Lean statement

Universal-cover product splitting

Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.

Theorem selected for verification: OAI.UniversalCoverSplitting.main

Lean statement file: KahlerSplitting.leanShow on the mapFamily 052. OpenAI lists this as a main result in its Lean catalogue.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem main (X : ComplexManifold.{u} n) [CompactSpace X.Carrier]    [ConnectedSpace X.Carrier] (hn : 2 ≤ n) (hr₁ : 0 < r₁) (hr₂ : 0 < r₂)    (g : KahlerMetric X) (S : HolomorphicSplitting X r₁ r₂)    (h₁ : Integrable S.projection) (h₂ : Integrable (complementaryProjection S))    (Z : ComplexManifold.{v} n) (π : OrdinaryUniversalCover X Z) :    ∃ (Y₁ : ComplexManifold.{v} r₁) (Y₂ : ComplexManifold.{v} r₂),      ConnectedSpace Y₁.Carrier ∧ ConnectedSpace Y₂.Carrier ∧      SimplyConnectedSpace Y₁.Carrier ∧ SimplyConnectedSpace Y₂.Carrier ∧      CompatibleProduct S π Y₁ Y₂ := by  sorry

A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.

Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

abbreviation OAI.UniversalCoverSplitting.Model line 14
```
abbrev Model (n : ℕ) := Fin n → ℂ
```
abbreviation OAI.UniversalCoverSplitting.model line 15
```
abbrev model (n : ℕ) := 𝓘(ℂ, Model n)
```

structure OAI.UniversalCoverSplitting.ComplexManifold line 17

structure ComplexManifold (n : ℕ) where  Carrier : Type u  [topology : TopologicalSpace Carrier]  [charted : ChartedSpace (Model n) Carrier]  [hausdorff : T2Space Carrier]  [secondCountable : SecondCountableTopology Carrier]  [complexSmooth : IsManifold (model n) ∞ Carrier]

abbreviation OAI.UniversalCoverSplitting.Tangent line 30

abbrev Tangent (X : ComplexManifold.{u} n) (x : X.Carrier) := TangentSpace (model n) x

abbreviation OAI.UniversalCoverSplitting.RealMetricField line 31

abbrev RealMetricField (X : ComplexManifold.{u} n) :=  ∀ x : X.Carrier, Tangent X x →ₗ[ℝ] Tangent X x →ₗ[ℝ] ℝ

definition OAI.UniversalCoverSplitting.metricInChart line 34

def metricInChart {X : ComplexManifold.{u} n} (g : RealMetricField X)    (c : X.Carrier) (z u v : Model n) : ℝ :=  let e := extChartAt (model n) c  let q := mfderiv (model n) (model n) e.symm z  g (e.symm z) (q u) (q v)

definition OAI.UniversalCoverSplitting.fundamentalInChart line 40

def fundamentalInChart {X : ComplexManifold.{u} n} (g : RealMetricField X)    (c : X.Carrier) (z u v : Model n) : ℝ :=  metricInChart g c z (Complex.I • u) v

structure OAI.UniversalCoverSplitting.KahlerMetric line 44

structure KahlerMetric (X : ComplexManifold.{u} n) where

Show the remaining 10 lines

  metric : RealMetricField X  symmetric : ∀ x u v, metric x u v = metric x v u  positive : ∀ x u, u ≠ 0 → 0 < metric x u u  hermitian : ∀ x u v, metric x (Complex.I • u) (Complex.I • v) = metric x u v  smooth : ∀ c u v, ContDiffOn ℝ ∞ (fun z ↦ metricInChart metric c z u v)    (extChartAt (model n) c).target  closed : ∀ c z, z ∈ (extChartAt (model n) c).target → ∀ u v w : Model n,    fderiv ℝ (fun y ↦ fundamentalInChart metric c y v w) z u +    fderiv ℝ (fun y ↦ fundamentalInChart metric c y w u) z v +    fderiv ℝ (fun y ↦ fundamentalInChart metric c y u v) z w = 0

