The projective-space base conjecture for Lagrangian fibrations of compact hyperkahler manifolds
Let be a fibration with connected fibres from a compact irreducible holomorphic symplectic Kahler manifold of dimension onto a normal base with . Matsushita showed and the fibres are Lagrangian; Hwang proved when is projective and is smooth, Greb-Lehn extended this to Kahler , and the conclusion was known for the , Kummer, OG6 and OG10 types and in dimension four (Ou, Huybrechts-Xu). The projective-space base conjecture asks: is the normal base of such a fibration always isomorphic to , without assuming smooth?
- 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
- Not named in the manuscript; it states the projective-space base conjecture in the setting of Matsushita (1999), Hwang (2008) and Greb-Lehn (2014)
- Year posed
- —
- Years open
- —
- 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
- 38 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: if is a projective surjective connected-fibre morphism from a compact IHS Kahler manifold of dimension onto a normal projective -dimensional , with every fibre component -dimensional and isotropic for , then , with no smoothness assumption on , no bound on and no restriction on deformation type. With Kim-Oguiso-Shinder it excludes codimension-one multiple fibres. Not shown: the case of a nonprojective (Kahler with a non-projective morphism).
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) uses the same-day companion 'The strong hyperkahler SYZ conjecture' (its Theorem 1.1 and Lemma 4.2) as an input.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 of 'Projective-space bases of Lagrangian fibrations' and its introduction were read against the conjecture as the paper states it. The proof (local cotangent tensors and a Deligne-Mumford stratum, an Euler-characteristic obstruction via reduction to positive characteristic, a second Lagrangian fibration from deformations, then Hwang's theorem) was not refereed. No Lean formalization. It depends on the unreviewed SYZ companion. Scope the paper itself states: projective fibrations (relatively ample bundle) onto a normal projective base with equidimensional Lagrangian fibres, so X is projective; the nonprojective Kahler case of the conjecture is not claimed.