Result 052, Algebraic and complex geometry

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 formalization Proof

The bigger picture

Why it matters

Splitting the possible directions of motion on a curved complex space need not obviously split the space itself. These manuscripts report conditions under which two such families of directions determine a global product.

What changes?

The first manuscript reports that, for a compact, connected Kähler manifold, a splitting of its tangent bundle into two positive-rank, integrable holomorphic subbundles yields a compatible product decomposition of its ordinary universal cover. The tangent bundle collects tangent directions; holomorphic means respecting the complex structure, and integrability means the directions fit together into leaves. The universal cover is the simply connected covering space. Compatibility says the two factors have exactly the lifted, specified tangent directions, not merely some unrelated splitting.

What does that help mathematicians do?

The second manuscript reports that integrability is automatic for every specified two-summand, positive-rank holomorphic tangent splitting on a smooth rationally connected projective complex manifold. Rational connectedness means that two general points can be joined by a rational curve. Höring's product theorem then supplies the compatible product decomposition. In this setting, researchers need not separately establish that each family of directions forms leaves: the splitting already forces that property and the resulting global structure.

Are there practical applications?

The immediate value is foundational: these claims connect a decomposition of infinitesimal directions to the geometry of an entire covering space. They give researchers a way to study qualifying manifolds through separate factors while tracking a chosen tangent splitting. The general Kähler conclusion concerns the universal cover, not necessarily a product decomposition of the original manifold.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Universal-cover splitting for compact Kähler manifolds

September 23, 2026 33 pages Main result formalized in Lean

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Universal-cover-splitting-for-compact-Kahler-manifolds-September-23-2026,
  author = {{OpenAI}},
  title = {{Universal-cover splitting for compact K"ahler manifolds}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Universal-cover-splitting-for-compact-Kahler-manifolds-September-23-2026/paper.pdf}{OAI:Universal-cover-splitting-for-compact-Kahler-manifolds-September-23-2026}},
  year = {2026}
}

Integrability of split tangent bundles on rationally connected manifolds

September 23, 2026 16 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Integrability-of-split-tangent-bundles-on-rationally-connected-manifolds-September-23-2026,
  author = {{OpenAI}},
  title = {{Integrability of split tangent bundles on rationally connected manifolds}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Integrability-of-split-tangent-bundles-on-rationally-connected-manifolds-September-23-2026/main.pdf}{OAI:Integrability-of-split-tangent-bundles-on-rationally-connected-manifolds-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/052.md.

Tangent splittings and product decompositions

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The splitting question asks whether a holomorphic decomposition of the tangent bundle comes from a product decomposition of the universal cover. The formalized result gives an affirmative answer for a compact connected Kähler manifold whose tangent bundle splits into two positive-rank, integrable holomorphic subbundles. The universal cover is a product of connected simply connected complex manifolds of the prescribed dimensions, and the differential identifies the two tangent factors with the original summands.

Both integrability assumptions are required. Automatic integrability and the paper's additional corollaries are not included.

The formalization proves integrability of both summands in a holomorphic splitting of the tangent bundle of a rationally connected projective manifold. More precisely, for a compact connected complex manifold of dimension at least two with a projective embedding witnessing rational connectedness, each positive-rank summand of a holomorphic tangent-bundle splitting is integrable. The paper's compatible product decomposition is a separate consequence, outside this selected statement.

Comparator links

Result Comparator statement
Universal-cover product splitting KahlerSplitting.lean
Integrability of both holomorphic tangent summands SplitTangentIntegrability.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.