abbreviation OAI.UniversalCoverSplitting.ProjectionField line 56

abbrev ProjectionField (X : ComplexManifold.{u} n) :=  ∀ x : X.Carrier, Tangent X x →L[ℂ] Tangent X x

definition OAI.UniversalCoverSplitting.projectionInChart line 59

def projectionInChart {X : ComplexManifold.{u} n} (P : ProjectionField X)    (c : X.Carrier) (z : Model n) : Model n →L[ℂ] Model n :=  let e := extChartAt (model n) c  (mfderiv (model n) (model n) e (e.symm z)).comp    ((P (e.symm z)).comp (mfderiv (model n) (model n) e.symm z))

definition OAI.UniversalCoverSplitting.HolomorphicProjection line 65

def HolomorphicProjection {X : ComplexManifold.{u} n} (P : ProjectionField X) : Prop :=  ∀ c, DifferentiableOn ℂ (projectionInChart P c) (extChartAt (model n) c).target

definition OAI.UniversalCoverSplitting.Integrable line 68

def Integrable {X : ComplexManifold.{u} n} (P : ProjectionField X) : Prop :=  ∀ c U, IsOpen U → U ⊆ (extChartAt (model n) c).target →    ∀ V W : Model n → Model n,      DifferentiableOn ℂ V U → DifferentiableOn ℂ W U →      (∀ z ∈ U, projectionInChart P c z (V z) = V z) →      (∀ z ∈ U, projectionInChart P c z (W z) = W z) →      ∀ z ∈ U, projectionInChart P c z (VectorField.lieBracket ℂ V W z) =        VectorField.lieBracket ℂ V W z

structure OAI.UniversalCoverSplitting.HolomorphicSplitting line 77

structure HolomorphicSplitting (X : ComplexManifold.{u} n) (r₁ r₂ : ℕ) where  projection : ProjectionField X  idempotent : ∀ x v, projection x (projection x v) = projection x v  holomorphic : HolomorphicProjection projection  first_rank : ∀ x, Module.finrank ℂ (LinearMap.range (projection x).toLinearMap) = r₁  second_rank : ∀ x, Module.finrank ℂ (LinearMap.ker (projection x).toLinearMap) = r₂

definition OAI.UniversalCoverSplitting.complementaryProjection line 84

def complementaryProjection {X : ComplexManifold.{u} n}    (S : HolomorphicSplitting X r₁ r₂) : ProjectionField X :=  fun x ↦ ContinuousLinearMap.id ℂ (Tangent X x) - S.projection x

definition OAI.UniversalCoverSplitting.LocalBiholomorphism line 88

def LocalBiholomorphism {X : ComplexManifold.{u} n} {Z : ComplexManifold.{v} n}    (π : Z.Carrier → X.Carrier) : Prop :=  ∀ z, ∃ e : OpenPartialHomeomorph Z.Carrier X.Carrier,    z ∈ e.source ∧ (∀ w ∈ e.source, e w = π w) ∧      MDifferentiableOn (model n) (model n) e e.source ∧      MDifferentiableOn (model n) (model n) e.symm e.target

structure OAI.UniversalCoverSplitting.OrdinaryUniversalCover line 95

structure OrdinaryUniversalCover (X : ComplexManifold.{u} n)    (Z : ComplexManifold.{v} n) where  projection : Z.Carrier → X.Carrier  covering : IsCoveringMap projection  [connected : ConnectedSpace Z.Carrier]  [simplyConnected : SimplyConnectedSpace Z.Carrier]  liftedComplexStructure : LocalBiholomorphism projection

definition OAI.UniversalCoverSplitting.CompatibleProduct line 103

def CompatibleProduct {X : ComplexManifold.{u} n} {Z : ComplexManifold.{v} n}    (S : HolomorphicSplitting X r₁ r₂) (π : OrdinaryUniversalCover X Z)    (Y₁ : ComplexManifold.{v} r₁) (Y₂ : ComplexManifold.{v} r₂) : Prop :=

