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. (Lean)

Manuscripts (2)

Universal-cover splitting for compact Kähler manifolds

September 23, 2026 · source · Lean formalization available · cite as \cite{OAI:Universal-cover-splitting-for-compact-Kahler-manifolds-September-23-2026}

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.

Integrability of split tangent bundles on rationally connected manifolds

September 23, 2026 · source · cite as \cite{OAI:Integrability-of-split-tangent-bundles-on-rationally-connected-manifolds-September-23-2026}

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.

Result 052 · Algebraic and complex geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.