---
title: "Anticanonical nonvanishing in every dimension"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 068; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/main.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "algebraic-and-complex-geometry"
  - "unformalized"
seed_rank: 1926
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "Anticanonical nonvanishing from smooth semipositivity"
    url: "https://github.com/openai/math/blob/main/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/main.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Invariant anticanonical indices and conversion of twisted differentials"
    url: "https://github.com/openai/math/blob/main/preprints/Invariant-anticanonical-indices-and-conversion-of-twisted-differentials-September-26-2026/main.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Bounded anticanonical metrics on klt pairs and torus quotients"
    url: "https://github.com/openai/math/blob/main/preprints/Bounded-anticanonical-metrics-on-klt-pairs-and-torus-quotients-September-26-2026/main.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Cohomological transfer and equivariant anticanonical sections"
    url: "https://github.com/openai/math/blob/main/preprints/Cohomological-transfer-and-equivariant-anticanonical-sections-September-26-2026/main.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Metric descent and rank-preserving contractions"
    url: "https://github.com/openai/math/blob/main/preprints/Metric-descent-and-rank-preserving-contractions-September-26-2026/main.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Integrable metrics and effectivity with controlled boundary"
    url: "https://github.com/openai/math/blob/main/preprints/Integrable-metrics-and-effectivity-with-controlled-boundary-September-26-2026/main.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Exact orders and invariant anticanonical linear systems"
    url: "https://github.com/openai/math/blob/main/preprints/Exact-orders-and-invariant-anticanonical-linear-systems-September-26-2026/main.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# Anticanonical nonvanishing in every dimension

## One-sentence takeaway

OpenAI's result family 068 (Algebraic and complex geometry) claims: If X is a smooth connected complex projective variety and $-K_X$ admits a smooth Hermitian metric with nonnegative curvature, then $H^0(X,-mK_X)\ne0$ for some m > 0.

## 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): Thus smooth semipositivity forces a nonzero section of a positive tensor power of the anticanonical bundle in every dimension.
- *Anticanonical nonvanishing from smooth semipositivity*: Every smooth connected projective complex variety with smoothly semipositive anticanonical bundle has a nonzero section of some positive anticanonical power.
- *Invariant anticanonical indices and conversion of twisted differentials*: For a holomorphic action of a compact torus on a compact complex manifold, we prove that the invariant Euler characteristic of naturally linearized anticanonical powers is polynomial on a divisible progression, with its actual value at exponent zero. Independently, on a smooth projective variety with smoothly semipositive anticanonical bundle, we remove a fixed pseudoeffective error from an unbounded sequence of effective twists. These results convert invariant cohomology into twisted differential forms and prove anticanonical nonvanishing on smooth projective varieties with smoothly semipositive anticanonical bundle, by descending the forms before conversion.
- *Bounded anticanonical metrics on klt pairs and torus quotients*: Let $(W,D)$ be a projective ℚ-factorial klt pair with effective rational boundary. We prove that a positive Cartier multiple of $-(K_W+D)$ has a nonzero section if this divisor is nef, its pullback to a resolution admits a semipositive metric with locally bounded weights, and the resolution has nonzero structure-sheaf Euler characteristic. Consequently a smooth rationally connected projective variety with smoothly semipositive anticanonical bundle has a nonzero naturally invariant anticanonical plurisection for every algebraic torus action.
- *Cohomological transfer and equivariant anticanonical sections*: Let X be a smooth projective complex variety with smoothly semipositive anticanonical bundle. For any torus linearization of that bundle, we show that invariant sections in unbounded degrees with one fixed negative pseudoeffective error yield an invariant section in a positive untwisted degree. Combining the natural invariant index with compact-monodromy structure, we deduce that every such X has a nonzero section of some positive anticanonical power.
- *Metric descent and rank-preserving contractions*: From a smooth semipositive anticanonical metric on a smooth projective complex variety, we construct an unweighted integrable semipositive metric on a corrected anticanonical line of a smooth base, using a normal equidimensional toroidal model and full generic adjoint rank one at exponent zero. The explicit rational boundary correction pulls back to an exceptional divisor. On varieties without positive-degree holomorphic forms, two-metric transfer gives sections with prescribed boundary poles from a finite-volume log-anticanonical metric and a target line with bounded semipositive weights.
- Plus 2 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: [Anticanonical nonvanishing from smooth semipositivity](https://github.com/openai/math/blob/main/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/main.pdf)
- Manuscript: [Invariant anticanonical indices and conversion of twisted differentials](https://github.com/openai/math/blob/main/preprints/Invariant-anticanonical-indices-and-conversion-of-twisted-differentials-September-26-2026/main.pdf)
- Manuscript: [Bounded anticanonical metrics on klt pairs and torus quotients](https://github.com/openai/math/blob/main/preprints/Bounded-anticanonical-metrics-on-klt-pairs-and-torus-quotients-September-26-2026/main.pdf)
- Manuscript: [Cohomological transfer and equivariant anticanonical sections](https://github.com/openai/math/blob/main/preprints/Cohomological-transfer-and-equivariant-anticanonical-sections-September-26-2026/main.pdf)
- Manuscript: [Metric descent and rank-preserving contractions](https://github.com/openai/math/blob/main/preprints/Metric-descent-and-rank-preserving-contractions-September-26-2026/main.pdf)
- Manuscript: [Integrable metrics and effectivity with controlled boundary](https://github.com/openai/math/blob/main/preprints/Integrable-metrics-and-effectivity-with-controlled-boundary-September-26-2026/main.pdf)
- Manuscript: [Exact orders and invariant anticanonical linear systems](https://github.com/openai/math/blob/main/preprints/Exact-orders-and-invariant-anticanonical-linear-systems-September-26-2026/main.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