Show the remaining 13 lines

  ∃ Φ : Z.Carrier ≃ₜ (Y₁.Carrier × Y₂.Carrier),    MDifferentiable (model n) ((model r₁).prod (model r₂)) Φ ∧    MDifferentiable ((model r₁).prod (model r₂)) (model n) Φ.symm ∧    (∀ z, Submodule.map      (mfderiv (model n) ((model r₁).prod (model r₂)) Φ z).toLinearMap      (Submodule.comap (mfderiv (model n) (model n) π.projection z).toLinearMap        (LinearMap.range (S.projection (π.projection z)).toLinearMap)) =      LinearMap.ker (ContinuousLinearMap.snd ℂ (Model r₁) (Model r₂)).toLinearMap) ∧    (∀ z, Submodule.map      (mfderiv (model n) ((model r₁).prod (model r₂)) Φ z).toLinearMap      (Submodule.comap (mfderiv (model n) (model n) π.projection z).toLinearMap        (LinearMap.ker (S.projection (π.projection z)).toLinearMap)) =      LinearMap.ker (ContinuousLinearMap.fst ℂ (Model r₁) (Model r₂)).toLinearMap)

Possible paper match, with no recorded review

Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.

Words of the Lean statement, its names split into words, that also appear in the theorem: pi, compact, biholomorphism, simply, universal, covering, ordinary, ranks and seven more. Found in the paper’s title, which its main theorem states: splitting. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: identifies, dimension and subbundles.

Statements in Universal-cover splitting for compact Kähler manifolds come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of Universal-cover splitting for compact Kähler manifolds

Theorem

Let X be a compact connected Kähler manifold of complex dimension n\geq2, and let

T_X=E_1\oplus E_2

be a specified decomposition into integrable holomorphic subbundles of positive ranks r_1,r_2. If \pi:\widetilde X\to X is the ordinary universal covering map, there are connected simply connected complex manifolds Y_1,Y_2 and a biholomorphism \Phi:\widetilde X\to Y_1\times Y_2 such that

\dd\Phi(\pi^*E_i)=\pr_i^*T_{Y_i}\qquad(i=1,2),

where \dd\pi identifies T_{\widetilde X} with \pi^*T_X. In particular, \dim_\C Y_i=r_i.

Section 1, Introduction Read it in the paper TeX source, line 100

Show the TeX

Let $X$ be a compact connected K\"ahler manifold of complex dimension
$n\geq2$, and let
\[
T_X=E_1\oplus E_2
\]
be a specified decomposition into integrable holomorphic subbundles of
positive ranks $r_1,r_2$. If $\pi:\widetilde X\to X$ is the ordinary
universal covering map, there are connected simply connected complex
manifolds $Y_1,Y_2$ and a biholomorphism
$\Phi:\widetilde X\to Y_1\times Y_2$ such that
\[
\dd\Phi(\pi^*E_i)=\pr_i^*T_{Y_i}\qquad(i=1,2),
\]
where $\dd\pi$ identifies $T_{\widetilde X}$ with $\pi^*T_X$.
In particular, $\dim_\C Y_i=r_i$.

Other possible paper matches, closest wording first

Corollary 1.2 of Universal-cover splitting for compact Kähler manifolds. Words of the Lean statement, its names split into words, that also appear in the theorem: universal, smooth, compatible, cover, ordinary, ranks and eight more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: summands and subbundles.
Corollary 1.3 of Universal-cover splitting for compact Kähler manifolds. Words of the Lean statement, its names split into words, that also appear in the theorem: universal, smooth, compatible, rank, cover, ordinary and eight more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: dimension, bundle and subbundles.
Proposition A.1 of Universal-cover splitting for compact Kähler manifolds. Words of the Lean statement, its names split into words, that also appear in the theorem: metric, real, continuous, splitting and product. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: resulting and hler.
Theorem C.1 of Universal-cover splitting for compact Kähler manifolds. Words of the Lean statement, its names split into words, that also appear in the theorem: charts, simply, ordinary, product, connected, map and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: subbundles.
Proposition D.3 of Universal-cover splitting for compact Kähler manifolds. Words of the Lean statement, its names split into words, that also appear in the theorem: lifts, universal, cover and tangent. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: factors and bundles.
Proposition 7.3 of Universal-cover splitting for compact Kähler manifolds. Words of the Lean statement, its names split into words, that also appear in the theorem: local, integrable, product and holomorphic. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: subbundles.

