Beauville's splitting conjecture for two integrable summands

Theorem 1.1: if a compact connected Kahler manifold has $T_X=E_1\oplus E_2$ with both summands integrable holomorphic subbundles of positive rank, its universal cover is biholomorphic to $Y_1\times Y_2$ (connected, simply connected) with $d\Phi(\pi^*E_i)=\mathrm{pr}_i^*T_{Y_i}$ . 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).

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Beauville's splitting conjecture for two integrable summands · Beauville splitting, two summands

Confidence: Not scored

Registry verification: lean checked · announcement · partial

Open the source record ↗

Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: Beauville's splitting conjecture for two integrable summands

specialist registry · passed

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 $n\ge2$ with a Kahler metric and a holomorphic splitting into ranks $r_1,r_2>0$ whose two projections are integrable, and any ordinary universal cover, there are connected simply connected complex manifolds $Y_1,Y_2$ of dimensions $r_1,r_2$ 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.

Source ↗

Universal-cover splitting for compact Kahler manifolds (companion)

paper · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Lean Comparator challenge KahlerSplitting (OpenAI math release)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

Hoering's conjecture: tangent-bundle summands on rationally connected manifolds are integrable

witness · pending

paper link supplied by the VibeMathed registry.

Source ↗

Artifacts and verifiers

Lean Comparator challenge KahlerSplitting (OpenAI math release)

lean artifact · pending

Artifact ↗

Compute record

No linked compute attempts recorded.

Lineage and corrections

This event attributed to Unreleased internal OpenAI model

Lean Comparator challenge KahlerSplitting (OpenAI math release) evidence for this event

Universal-cover splitting for compact Kahler manifolds (companion) evidence for this event

Hoering's conjecture: tangent-bundle summands on rationally connected manifolds are integrable evidence for this event

VibeMathed record: Beauville's splitting conjecture for two integrable summands evidence for this event

Beauville's splitting conjecture for two integrable summands parent of this event

Act on this frontier

Verify, challenge, or extend the result.

Help verify Open canonical problem Explore evidence graph