Hoering's conjecture: tangent-bundle summands on rationally connected manifolds are integrable
Let be a smooth rationally connected projective complex manifold with a holomorphic splitting into positive-rank subbundles. Hoering (2007) proved that if one summand is integrable then compatibly, and conjectured (2008, Conjecture 1.2) that at least one summand is always integrable; Campana-Peternell had settled small ranks on Fano manifolds, and Hoering (2026, Conjecture 1.5) states that both summands are integrable in the smooth rationally connected case, after proving it for Fano type. Is every summand of such a splitting integrable, so that is a compatible product?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Algebraic geometry; rationally connected varieties, foliations
- Posed by
- A. Hoering, The structure of uniruled manifolds with split tangent bundle, Osaka J. Math. 45 (2008), Conjecture 1.2; both-summand form: Hoering, arXiv:2602.15427 (2026), Conjecture 1.5
- Year posed
- 2008
- Years open
- 18y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-checked, statement unaudited
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 22 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: on a smooth connected rationally connected projective manifold of dimension at least two, both summands of every holomorphic splitting into positive-rank subbundles are integrable. Corollary 1.2 (with Hoering's 2007 product theorem): with . Smoothness is needed (Hoering 2026 gives a singular counterexample); splittings with more than two summands are not treated.
What the AI did
Produced by an unreleased internal OpenAI model as part of the openai/math release (pinned commit adc7f12). The release README says results were produced by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute each; this result is not among the README exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored as OpenAI with no human author named. A Lean formalization accompanies it (Comparator challenge SplitTangentIntegrability).
Verification
No independent mathematician has checked this yet. Checked here: abstract, introduction, Theorem 1.1 and Corollary 1.2 of the TeX source, read against Hoering's conjectures as cited. Lean: Comparator challenge SplitTangentIntegrability, declaration OAI.SplitTangent.integrability_main. This challenge is not in lean/formalization.yaml; its JSON config and solution module exist at the pinned commit. Its statement was read here: for a compact connected complex manifold of dimension at least 2 with a projective embedding for which a dense Zariski-open set of pairs of points lie on rational curves, both summands of every holomorphic tangent splitting with positive ranks are integrable. That is the headline integrability claim; the product corollary (via Hoering 2007, Theorem 1.4) is not formalized. Not rebuilt here.