How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.

Reported Comparator runs

three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.

For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.

Carlos Toledo’s cert-machine

How they ran it, in their words

Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners (.github/workflows/openai-math-kernel.yml), one j…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory; the run was still in progress at this snapshot (192 accepted, 224 pending).

Accepted Their term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 216 s; peak memory 6.5 GiB).
Reported 11 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.

This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.

Allowed axioms: propext, Quot.sound, Classical.choice

Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.

Do family 052’s Lean statements express the advertised result?

These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.

MathVet and cert-machine disagree on how much of the advertised result the Lean statements express.

MathVet
Finds the whole advertised result stated or implied Their term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: Both headline theorems are stated; the 'corresponding product decomposition' for rationally connected manifolds (composition of the two, needing a Kahler metric from the embedding) is not a single statement. Complex manifold, Kahler metric, integrability and rational connectedness are all bespoke encodings.
Definitions the group asks readers to compare with the paper: Bespoke ComplexManifold/KahlerMetric (closedness of the fundamental form via a cyclic derivative sum in charts), Integrable (Frobenius condition through chartwise Lie brackets of holomorphic local sections), OrdinaryUniversalCover, CompatibleProduct, ProjectiveEmbedding (injective holomorphic immersion into P^N) and RationallyConnected (dense Zariski-open set of pairs joined by a RationalCurve); plausible but unchecked against Mathlib/standard notions
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — DECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note says so. Their record
Their note: The product decomposition for rationally connected projective manifolds (the last clause) is not stated; neither are the Kahler paper's further corollaries.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“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.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

139 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.

Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.

Links in the file: 110 to its own declarations, 62 to Mathlib’s documentation. 18 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves main exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches all 17 other declarations of the file; with the theorem itself they span 96 lines. 7 of them are named in the statement itself.

Names in the namespace OAI.UniversalCoverSplitting are listed without it.

abbrev Modelline 14library
abbrev modelline 15library
structure ComplexManifoldline 17in the statementlibrary
abbrev Tangentline 30library
abbrev RealMetricFieldline 31library
def metricInChartline 34library
def fundamentalInChartline 40library
structure KahlerMetricline 44in the statementlibrary
abbrev ProjectionFieldline 56library
def projectionInChartline 59library
def HolomorphicProjectionline 65library
def Integrableline 68in the statementlibrary
structure HolomorphicSplittingline 77in the statementlibrary
def complementaryProjectionline 84in the statementlibrary
def LocalBiholomorphismline 88library
structure OrdinaryUniversalCoverline 95in the statementlibrary
def CompatibleProductline 103in the statementlibrary
theorem mainline 124selected theoremlibrary

Other Lean rules that may affect the statement 1

Lean can use instances, notation and attributes without naming them in the statement. These are rules for interpreting or finding mathematical objects. Atlas’s scan of names cannot follow all such uses; read the listed source as well.

