Donaldson's hypersymplectic deformation conjecture: deforming hypersymplectic four-manifolds to hyperkahler triples in fixed cohomology
A hypersymplectic structure on a closed oriented four-manifold is a triple of closed two-forms whose pointwise span is three-dimensional and positive definite for the wedge product. Donaldson (2006, Section 5.3, Question 3) asked whether a compact oriented four-manifold carrying such a triple admits a hyperkahler structure, and proposed a continuity method keeping the cohomology classes fixed. Fine and Yao formulated the normalized deformation problem as a conjecture: after normalizing , can every hypersymplectic triple be deformed through hypersymplectic triples, keeping each class fixed, to a hyperkahler triple, i.e. one with for a volume form ? Known before this work: convergence of the hypersymplectic flow for simple-type structures on (Huang-Wang-Yao), -invariant triples on (Fine-He-Yao) and triples preserved by a circle action. Does every normalized hypersymplectic triple on a closed four-manifold deform, with its cohomology classes fixed, to a hyperkahler triple?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Differential geometry: hypersymplectic and hyperkahler four-manifolds
- Posed by
- S. K. Donaldson, Two-forms on four-manifolds and elliptic equations (2006), Section 5.3, Question 3; deformation form stated as Conjecture 1.1 by J. Fine and C. Yao (Duke Math. J. 2018)
- Year posed
- 2006
- Years open
- 20y
- 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: for a closed connected oriented smooth four-manifold and a smooth hypersymplectic triple with , there is a smooth path of hypersymplectic triples with , , ending at forms with , parallel and self-dual for a hyperkahler metric. An arbitrary positive triple is reduced to this case by a constant linear change. Corollaries: every nonzero combination is Kahler for some hyperkahler metric, and any closed four-manifold with a positive closed triple is diffeomorphic to the K3 manifold or . Not shown: long-time existence or convergence of the hypersymplectic flow, or any statement in higher dimensions.
What the AI did
The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The manuscript reuses the current-estimate and density-splitting machinery of the companion release paper on Donaldson's tamed-to-compatible conjecture, reproving the parts it needs.
Verification
No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 of the TeX source, read against Donaldson's Question 3 and Fine-Yao Conjecture 1.1; the theorem states the full normalized deformation with all three classes fixed and a hyperkahler endpoint, and the paper says it resolves that conjecture. The proof was not refereed. It does not go through the hypersymplectic flow; convergence or uniqueness of that flow is left open. Key inputs are Bauer's bound for symplectic four-manifolds with torsion first Chern class, Preiss's rectifiability theorem, Riviere-Tian regularity of integral cycles, and the companion release paper Taming implies compatibility on four-manifolds (itself unreviewed), whose needed proofs are reproduced in Sections 3 and 4. No Lean formalization is supplied for this family.