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A Smooth Counterexample to the Trautman Conjecture

The Trautman conjecture asserted that a smooth three-dimensional CR manifold admitting a nowhere-zero closed section of its canonical bundle must be locally embeddable in C2\mathbb{C}^2. We modify a standard construction of nonembeddable smooth strongly pseudoconvex CR 3-manifolds so that the condition of having a nowhere-zero closed section of the canonical bundle is preserved, thus providing a strongly pseudoconvex counterexample to the Trautman conjecture.

Result
Disproved(see note)
Status
Resolved
AI contribution
AI-discovered
Method
Construction
Field
CR geometry
Posed by
Andrzej Trautman
Year posed
1998
Years open
28y
Solved
2026-09-02
Model
ChatGPT (model not named; the paper says ChatGPT Plus)
Vendor
OpenAI
Collaborators
Sean N. Curry
Verification
Unreviewed
Publication
Preprint
Significance
25 / 100
Disclosed cost
—
Wikipedia
Not counted (article postdates the solution)

What was actually shown

For every neighborhood UU of the origin in C×R\mathbb{C}\times\mathbb{R} and every ε>0\varepsilon>0, Curry constructs a smooth nonnegative perturbation ϕ\phi, supported in UU with ∥ϕ∥C2<ε\|\phi\|_{C^2}<\varepsilon, such thatTϕ0,1=span⁡C{Lϕ} T^{0,1}_\phi=\operatorname{span}_{\mathbb C}\{L_\phi\} is strongly pseudoconvex, its canonical bundle admits a nowhere-zero closed section, yet every C1C^1 CR function near the origin satisfies dh(0)=0dh(0)=0. Hence the CR structure is not locally embeddable at the origin. The construction can also be globalized to S3S^3 as an arbitrarily small C1C^1 perturbation of the standard spherical CR structure.

What the AI did

Sean Curry states that the counterexample construction itself was discovered through experimentation with ChatGPT Plus on August 18, 2026. ChatGPT was also used to produce an initial rough draft and later for proofreading. Curry independently checked all calculations and arguments and takes responsibility for the paper.

Verification

Unreviewed. A single-author arXiv preprint two days old, labelled by its author "Preliminary version. Comments welcome." The author states he independently checked all calculations and arguments. The construction is a modification of a standard nonembeddable strongly pseudoconvex CR 3-manifold, arranged so that the canonical bundle keeps a nowhere-zero closed section, so it is checkable by anyone who knows the Rosay-type examples; three CR specialists are thanked in the paper but none is on the record as having checked it. No peer review, no formalisation.

Source

Submitted by VibeGene on

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