---
title: "Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 034; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-4-2026/main.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "algebraic-and-complex-geometry"
  - "unformalized"
seed_rank: 1894
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity"
    url: "https://github.com/openai/math/blob/main/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-4-2026/main.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Uniform indices for semi-log-canonical log Calabi–Yau pairs"
    url: "https://github.com/openai/math/blob/main/preprints/Uniform-indices-for-semi-log-canonical-log-Calabi-Yau-pairs-October-5-2026/uniform-slc-index.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Conditional good minimal models for compact Kähler fourfolds"
    url: "https://github.com/openai/math/blob/main/preprints/Conditional-good-minimal-models-for-compact-Kahler-fourfolds-October-5-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Log abundance in characteristic zero"
    url: "https://github.com/openai/math/blob/main/preprints/Log-abundance-in-characteristic-zero-September-24-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Minimal metrics and interior injectivity for nef adjoints"
    url: "https://github.com/openai/math/blob/main/preprints/Minimal-metrics-and-interior-injectivity-for-nef-adjoints-September-27-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Fourfold nonvanishing by minimal metrics and moving jets"
    url: "https://github.com/openai/math/blob/main/preprints/Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets-September-27-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Lifting sections from the reduced support of an adjoint"
    url: "https://github.com/openai/math/blob/main/preprints/Lifting-sections-from-the-reduced-support-of-an-adjoint-September-27-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Schnell fiber spaces and good canonical models"
    url: "https://github.com/openai/math/blob/main/preprints/Schnell-fiber-spaces-and-good-canonical-models-September-24-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Uniform log Iitaka fibrations and bounded moduli denominators"
    url: "https://github.com/openai/math/blob/main/preprints/Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators-October-4-2026/uniform-log-iitaka.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Uniform Pluricanonical Iitaka Fibrations"
    url: "https://github.com/openai/math/blob/main/preprints/Uniform-Pluricanonical-Iitaka-Fibrations-October-3-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Relative denominators and effective systems for log Calabi-Yau fibrations"
    url: "https://github.com/openai/math/blob/main/preprints/Relative-denominators-and-effective-systems-for-log-Calabi-Yau-fibrations-September-27-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Arithmetic Stein-degree bounds for log Calabi–Yau pairs"
    url: "https://github.com/openai/math/blob/main/preprints/Arithmetic-Stein-degree-bounds-for-log-Calabi-Yau-pairs-September-25-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Uniform effective log Iitaka fibrations for fourfolds"
    url: "https://github.com/openai/math/blob/main/preprints/Uniform-effective-log-Iitaka-fibrations-for-fourfolds-September-26-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Abundance after nonvanishing for compact Kähler fourfolds"
    url: "https://github.com/openai/math/blob/main/preprints/Abundance-after-nonvanishing-for-compact-Kahler-fourfolds-September-27-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity

## One-sentence takeaway

OpenAI's result family 034 (Algebraic and complex geometry) claims: Using logarithmic Iitaka subadditivity, proves log abundance in every dimension for normal compact Kähler log canonical pairs with effective rational boundary: an analytically nef ℚ-Cartier adjoint is semiample.

