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Beauville's splitting conjecture for compact Kähler manifolds

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

TX=⨁i∈IFi\mathcal{T}_{\mathsf{X}} = \bigoplus_{i\in I} \mathcal{F}_i

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

T∏i∈IUi≅⨁i∈ITUi.\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.

Equivalent formulations 1

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

  1. Beauville's splitting conjecture for compact Kähler manifolds

    Let XX be a compact Kähler manifold with a splitting

    TX=V1⊕V2,T_X=V_1\oplus V_2,

    where V1V_1 and V2V_2 are integrable subbundles. Let μ:X~→X\mu:\widetilde{X}\rightarrow X be the universal covering of XX. Beauville's conjecture. There is a decomposition

    X~≃X1×X2\widetilde{X}\simeq X_1\times X_2

    and the decomposition TX=V1⊕V2T_X=V_1\oplus V_2 lifts to the canonical decomposition pX1∗TX1⊕pX2∗TX2p_{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).

Sources & referencesView supporting material

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).

complex-geometryfoliationskahler-geometrysplit-tangent-sheaf

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