Integrability of split tangent bundles on rationally connected manifolds
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If the holomorphic tangent bundle of a smooth connected rationally connected projective complex manifold splits as TX=E₁⊕E₂ into positive-rank subbundles, must both summands be integrable?
Why it matters and what counts as progress
Why it matters: Automatic integrability would remove a premise from known compatible product-decomposition results.
What counts as progress: A proof under the splitting and rational-connectedness assumptions alone, or a nonintegrable summand on a manifold satisfying them.
How the problem developed
- 16 May 2005 · Partial result [1]
Andreas Höring Höring studied split tangent bundles on uniruled varieties and proved product-decomposition results for rationally connected varieties under the additional hypothesis that one summand is integrable.
- 23 Sept 2026 · Reported claim [2]
OpenAI The manuscript claims that both summands are automatically integrable on a smooth rationally connected projective manifold with a split tangent bundle, and derives the corresponding product decomposition.
AI contribution
Integrability of split tangent bundles on rationally connected manifolds
If a smooth connected rationally connected projective complex manifold has a specified holomorphic splitting TX=E₁⊕E₂ into positive-rank subbundles, both summands are integrable.
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