The strong hyperkahler SYZ conjecture
Let be a compact irreducible holomorphic symplectic (hyperkahler) Kahler manifold of dimension with Beauville-Bogomolov-Fujiki form . A nonzero class that is nef and isotropic, , has numerical dimension , and should come from a Lagrangian fibration with . This was known for all known deformation types (Bayer-Macri, Markman, Yoshioka, Mongardi-Rapagnetta, Mongardi-Onorati, with Soldatenkov-Verbitsky's deformation theorem), and Verbitsky proved nonvanishing under a semipositive metric. Is every holomorphic line bundle on such an with nef and isotropic semiample?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Complex and algebraic geometry; hyperkahler manifolds, Lagrangian fibrations
- Posed by
- Misha Verbitsky (GAFA 2010, Conjecture 1.7); earlier nef-isotropic forms in Hassett-Tschinkel (2001, Remark 3.12) and Sawon (2003, Conjecture 4.1)
- Year posed
- 2001
- Years open
- 25y
- 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 in the closure of the Kahler cone and , then is semiample: some for a surjective connected-fibre map onto a normal projective -dimensional with ample, every fibre component Lagrangian of dimension . Corollary 1.2: in each dimension there are finitely many deformation types with . Corollary 1.3 (projective , via the companion): . Not shown: existence of a nef isotropic class on a given , or anything for in the finiteness corollary.
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 manuscripts are authored 'OpenAI' and name no human author. The manuscript (September 23, 2026) proves the main theorem without companion inputs; its projective-case corollary uses the same-day companion 'Projective-space bases of Lagrangian fibrations', which in turn uses this paper's main theorem.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollaries 1.2-1.3 of 'The strong hyperkahler SYZ conjecture' were read against Verbitsky's Conjecture 1.7 as cited. The proof (a degeneration to a flat cylinder in logarithmic coordinates, a special Lagrangian torus by the Zhang-McLean method, hyperkahler rotation to a nonprojective manifold, Greb-Lehn-Rollenske, and Soldatenkov-Verbitsky deformation back to the given pair) was not refereed. No Lean formalization. The theorem has no restriction on dimension, deformation type or second Betti number; the finiteness corollary for deformation types with b2 >= 5 also uses Engel-Filipazzi-Greer-Mauri-Svaldi. The paper does not use the release's contested log abundance results.