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Minimal models in numerical dimension one

OpenAI · September 24, 2026

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Abstract

We resolve the numerical-dimension-one case of the minimal-model conjecture for smooth connected complex projective varieties of dimension at least three. If KX is pseudo-effective and $\kappa_\sigma(X,K_X)=1$, with κσ defined by section growth with a fixed ample twist, then X admits a projective ℚ-factorial terminal minimal model.

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

  • V. Lazić and T. Peternell, On Generalised Abundance, II (arXiv:1809.02500; Peking Math. J. 3 (2020)). Section 7 ("Numerical dimension 1") proves Theorem B, num-effectivity of $K_X+\Delta+tL$ for terminal pairs with $K_X+\Delta$ nef, $\nu(K_X+\Delta+L)=1$ and $\chi(\mathcal O_X)\neq0$; this is a numerical-dimension-one result next to the LP18 and Liu–Xu comparisons in the introduction.

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Cite

@misc{OAI:Minimal-models-in-numerical-dimension-one-September-24-2026,
  author = {{OpenAI}},
  title = {{Minimal models in numerical dimension one}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Minimal-models-in-numerical-dimension-one-September-24-2026/paper.pdf}{OAI:Minimal-models-in-numerical-dimension-one-September-24-2026}},
  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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