---
title: "Numerical semiampleness and generalized minimal models"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 036; unreviewed"
arxiv: null
doi: null
source: "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"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "algebraic-and-complex-geometry"
  - "unformalized"
seed_rank: 1896
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "Numerical semiampleness of nef adjoint classes on compact Kähler manifolds"
    url: "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"
    year: 2026
    arxiv: null
    doi: null
  - title: "Numerical Semiampleness of Nef Adjoint Divisors"
    url: "https://github.com/openai/math/blob/main/preprints/Numerical-Semiampleness-of-Nef-Adjoint-Divisors-October-3-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Minimal models and Mori fibre spaces for generalized log canonical Q-pairs"
    url: "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"
    year: 2026
    arxiv: null
    doi: null
  - title: "Minimal models in numerical dimension one"
    url: "https://github.com/openai/math/blob/main/preprints/Minimal-models-in-numerical-dimension-one-September-24-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# Numerical semiampleness and generalized minimal models

## One-sentence takeaway

OpenAI's result family 036 (Algebraic and complex geometry) claims: 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.

## Why it matters here

Pure mathematics with no direct engine use; it is in the library as part of the openai/math corpus, which records what an AI research model claims to have proved. More broadly, every card in this batch is evidence for the `ai-mathematical-reasoning` thread: per the repo README, an unreleased internal OpenAI model produced it, using about three hours of ChatGPT Pro thinking compute per result on average, across roughly 4,000 posed problems.

## Key ideas

- Family summary (repo): 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.
- *Numerical semiampleness of nef adjoint classes on compact Kähler manifolds*: 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.
- *Numerical Semiampleness of Nef Adjoint Divisors*: 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.
- *Minimal models and Mori fibre spaces for generalized log canonical Q-pairs*: 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.
- *Minimal models in numerical dimension one*: 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.

## Caveats

- Unreviewed: these are AI-generated manuscripts posted to GitHub with no peer review or arXiv deposit. The repo README says results are "at different stages of verification" and that "some of the unformalized results could have issues."
- This card restates the repo's family summary, manuscript abstracts and Lean scope notes. It is not an independent check of the proofs, and the scope of each theorem is exactly as stated there.
- Lean: not formalized. The repo has no Lean development for this family, so the claim rests on the prose manuscript(s) alone.

## Links

- Manuscript: [Numerical semiampleness of nef adjoint classes on compact Kähler manifolds](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)
- Manuscript: [Numerical Semiampleness of Nef Adjoint Divisors](https://github.com/openai/math/blob/main/preprints/Numerical-Semiampleness-of-Nef-Adjoint-Divisors-October-3-2026/paper.pdf)
- Manuscript: [Minimal models and Mori fibre spaces for generalized log canonical Q-pairs](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)
- Manuscript: [Minimal models in numerical dimension one](https://github.com/openai/math/blob/main/preprints/Minimal-models-in-numerical-dimension-one-September-24-2026/paper.pdf)
- Family entry: https://github.com/openai/math/blob/main/CONTENTS.md
- Overview PDF: https://github.com/openai/math/blob/main/overview.pdf
- Repo: https://github.com/openai/math