## 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): It also proves projective log abundance over every algebraically closed field of characteristic zero and the effective Iitaka fibration conjecture for smooth projective varieties of nonnegative Kodaira dimension in that setting. A further result resolves the finite-rational-coefficient index conjecture for connected projective semi-log-canonical log Calabi–Yau pairs over such fields in each fixed dimension and for each fixed finite set of rational boundary coefficients, with a uniform index independent of the number of components.
- *Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity*: Assume logarithmic Iitaka subadditivity for surjective morphisms with connected fibers between smooth projective complex varieties with compatible reduced simple normal crossing boundaries. We prove log abundance for normal irreducible compact Kähler spaces in every dimension: for a log canonical pair $(X,\Delta)$ with effective rational boundary and $K_X+\Delta$ ℚ-Cartier, analytic nefness of $K_X+\Delta$ implies semiampleness.
- *Uniform indices for semi-log-canonical log Calabi–Yau pairs*: We prove a uniform index theorem for connected projective semi-log-canonical log Calabi–Yau pairs in every fixed dimension at least four over an algebraically closed field of characteristic zero. For boundary coefficients in a fixed finite rational set, a single multiple of the log canonical divisor is Cartier and linearly trivial. The multiple depends only on the dimension and coefficient set, not on the number of irreducible components.
- *Conditional good minimal models for compact Kähler fourfolds*: Assuming orbifold Iitaka subadditivity, the specified pseudo-effective fourfold minimal model program, and abundance for nef fourfold adjoints of nonnegative Kodaira dimension, we prove the existence of good minimal models for globally strongly ℚ-factorial compact Kähler klt fourfold pairs with effective rational boundary and analytically pseudo-effective actual ℚ-Cartier adjoint. The additional step is nonvanishing. We prove it by fibration arguments and, in algebraic dimension zero, by singular metrics, holomorphic foliations, and extension from a reduced boundary.
- *Log abundance in characteristic zero*: We prove the rational-boundary log abundance conjecture in every dimension over algebraically closed fields of characteristic zero: every nef ℚ-Cartier log canonical divisor on a projective log canonical pair with effective rational boundary is semiample. Over ℂ, the proof also establishes canonical nonvanishing for smooth projective varieties in every dimension.
- *Minimal metrics and interior injectivity for nef adjoints*: Let $(H,\Theta)$ be a projective complex klt pair with effective rational boundary and nef ℚ-Cartier adjoint. On any projective log resolution, minimal semipositive metrics on the pulled-back adjoint exist and have zero Lelong numbers everywhere. On smooth projective complex varieties, we also prove an H1-injectivity theorem for rational interior boundaries with simple-normal-crossing support when both endpoint bundles carry zero-Lelong semipositive metrics.
- Plus 9 more companion manuscripts in this family; see Links.

## 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: [Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity](https://github.com/openai/math/blob/main/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-4-2026/main.pdf)
- Manuscript: [Uniform indices for semi-log-canonical log Calabi–Yau pairs](https://github.com/openai/math/blob/main/preprints/Uniform-indices-for-semi-log-canonical-log-Calabi-Yau-pairs-October-5-2026/uniform-slc-index.pdf)
- Manuscript: [Conditional good minimal models for compact Kähler fourfolds](https://github.com/openai/math/blob/main/preprints/Conditional-good-minimal-models-for-compact-Kahler-fourfolds-October-5-2026/paper.pdf)
- Manuscript: [Log abundance in characteristic zero](https://github.com/openai/math/blob/main/preprints/Log-abundance-in-characteristic-zero-September-24-2026/paper.pdf)
- Manuscript: [Minimal metrics and interior injectivity for nef adjoints](https://github.com/openai/math/blob/main/preprints/Minimal-metrics-and-interior-injectivity-for-nef-adjoints-September-27-2026/paper.pdf)
- Manuscript: [Fourfold nonvanishing by minimal metrics and moving jets](https://github.com/openai/math/blob/main/preprints/Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets-September-27-2026/paper.pdf)
- Manuscript: [Lifting sections from the reduced support of an adjoint](https://github.com/openai/math/blob/main/preprints/Lifting-sections-from-the-reduced-support-of-an-adjoint-September-27-2026/paper.pdf)
- Manuscript: [Schnell fiber spaces and good canonical models](https://github.com/openai/math/blob/main/preprints/Schnell-fiber-spaces-and-good-canonical-models-September-24-2026/paper.pdf)
- Manuscript: [Uniform log Iitaka fibrations and bounded moduli denominators](https://github.com/openai/math/blob/main/preprints/Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators-October-4-2026/uniform-log-iitaka.pdf)
- Manuscript: [Uniform Pluricanonical Iitaka Fibrations](https://github.com/openai/math/blob/main/preprints/Uniform-Pluricanonical-Iitaka-Fibrations-October-3-2026/paper.pdf)
- Manuscript: [Relative denominators and effective systems for log Calabi-Yau fibrations](https://github.com/openai/math/blob/main/preprints/Relative-denominators-and-effective-systems-for-log-Calabi-Yau-fibrations-September-27-2026/paper.pdf)
- Manuscript: [Arithmetic Stein-degree bounds for log Calabi–Yau pairs](https://github.com/openai/math/blob/main/preprints/Arithmetic-Stein-degree-bounds-for-log-Calabi-Yau-pairs-September-25-2026/paper.pdf)
- Manuscript: [Uniform effective log Iitaka fibrations for fourfolds](https://github.com/openai/math/blob/main/preprints/Uniform-effective-log-Iitaka-fibrations-for-fourfolds-September-26-2026/paper.pdf)
- Manuscript: [Abundance after nonvanishing for compact Kähler fourfolds](https://github.com/openai/math/blob/main/preprints/Abundance-after-nonvanishing-for-compact-Kahler-fourfolds-September-27-2026/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