Result 036, Algebraic and complex geometry

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.

Proof

The bigger picture

Why it matters

Nef divisors satisfy a weak positivity condition, but need not have enough sections to define a map. The manuscripts report when canonical divisors with added boundary and nef data have a numerically equivalent representative whose multiple defines a map everywhere.

What changes?

The numerical-semiampleness claim assumes that the canonical-plus-boundary divisor is pseudo-effective, a limit of effective classes, and its sum with a nef rational Cartier divisor is nef. This covers projective rational klt pairs, with controlled singularities, over algebraically closed characteristic-zero fields. It also covers smooth connected compact Kähler manifolds with effective rational simple-normal-crossing boundaries, coefficients below one, and nef rational holomorphic line bundles. There equality means equality of real Bott–Chern classes, allowing a flat change rather than proving semiampleness of the original adjoint.

What does that help mathematicians do?

Separately, the generalized-pair manuscript reports that every projective generalized log canonical rational pair over an algebraically closed characteristic-zero field has a minimal model if its adjoint is pseudo-effective, and a Mori fiber space otherwise, keeping its nef rational Cartier b-divisor fixed. Thus researchers could obtain either a birational model with nef adjoint or a fiber structure governed by negativity, without changing the prescribed nef data. Setting that data to zero recovers ordinary log canonical pairs.

Are there practical applications?

The immediate value is foundational: numerical positivity can be studied using a representative whose sections define a geometric map, connecting numerical information to geometric structure. The model-existence claim would also supply endpoints for classifying varieties through birational transformations. It is not, by itself, a computational procedure for finding those models.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

4 manuscripts

Numerical semiampleness of nef adjoint classes on compact Kähler manifolds

October 4, 2026 52 pages

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.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-4-2026,
  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-4-2026/numerical-generalized-abundance.pdf}{OAI:Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-4-2026}},
  year = {2026}
}

Numerical Semiampleness of Nef Adjoint Divisors

October 3, 2026 37 pages

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.

PDF LaTeX source

Cite (BibTeX)

@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}
}

Minimal models and Mori fibre spaces for generalized log canonical Q-pairs

September 24, 2026 79 pages

We resolve the existence form of the minimal-model conjecture for projective generalized log canonical ℚ-pairs over algebraically closed fields of characteristic zero. Such a pair admits a minimal model when its adjoint divisor is pseudo-effective, and a Mori fibre space otherwise, while keeping the nef ℚ-Cartier b-divisor fixed. Taking the nef part to be zero gives the corresponding result for ordinary log canonical ℚ-pairs.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Minimal-models-and-Mori-fibre-spaces-for-generalized-log-canonical-Q-pairs-September-24-2026,
  author = {{OpenAI}},
  title = {{Minimal models and Mori fibre spaces for generalized log canonical $\mathbb{Q}$-pairs}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Minimal-models-and-Mori-fibre-spaces-for-generalized-log-canonical-Q-pairs-September-24-2026/paper.pdf}{OAI:Minimal-models-and-Mori-fibre-spaces-for-generalized-log-canonical-Q-pairs-September-24-2026}},
  year = {2026}
}

Minimal models in numerical dimension one

September 24, 2026 14 pages

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

PDF LaTeX source

Cite (BibTeX)

@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}
}

At a glance

Subject: Algebraic and complex geometry
Manuscripts: 4
Pages: 182
Dated: Sep 24 to Oct 4, 2026
Lean formalization: No
Reasoning summary: No

Named problems addressed

Indexed by Emergent Mind with an LLM from the text above; check the manuscripts for the precise statements.

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