Browse › Algebraic and complex geometry › 052 Tangent splittings and product decompositions
Universal-cover splitting for compact Kähler manifolds
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Abstract
We prove the two-summand form of Beauville's compatible splitting conjecture. If the tangent bundle of a compact connected Kähler manifold decomposes into two integrable holomorphic subbundles of positive rank, then its ordinary universal cover admits a product decomposition whose factor tangent bundles are the lifted specified summands.
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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 the two-integrable-summand universal-cover splitting; automatic integrability and the paper's further corollaries are not included. [family-level]
- OpenAI math release, KahlerSplitting.lean (comparator challenge, declaration
OAI.UniversalCoverSplitting.main, October 2026). Comparator statement of Theorem 1.1 (compact connected Kähler, two positive-rank integrable summands, compatible product on the universal cover); the proof body in this file issorry, as for a comparator statement.
Possibly relevant literature (found by the agent, not checked)
- Andreas Höring, The structure of uniruled manifolds with split tangent bundle (arXiv:math/0612635, 2006; Osaka J. Math. 45, 2008). Studies integrability of the factors and the universal cover for uniruled manifolds with split tangent bundle, a case covered by Theorem 1.1 when both summands are integrable; not in the bibliography (only Höring 2007 and 2026 are cited).
- Frédéric Campana and Thomas Peternell, Projective manifolds with splitting tangent bundles, I (arXiv:math/0101198, 2001; Math. Z. 241, 2002). Investigates the universal cover of projective threefolds with split tangent bundle, the three-dimensional projective case of the question addressed by Theorem 1.1.
- Fabrizio Catanese and Marco Franciosi, On varieties whose universal cover is a product of curves (arXiv:0812.4317, 2008). Classifies Kähler surfaces whose universal cover is a product of two curves, related to the surface case (Beauville's Theorem C) recalled in the introduction.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 052
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Universal-cover-splitting-for-compact-Kahler-manifolds-September-23-2026,
author = {{OpenAI}},
title = {{Universal-cover splitting for compact K"ahler manifolds}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Universal-cover-splitting-for-compact-Kahler-manifolds-September-23-2026/paper.pdf}{OAI:Universal-cover-splitting-for-compact-Kahler-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:
- Integrability of split tangent bundles on rationally connected manifolds · September 23, 2026