020). The AI Proved What? exposition identifies this as the source of the generalized abundance conjecture addressed by the paper and its known low-dimensional cases. [family-level]

Possibly relevant literature (found by the agent, not checked)

  • Priyankur Chaudhuri, Semiampleness for generalized pairs (arXiv:2209.01452, 2022; Int. J. Math. 2023). Proves that a nef and abundant generalized klt adjoint is semiample, an actual-semiampleness statement next to the paper's numerical semiampleness for $K_X+B+M$ (the paper cites a different Chaudhuri article, on nef reduction).
  • A. Höring, V. Lazić, C. Lehn, Nonvanishing results for Kähler varieties (arXiv:2508.14634, 2025). Proves nonvanishing for nef line bundles of numerical dimension one on K-trivial varieties under non-uniruledness and $\chi\neq0$ hypotheses, a special case related to Corollary 1.2 (every nef divisor on a klt pair with $K_X+B\equiv0$ is numerically semiample).

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Cite

@misc{OAI:Numerical-Semiampleness-of-Nef-Adjoint-Divisors-October-3-2026,
  author = {{OpenAI}},
  title = {{Numerical Semiampleness of Nef Adjoint Divisors}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Numerical-Semiampleness-of-Nef-Adjoint-Divisors-October-3-2026/paper.pdf}{OAI:Numerical-Semiampleness-of-Nef-Adjoint-Divisors-October-3-2026}},
  year = {2026}
}

Family 036: Numerical semiampleness and generalized minimal models

Proves numerical semiampleness for nef adjoints $K_X+B+M$ with $K_X+B$ pseudo-effective and M nef rational, for projective klt rational pairs over algebraically closed characteristic-zero fields and smooth compact Kähler rational klt simple-normal-crossing pairs, using Bott–Chern cohomology in the latter case. Separately, projective generalized log canonical rational pairs over such fields admit minimal models for pseudo-effective adjoints and Mori fiber spaces otherwise, with nef b-data fixed.

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