Tangent splittings and product decompositions

What OpenAI claims

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)

Papers (2)

1. Universal-cover splitting for compact Kähler manifolds

   23 September 2026 · Preprint, not peer reviewed · PDF

   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.

2. Integrability of split tangent bundles on rationally connected manifolds

   23 September 2026 · Preprint, not peer reviewed · PDF

   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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.