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Numerical semiampleness of nef adjoint classes on compact Kähler manifolds

OpenAI · October 6, 2026

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Abstract

Let X be a smooth connected compact Kähler manifold, let B be an effective rational simple normal crossing divisor with coefficients less than one, and let M be a nef rational holomorphic line bundle on X. If $K_X+B$ is pseudo-effective and $K_X+B+M$ is nef, we prove that its first Chern class in real Bott–Chern cohomology is represented by a semiample rational line bundle. This numerical statement allows a flat change of line bundle; it does not assert semiampleness of the original adjoint.

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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.

  • 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 its projective generalized abundance conjecture as background for this paper’s compact Kähler extension. [family-level]

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

  • A. Höring, V. Lazić, C. Lehn, Nonvanishing results for Kähler varieties (arXiv:2508.14634, 2025). Proves nonvanishing for numerical-dimension-one adjoint bundles on Kähler klt pairs and for nef line bundles on K-trivial Kähler varieties, under non-uniruledness and $\chi\neq0$; this relates to the meromorphic nonvanishing for $K_T+M$ needed in the algebraic-dimension-zero case of Theorem 1.2.

Digests and expositions

Elsewhere

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Cite

@misc{OAI:Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026,
  author = {{OpenAI}},
  title = {{Numerical semiampleness of nef adjoint classes on compact K{\"a}hler manifolds}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/numerical-generalized-abundance.pdf}{OAI:Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026}},
  year = {2026}
}

Versions

This version states the Iitaka, model, and contraction inputs used in the divisorial-decomposition argument explicitly; see the previous version.

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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