attribute [instance] ComplexManifold.topology ComplexManifold.charted line 25

import Mathlibnamespace OAInoncomputable sectionopen scoped Manifold ContDiffopen Setnamespace UniversalCoverSplittinguniverse u vabbrev Model (n : ℕ) := Fin n → ℂabbrev model (n : ℕ) := 𝓘(ℂ, Model n)structure ComplexManifold (n : ℕ) where  Carrier : Type u  [topology : TopologicalSpace Carrier]  [charted : ChartedSpace (Model n) Carrier]  [hausdorff : T2Space Carrier]  [secondCountable : SecondCountableTopology Carrier]  [complexSmooth : IsManifold (model n) ∞ Carrier]attribute [instance] ComplexManifold.topology ComplexManifold.charted  ComplexManifold.hausdorff ComplexManifold.secondCountable ComplexManifold.complexSmoothvariable {n r₁ r₂ : ℕ}abbrev Tangent (X : ComplexManifold.{u} n) (x : X.Carrier) := TangentSpace (model n) xabbrev RealMetricField (X : ComplexManifold.{u} n) :=  ∀ x : X.Carrier, Tangent X x →ₗ[ℝ] Tangent X x →ₗ[ℝ] ℝdef metricInChart {X : ComplexManifold.{u} n} (g : RealMetricField X)    (c : X.Carrier) (z u v : Model n) : ℝ :=  let e := extChartAt (model n) c  let q := mfderiv (model n) (model n) e.symm z  g (e.symm z) (q u) (q v)def fundamentalInChart {X : ComplexManifold.{u} n} (g : RealMetricField X)    (c : X.Carrier) (z u v : Model n) : ℝ :=  metricInChart g c z (Complex.I • u) vstructure KahlerMetric (X : ComplexManifold.{u} n) where  metric : RealMetricField X  symmetric : ∀ x u v, metric x u v = metric x v u  positive : ∀ x u, u ≠ 0 → 0 < metric x u u  hermitian : ∀ x u v, metric x (Complex.I • u) (Complex.I • v) = metric x u v  smooth : ∀ c u v, ContDiffOn ℝ ∞ (fun z ↦ metricInChart metric c z u v)    (extChartAt (model n) c).target  closed : ∀ c z, z ∈ (extChartAt (model n) c).target → ∀ u v w : Model n,    fderiv ℝ (fun y ↦ fundamentalInChart metric c y v w) z u +    fderiv ℝ (fun y ↦ fundamentalInChart metric c y w u) z v +    fderiv ℝ (fun y ↦ fundamentalInChart metric c y u v) z w = 0abbrev ProjectionField (X : ComplexManifold.{u} n) :=  ∀ x : X.Carrier, Tangent X x →L[ℂ] Tangent X xdef projectionInChart {X : ComplexManifold.{u} n} (P : ProjectionField X)    (c : X.Carrier) (z : Model n) : Model n →L[ℂ] Model n :=  let e := extChartAt (model n) c  (mfderiv (model n) (model n) e (e.symm z)).comp    ((P (e.symm z)).comp (mfderiv (model n) (model n) e.symm z))def HolomorphicProjection {X : ComplexManifold.{u} n} (P : ProjectionField X) : Prop :=  ∀ c, DifferentiableOn ℂ (projectionInChart P c) (extChartAt (model n) c).targetdef Integrable {X : ComplexManifold.{u} n} (P : ProjectionField X) : Prop :=  ∀ c U, IsOpen U → U ⊆ (extChartAt (model n) c).target →    ∀ V W : Model n → Model n,      DifferentiableOn ℂ V U → DifferentiableOn ℂ W U →      (∀ z ∈ U, projectionInChart P c z (V z) = V z) →      (∀ z ∈ U, projectionInChart P c z (W z) = W z) →      ∀ z ∈ U, projectionInChart P c z (VectorField.lieBracket ℂ V W z) =        VectorField.lieBracket ℂ V W zstructure HolomorphicSplitting (X : ComplexManifold.