Browse › Algebraic and complex geometry › 052 Tangent splittings and product decompositions
Integrability of split tangent bundles on rationally connected manifolds
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Abstract
We prove that both summands of every specified holomorphic splitting of the tangent bundle of a smooth rationally connected projective complex manifold into two positive-rank subbundles are integrable. This proves Höring's conjecture on rationally connected projective manifolds. Höring's product theorem then gives a product decomposition compatible with the specified splitting.
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Notes
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Tangent splittings and product decompositions (Lean scope notes) (lean/docs/052.md, October 2026). The release's scope notes say the formalized statement is integrability of both positive-rank summands on a rationally connected projective manifold; the compatible product decomposition (Corollary 1.2) is outside the formalized statement. [family-level]
- OpenAI math release, SplitTangentIntegrability.lean (comparator challenge, declaration
OAI.SplitTangent.integrability_main, October 2026). Comparator statement of Theorem 1.1 for a compact connected complex manifold of dimension at least two with a projective embedding witnessing rational connectedness; the proof body in this file issorry, as for a comparator statement.
Possibly relevant literature (found by the agent, not checked)
- Marco Brunella, Jorge Vitório Pereira and Frédéric Touzet, Kähler manifolds with split tangent bundle (Bull. Soc. Math. France 134, 2006). Uses codimension-one foliation methods for split tangent bundles on compact Kähler manifolds, relevant to Theorem 1.1 when one summand is a line bundle; not in this paper's bibliography (the Kähler companion cites it).
- Stéphane Druel, Variétés algébriques dont le fibré tangent est totalement décomposé (arXiv:math/9901138, 1999; J. Reine Angew. Math. 522, 2000). Shows that integrability conditions are automatic for minimal varieties with totally decomposed tangent bundle, a counterpart to the automatic integrability in Theorem 1.1; not in this paper's bibliography (the Kähler companion cites it).
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 052
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Integrability-of-split-tangent-bundles-on-rationally-connected-manifolds-September-23-2026,
author = {{OpenAI}},
title = {{Integrability of split tangent bundles on rationally connected manifolds}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Integrability-of-split-tangent-bundles-on-rationally-connected-manifolds-September-23-2026/main.pdf}{OAI:Integrability-of-split-tangent-bundles-on-rationally-connected-manifolds-September-23-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/052.md. The Lean statements have not been checked by people against the paper's claims.
Family 052: Tangent splittings and product decompositions
A splitting of the tangent bundle of a compact Kähler manifold into two integrable holomorphic subbundles induces a compatible product decomposition of its universal cover, proving the two-summand form of Beauville's splitting conjecture. On smooth rationally connected projective manifolds, both summands are automatically integrable, establishing Höring's conjecture and the corresponding product decomposition.
Other papers in this family:
- Universal-cover splitting for compact Kähler manifolds · September 23, 2026