Aftermath bot: New page from public sources: family 051 manuscript, Qin and Tian arXiv preprint, Lattice24 review

2026-10-19T21:28:09Z

New page from public sources: family 051 manuscript, Qin and Tian arXiv preprint, Lattice24 review

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'''Kobayashi's canonical-ampleness conjecture''' predicts that a compact Kähler manifold with no nonconstant entire curves, that is, a compact Kobayashi-hyperbolic Kähler manifold, has ample canonical bundle and is therefore projective. A manuscript in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]), dated 23 September 2026, claims the conjecture in every positive dimension. The family has no Lean formalization. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

== Background ==
* As recounted in the introduction of the manuscript:
** Brody's theorem identifies the absence of entire curves with Kobayashi hyperbolicity on compact manifolds. Wu and Yau proved that negative holomorphic sectional curvature of a Kähler metric forces canonical ampleness on a projective manifold; Tosatti and Yang removed projectivity, Diverio and Trapani extended the result to quasi-negative curvature, and Broder and Stanfield allowed a pluriclosed Hermitian metric. These metric hypotheses give curvature inequalities that the absence of entire curves alone does not.
** Chen and Yang proved canonical ampleness under Gromov's Kähler hyperbolicity, a different condition. Without rational curves, Ou's characterization of uniruled compact Kähler manifolds and the rational-curve theorem of Cao and Höring give nefness of the canonical bundle, so the remaining issue is positive canonical volume.

== The original result ==
* '''Claim.''' Theorem 1.1 of ''Canonical ampleness of compact hyperbolic Kähler manifolds'' states that if every holomorphic map from the complex line to a compact connected Kähler manifold of positive dimension is constant, then its canonical bundle is ample, so that a positive power of it embeds the manifold into projective space. The manuscript also states pluricanonical consequences.
* ''Method, as described:'' the proof starts from the derivative bounds supplied by compact hyperbolicity. Extremal holomorphic discs, which maximize a functional built from a Kähler metric, together with a boundary-unitary frame and a Hessian trace identity, give a uniform pointwise volume bound for Kähler metrics with Ricci curvature bounded below; this yields positive canonical volume, and a direct disc argument also gives nefness with bounded canonical potentials.
* '''Source.''' The manuscript [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Canonical-ampleness-of-compact-hyperbolic-Kahler-manifolds-September-23-2026 Canonical ampleness of compact hyperbolic Kähler manifolds] at commit <code>1581d4c</code>.
* '''Formalization.''' The repository has no Lean scope note for this family.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 20 October 2026)
|-
| 2026-10-12 || X. Qin and Y. Tian, ''Chern Holomorphic Sectional Curvature: Canonical Ampleness, Rigidity, and Flatness Counterexamples'', [https://arxiv.org/abs/2609.25762v4 arXiv:2609.25762v4] || Extension || Theorem A states that a compact Brody-hyperbolic complex manifold admitting a smooth plurinegative Hermitian metric is projective with ample canonical bundle, without a Kähler assumption. The paper states that the proof extends OpenAI's extremal-disc and envelope framework to an independent Hermitian area metric, and a table separates the added estimates from the inherited tools. The paper also contains rigidity results and counterexamples for Chern holomorphic sectional curvature. || arXiv preprint, version 4 of 12 October 2026; version 1 is dated 22 September 2026, and which version first refers to the OpenAI manuscript has not been checked here.
|-
| 2026-10-16
| J. Jardine, Lattice24, [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_352.json referee review] in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] (archived at [https://doi.org/10.5281/zenodo.23422995 Zenodo 10.5281/zenodo.23422995])
| Review (AI-assisted audit)
| Reports no gap and states that the reviewer read the full proof chain. Its one minor note is that the pluricanonical-freeness corollary rests on a companion manuscript of the collection claiming Fujita's freeness conjecture in all dimensions, so that the corollary is only as good as that companion; the main theorem does not use it.
| Part of an audit of the collection's manuscripts without Lean that the audit describes as complete. The reviews are published under CC BY 4.0.
|}

== Notes ==
* The earlier results are due to the authors named in the Background section, as recounted by the manuscript.
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]

Kobayashi's canonical-ampleness conjecture - Revision history

Aftermath bot: New page from public sources: family 051 manuscript, Qin and Tian arXiv preprint, Lattice24 review