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

Theorem 1.1: on a smooth connected rationally connected projective manifold of dimension at least two, both summands of every holomorphic splitting $T_X=E_1\oplus E_2$ into positive-rank subbundles are integrable. Corollary 1.2 (with Hoering's 2007 product theorem): $X\cong X_1\times X_2$ with $E_i=\mathrm{pr}_i^*T_{X_i}$ . Smoothness is needed (Hoering 2026 gives a singular counterexample); splittings with more than two summands are not treated.

Exact FrontierDelta

Prior state unknown→proved

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

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

specialist registry · passed

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.

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

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Lean Comparator challenge SplitTangentIntegrability (OpenAI math release)

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Universal-cover splitting for compact Kahler manifolds (companion)

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Lean Comparator challenge SplitTangentIntegrability (OpenAI math release)

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

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

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

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