{u} n) (r₁ r₂ : ℕ) where  projection : ProjectionField X  idempotent : ∀ x v, projection x (projection x v) = projection x v  holomorphic : HolomorphicProjection projection  first_rank : ∀ x, Module.finrank ℂ (LinearMap.range (projection x).toLinearMap) = r₁  second_rank : ∀ x, Module.finrank ℂ (LinearMap.ker (projection x).toLinearMap) = r₂def complementaryProjection {X : ComplexManifold.{u} n}    (S : HolomorphicSplitting X r₁ r₂) : ProjectionField X :=  fun x ↦ ContinuousLinearMap.id ℂ (Tangent X x) - S.projection xdef LocalBiholomorphism {X : ComplexManifold.{u} n} {Z : ComplexManifold.{v} n}    (π : Z.Carrier → X.Carrier) : Prop :=  ∀ z, ∃ e : OpenPartialHomeomorph Z.Carrier X.Carrier,    z ∈ e.source ∧ (∀ w ∈ e.source, e w = π w) ∧      MDifferentiableOn (model n) (model n) e e.source ∧      MDifferentiableOn (model n) (model n) e.symm e.targetstructure OrdinaryUniversalCover (X : ComplexManifold.{u} n)    (Z : ComplexManifold.{v} n) where  projection : Z.Carrier → X.Carrier  covering : IsCoveringMap projection  [connected : ConnectedSpace Z.Carrier]  [simplyConnected : SimplyConnectedSpace Z.Carrier]  liftedComplexStructure : LocalBiholomorphism projectiondef CompatibleProduct {X : ComplexManifold.{u} n} {Z : ComplexManifold.{v} n}    (S : HolomorphicSplitting X r₁ r₂) (π : OrdinaryUniversalCover X Z)    (Y₁ : ComplexManifold.{v} r₁) (Y₂ : ComplexManifold.{v} r₂) : Prop :=  ∃ Φ : Z.Carrier ≃ₜ (Y₁.Carrier × Y₂.Carrier),    MDifferentiable (model n) ((model r₁).prod (model r₂)) Φ ∧    MDifferentiable ((model r₁).prod (model r₂)) (model n) Φ.symm ∧    (∀ z, Submodule.map      (mfderiv (model n) ((model r₁).prod (model r₂)) Φ z).toLinearMap      (Submodule.comap (mfderiv (model n) (model n) π.projection z).toLinearMap        (LinearMap.range (S.projection (π.projection z)).toLinearMap)) =      LinearMap.ker (ContinuousLinearMap.snd ℂ (Model r₁) (Model r₂)).toLinearMap) ∧    (∀ z, Submodule.map      (mfderiv (model n) ((model r₁).prod (model r₂)) Φ z).toLinearMap      (Submodule.comap (mfderiv (model n) (model n) π.projection z).toLinearMap        (LinearMap.ker (S.projection (π.projection z)).toLinearMap)) =      LinearMap.ker (ContinuousLinearMap.fst ℂ (Model r₁) (Model r₂)).toLinearMap)open Setopen scoped Manifold ContDiffvariable {n r₁ r₂ : ℕ}theorem main (X : ComplexManifold.{u} n) [CompactSpace X.Carrier]    [ConnectedSpace X.Carrier] (hn : 2 ≤ n) (hr₁ : 0 < r₁) (hr₂ : 0 < r₂)    (g : KahlerMetric X) (S : HolomorphicSplitting X r₁ r₂)    (h₁ : Integrable S.projection) (h₂ : Integrable (complementaryProjection S))    (Z : ComplexManifold.{v} n) (π : OrdinaryUniversalCover X Z) :    ∃ (Y₁ : ComplexManifold.{v} r₁) (Y₂ : ComplexManifold.{v} r₂),      ConnectedSpace Y₁.Carrier ∧ ConnectedSpace Y₂.Carrier ∧      SimplyConnectedSpace Y₁.Carrier ∧ SimplyConnectedSpace Y₂.Carrier ∧      CompatibleProduct S π Y₁ Y₂ := by  sorryend UniversalCoverSplittingendend OAI

