Beauville's splitting conjecture for compact Kähler manifolds

At least 4 years old · documented by 2021

Let $\mathsf{X}$ be a compact Kähler manifold equipped with a holomorphic decomposition

\mathcal{T}_{\mathsf{X}} = \bigoplus_{i\in I} \mathcal{F}_i

of the tangent bundle such that each subbundle $\bigoplus_{j\in J} \mathcal{F}_j$ , for $J \subset I$ , is involutive. Then the universal cover of $\mathsf{X}$ is isomorphic to a product $\prod_{i \in I} \mathsf{U}_i$ in such a way that the given decomposition of $\mathcal{T}_{\mathsf{X}}$ corresponds to the natural decomposition

\mathcal{T}_{\prod_{i \in I}\mathsf{U}_i} \cong \bigoplus_{i \in I} \mathcal{T}_{\mathsf{U}_i}.

This conjecture predicts that compatible involutive splittings of the tangent bundle of a compact Kähler manifold arise from a product decomposition of its universal cover. It is presented as an open conjecture of Beauville (2000); the general statement remains unresolved.

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Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Beauville's splitting conjecture for compact Kähler manifolds
Let $X$ be a compact Kähler manifold with a splitting
$T_X=V_1\oplus V_2,$
where $V_1$ and $V_2$ are integrable subbundles. Let $\mu:\widetilde{X}\rightarrow X$ be the universal covering of $X$ . Beauville's conjecture. There is a decomposition
$\widetilde{X}\simeq X_1\times X_2$
and the decomposition $T_X=V_1\oplus V_2$ lifts to the canonical decomposition $p_{X_1}^*T_{X_1}\oplus p_{X_2}^*T_{X_2}$ . This conjecture predicts that integrable splittings arise from product structures on universal covers; it is known in several settings but remains open in general, with counterexamples showing that integrability cannot be omitted.
source: Andreas Höring, “Fano varieties with split tangent sheaf”, arXiv:2602.15427 (2026).

References

Primary source

Stéphane Druel, Jorge Vitório Pereira, Brent Pym and Frédéric Touzet, “A global Weinstein splitting theorem for holomorphic Poisson manifolds”, arXiv:2102.12641 (2021).

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RemarkAI-assistedClaimed by OpenAI. The manuscript claims compatible product splitting of the ordinary universal cover of a compact connected Kahler manifold whose tangent bundle has two prescribed integrable positive-rank holomorphic summands. This is the two-summand subcase of the target, retaining the given summands rather than asserting only an abstract product.See full solutionHide full solution

Claimed by OpenAI. The manuscript claims compatible product splitting of the ordinary universal cover of a compact connected Kahler manifold whose tangent bundle has two prescribed integrable positive-rank holomorphic summands. This is the two-summand subcase of the target, retaining the given summands rather than asserting only an abstract product.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Universal-cover-splitting-for-compact-Kahler-manifolds-September-23-2026/paper.pdf

OpenAI-052-01-Universal-cover-splitting-for-compact-K-hler-manifolds.pdf479,820 bytesOpen

RemarkAI-assistedClaimed by OpenAI. For smooth rationally connected complex projective manifolds with a prescribed two-summand positive-rank tangent-bundle splitting, the manuscript claims automatic integrability of both summands and a compatible product decomposition of the manifold itself. This gives the corresponding universal-cover splitting in this special case of the target.See full solutionHide full solution

Claimed by OpenAI. For smooth rationally connected complex projective manifolds with a prescribed two-summand positive-rank tangent-bundle splitting, the manuscript claims automatic integrability of both summands and a compatible product decomposition of the manifold itself. This gives the corresponding universal-cover splitting in this special case of the target.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Integrability-of-split-tangent-bundles-on-rationally-connected-manifolds-September-23-2026/main.pdf

OpenAI-052-02-Integrability-of-split-tangent-bundles-on-rationally-connected-manifolds.pdf362,906 bytesOpen