Kobayashi's conjecture: a compact Kahler manifold with no entire curves has ample canonical bundle
A compact complex manifold is Kobayashi hyperbolic exactly when every holomorphic map is constant (Brody). Kobayashi conjectured that hyperbolicity forces positivity of the canonical bundle: a compact Kahler hyperbolic manifold should have ample , and hence be projective. Known results needed metric hypotheses: negative or quasi-negative holomorphic sectional curvature (Wu-Yau, Tosatti-Yang, Diverio-Trapani), or Kahler hyperbolicity in Gromov's sense (Chen-Yang); without rational curves is known to be nef (Ou, Cao-Horing), but bigness was open. Is ample for every compact Kahler manifold of positive dimension that contains no nonconstant entire curve?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Complex geometry; Kobayashi hyperbolicity
- Posed by
- Shoshichi Kobayashi
- Year posed
- 1970
- Years open
- 56y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 48 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: if is a compact connected Kahler manifold of positive dimension and every holomorphic map is constant, then is ample and is projective. Corollaries: is globally generated for (using the release's Fujita freeness result), and a finite cover splits as a ball-quotient-type factor times a hyperbolic factor when the universal cover is semialgebraic. It does not cover singular spaces, non-Kahler compact complex manifolds, or the converse direction (Green-Griffiths-Lang type statements).
What the AI did
The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family is a single manuscript (September 23, 2026). Its two corollaries cite other results from the same release (Fujita freeness and a semialgebraic-cover classification); the main theorem does not use them.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Kobayashi's conjecture. It states that every compact connected Kahler manifold of positive dimension in which every holomorphic map from is constant has ample , so a power of embeds projectively. That is the conjecture in the smooth compact Kahler category. The proof (extremal holomorphic discs with an area penalty, a boundary-normalised frame, a Hessian identity giving a volume lower bound, then Aubin-Yau and Morse inequalities) was not refereed. There is no Lean formalization. The singular and non-Kahler settings are outside its scope, as the paper says.