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Math-2578 · mathematics · Algebraic geometry

Split tangents on rationally connected projective manifolds

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For a smooth connected rationally connected complex projective X with holomorphic $T_X=V_{1}\oplus V_{2}$, must both summands be integrable and X be biholomorphic to a product whose two factor tangents correspond to the given summands?

Short English statement

For a smooth connected rationally connected complex projective X with holomorphic $T_X=V_{1}\oplus V_{2}$, must both summands be integrable and X be biholomorphic to a product whose two factor tangents correspond to the given summands?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2578-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2578

Automatic integrability proves the first requested conclusion for the specified splitting. The same manuscript's compatible-product corollary then gives the two smooth projective factors and identifies their tangent directions with the original embedded summands. Together these results cover both assertions on this page.

Split tangents on rationally connected projective manifolds

OpenAI's claim

Both summands of a specified two-part tangent splitting on a rationally connected projective complex manifold are claimed to be integrable, yielding a compatible product decomposition. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $X$ be smooth connected projective complex of dimension at least two and rationally connected, meaning a morphism $\mathbb P^1\to X$ can contain two general points in its image. Specify any holomorphic decomposition into subbundles of positive rank \[ T_X=E_1\oplus E_2, \] OpenAI claims both summands are integrable: $[s,t]$ is a local section of $E_i$ whenever $s,t$ are local sections of $E_i$. The embeddings into the tangent bundle are fixed data, so this concerns their actual distributions.

The manuscript's product corollary also gives smooth projective factors $X_1,X_2$ and an isomorphism $X\simeq X_1\times X_2$ taking each specified tangent summand to the tangent direction of its corresponding factor. It derives this global conclusion by applying the cited earlier product theorem after establishing integrability.

Outline of the argument

Manuscript 1 (family 052, 2026-09-23): A formal diagonal solution separates derivatives through the two holomorphic projectors. Positive normal-bundle section counts force the formal projective ratios to be rational, and tangency contracts the exceptional curves of a resolution, giving a global holomorphic map whose factor directions certify integrability.

Formal verification

For family 052, the scope page documents the following selections. One selected statement splits the universal cover assuming integrability of both supplied tangent summands. Another establishes that integrability for rationally connected projective manifolds; its compatible-product consequence is not itself that selected statement. The documented statement covers the main claim of manuscript 1. It does not establish formal coverage of the complete problem clause. Comparator files: KahlerSplitting.lean, SplitTangentIntegrability.lean. Declarations: OAI.UniversalCoverSplitting.main, OAI.SplitTangent.integrability_main.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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