Beauville's splitting conjecture for two integrable summands
Let be a compact Kahler manifold with a holomorphic splitting such that partial sums of the summands are integrable. Beauville (2000, Section 2.3) conjectured that the universal cover of then splits as a product compatibly with the decomposition, and proved it for Kahler-Einstein manifolds and surfaces. Later work covered line-bundle summands with integrable complement (Brunella-Pereira-Touzet), Hermitian-flat summands (Pereira-Touzet, Druel-Pereira-Pym-Touzet) and foliations with a compact leaf of finite holonomy. For two summands the hypothesis is that both are integrable. If with both summands integrable of positive rank, is the universal cover a compatible product ?
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Complex and Kahler geometry; split tangent bundles and foliations
- Posed by
- A. Beauville, Complex manifolds with split tangent bundle, in Complex Analysis and Algebraic Geometry (de Gruyter 2000), Section 2.3
- Year posed
- 2000
- Years open
- 26y
- 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
- 30 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: if a compact connected Kahler manifold has with both summands integrable holomorphic subbundles of positive rank, its universal cover is biholomorphic to (connected, simply connected) with . Corollary: non-uniruled projective manifolds with a two-summand splitting (integrability by Hoering 2007). It does not treat three or more summands, gives no finite-cover product, and assumes integrability (automatic integrability is the companion entry, for rationally connected manifolds).
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 KahlerSplitting, listed in lean/formalization.yaml).
Verification
No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the TeX source, read against Beauville's Section 2.3 as cited. Lean: Comparator challenge KahlerSplitting, declaration OAI.UniversalCoverSplitting.main (listed in lean/formalization.yaml). Its statement was read here: for a compact connected complex manifold of dimension with a Kahler metric and a holomorphic splitting into ranks whose two projections are integrable, and any ordinary universal cover, there are connected simply connected complex manifolds of dimensions forming a compatible product. That is the headline; manifolds, metrics and covers are encoded in the challenge file itself and were read but not audited. Not rebuilt here.