Kobayashi's canonical-ampleness conjecture
From OpenAI Aftermath
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 (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 Canonical ampleness of compact hyperbolic Kähler manifolds at commit
1581d4c. - Formalization. The repository has no Lean scope note for this family.
Public follow-up work
| 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, 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, referee review in JJardine919/openai-math-audit (archived at 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.
