Hartshorne Conjecture

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import FormalConjecturesUtil

/-! # Hartshorne's conjecture on Vector Bundles

*References:*
* [Har1974] R. Hartshorne, [Varieties of small codimension in projective space](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-80/issue-6/Varieties-of-small-codimension-in-projective-space/bams/1183535999.full).
* [MO2010] [Evidences on Hartshorne's conjecture? References?](https://mathoverflow.net/questions/13990/evidences-on-hartshornes-conjecture-references)
-/

open CategoryTheory Limits MvPolynomial AlgebraicGeometry

universe u
variable (S : Scheme.{u})

namespace AlgebraicGeometry.Scheme

/--
A vector bundle over a scheme `S` is a locally free $\mathcal{O}_S$-module of finite rank.
-/
structure VectorBundles where
  carrier : S.Modules
  rank : ℕ
  isLocallyFreeFiniteConstantRank : SheafOfModules.IsVectorBundleWithRank
    (J := Opens.grothendieckTopology S) carrier rank

instance (S : Scheme) : Coe S.VectorBundles S.Modules where
  coe 𝓕 := 𝓕.carrier

/--
Vector bundles form a category.
-/
instance : Category S.VectorBundles :=
  inferInstanceAs <| Category <| InducedCategory _ VectorBundles.carrier

def VectorBundles.toModule : S.VectorBundles ⥤ S.Modules where
  obj 𝓕 := 𝓕.carrier
  map f := f.hom

@[category API, AMS 14]
theorem hasFiniteCoproductsVectorBundles : HasFiniteCoproducts S.VectorBundles := by
  sorry

instance : HasFiniteCoproducts S.VectorBundles :=
  hasFiniteCoproductsVectorBundles S

variable {S} in
/--
A splitting of a vector bundle `𝓕` is a non-trivial direct sum decomposition of `𝓕`
-/
structure VectorBundles.Splitting (𝓕 : S.VectorBundles) (ι : Type) [Fintype ι] [Nonempty ι] where
  components : ι → S.VectorBundles
  iso : 𝓕 ≅ ∐ components
  non_trivial : ∀ i, IsEmpty (components i ≅ 𝓕)

instance {S : Scheme} (𝓕 : S.VectorBundles) (ι : Type) [Fintype ι] [Nonempty ι] :
    CoeOut (𝓕.Splitting ι) (ι → S.VectorBundles) where
  coe s := s.components

-- TODO(lezeau): here we would really need some sanity checks and easier results.

/--
There are no indecomposable vector bundles of rank 2 on $\mathbb{P}^n$ for $n \ge 7$.
This is Conjecture 6.3 in [Har1974].
-/
@[category research open, AMS 14]
theorem harthshorne_conjecture (n : ℕ) (hn : 7 ≤ n)
    (𝓕 : VectorBundles ℙ(Fin (n + 1); Spec (.of ℂ)))
    (h𝓕 : 𝓕.rank = 2) :
    Nonempty (𝓕.Splitting (Fin 2)) := by
  sorry

end AlgebraicGeometry.Scheme

• Formalization source ↗

On-site verification needs an executable Lean target. None is attached yet.

A source attachment or link is not a verified proof. The statement’s correspondence with the original question needs human review.