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Math-2587 · mathematics · Algebraic geometry

Universal-cover products for integrable tangent summands

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Let X be a connected compact Kähler manifold with $T_X=V_{1}\oplus V_{2}$ a decomposition into holomorphic integrable subbundles. Must its universal cover split biholomorphically as $Y_{1}×Y_{2}$ so that the lifted $V_i$ is the tangent bundle along the corresponding factor?

Short English statement

Let X be a connected compact Kähler manifold with $T_X=V_{1}\oplus V_{2}$ a decomposition into holomorphic integrable subbundles. Must its universal cover split biholomorphically as $Y_{1}×Y_{2}$ so that the lifted $V_i$ is the tangent bundle along the corresponding factor?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2587-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2587

Manuscript 1 (family 052, 2026-09-23): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Universal-cover products for integrable tangent summands

OpenAI's claim

If a compact Kähler tangent bundle splits into two integrable positive-rank subbundles, its ordinary universal cover is claimed to split compatibly with those particular distributions. 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 compact connected Kähler of dimension at least two with a specified positive-rank holomorphic splitting $T_X=E_1\oplus E_2$ whose summands are integrable. For its ordinary universal cover $\pi:\widetilde X\to X$, OpenAI claims simply connected connected complex manifolds $Y_1,Y_2$ and a biholomorphism \[ \Phi:\widetilde X\xrightarrow{\sim}Y_1\times Y_2,\qquad d\Phi(\pi^*E_i)=\operatorname{pr}_i^*T_{Y_i}. \] In particular $\dim Y_i=\operatorname{rank}E_i$. Compatibility is with the given embedded distributions, and integrability is an explicit premise.

Outline of the argument

Manuscript 1 (family 052, 2026-09-23): Separate tangency identities remove fibers of the meromorphic graph when a box is extended across a Hartogs figure. Kähler area bounds on annuli then extend adapted boxes across convex boundaries; continuation produces the two global leaf factors with the original summands as their tangent bundles.

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. 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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