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

Universal-cover splitting for compact Kähler manifolds

Theorem

Let X be a compact connected Kähler manifold of complex dimension n\geq2, and let

T_X=E_1\oplus E_2

\dd\Phi(\pi^*E_i)=\pr_i^*T_{Y_i}\qquad(i=1,2),

where \dd\pi identifies T_{\widetilde X} with \pi^*T_X. In particular, \dim_\C Y_i=r_i.

Section 1, Introduction Read it in the paper TeX source, line 100

Show the TeX

Let $X$ be a compact connected K\"ahler manifold of complex dimension
$n\geq2$, and let
\[
T_X=E_1\oplus E_2
\]
be a specified decomposition into integrable holomorphic subbundles of
positive ranks $r_1,r_2$. If $\pi:\widetilde X\to X$ is the ordinary
universal covering map, there are connected simply connected complex
manifolds $Y_1,Y_2$ and a biholomorphism
$\Phi:\widetilde X\to Y_1\times Y_2$ such that
\[
\dd\Phi(\pi^*E_i)=\pr_i^*T_{Y_i}\qquad(i=1,2),
\]
where $\dd\pi$ identifies $T_{\widetilde X}$ with $\pi^*T_X$.
In particular, $\dim_\C Y_i=r_i$.

Corollary (Non-uniruled projective manifolds)

Let X be a smooth connected non-uniruled projective complex manifold, and let T_X=E_1\oplus E_2 be a specified holomorphic splitting into subbundles of positive ranks. Then both summands are integrable and the ordinary universal cover has a compatible product decomposition as in Theorem thm:main.

Section 1, Introduction Read it in the paper TeX source, line 126

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Let $X$ be a smooth connected non-uniruled projective complex manifold,
and let $T_X=E_1\oplus E_2$ be a specified holomorphic splitting into
subbundles of positive ranks. Then both summands are integrable and the
ordinary universal cover has a compatible product decomposition as in
Theorem~\ref{thm:main}.

Corollary

Let X be a smooth connected projective complex manifold of dimension n\geq2 with nef and big canonical bundle K_X. Every specified holomorphic splitting T_X=E_1\oplus E_2 into positive-rank subbundles is integrable and induces a compatible product decomposition of the ordinary universal cover as in Theorem thm:main.

Section 1, Introduction Read it in the paper TeX source, line 156

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Let $X$ be a smooth connected projective complex manifold of dimension
$n\geq2$ with nef and big canonical bundle $K_X$. Every specified
holomorphic splitting $T_X=E_1\oplus E_2$ into positive-rank subbundles
is integrable and induces a compatible product decomposition of the
ordinary universal cover as in Theorem~\ref{thm:main}.

Integrability of split tangent bundles on rationally connected manifolds

Theorem (Automatic integrability)

Let X be a smooth connected projective complex manifold of dimension at least two. Suppose that two general points of X lie on the image of a morphism \PP^1\to X. For every specified holomorphic decomposition

T_X=E_1\oplus E_2

into subbundles of positive rank, both E_1 and E_2 are integrable.

Section 1, Introduction Read it in the paper TeX source, line 21

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\label[theorem]{thm:main}
Let \(X\) be a smooth connected projective complex manifold of dimension
at least two. Suppose that two general points of \(X\) lie on the image
of a morphism \(\PP^1\to X\). For every specified holomorphic decomposition
\[
T_X=E_1\oplus E_2
\]
into subbundles of positive rank, both \(E_1\) and \(E_2\) are integrable.

Corollary (Compatible product)

Under the hypotheses of thm:main, there are smooth projective manifolds X_1,X_2 and an isomorphism X\simeq X_1\times X_2 under which

E_i=\pr_i^*T_{X_i}\qquad (i=1,2).

Section 1, Introduction Read it in the paper TeX source, line 37

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\label[corollary]{cor:product}
Under the hypotheses of \cref{thm:main}, there are smooth projective
manifolds \(X_1,X_2\) and an isomorphism \(X\simeq X_1\times X_2\)
under which
\[
E_i=\pr_i^*T_{X_i}\qquad (i=1,2).
\]

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