Challenge.lean

/-
Copyright 2025 The Formal Conjectures Authors.

This file has been modified from the original Formal Conjectures source.
The project-specific import and conjecture metadata were removed, and the
theorem statement was aligned with Solution.lean.

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
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-/

import Mathlib

/-!
# Mathoverflow 1973

Does the 6-sphere $S^6$ admit the structure of a complex manifold?

*Reference:* [mathoverflow/1973](https://mathoverflow.net/questions/1973/)
asked by user [*Fetchinson0234*](https://mathoverflow.net/users/41312/victor-ramos)
-/
open scoped Manifold
namespace Mathoverflow1973

/-- The unit `n`-sphere, defined as `Metric.sphere 0 1` in `EuclideanSpace ℝ (Fin (n + 1))`. -/
abbrev unitSphere (n : ℕ) :=
  Metric.sphere (0 : EuclideanSpace ℝ (Fin (n + 1))) 1

/--
Does the 6-sphere admit a complex structure, i.e. an atlas of holomorphically compatible charts
relating it to `EuclideanSpace ℂ (Fin 3)`?
-/
theorem mathoverflow_1973 :
    ∃ atlas : ChartedSpace (EuclideanSpace ℂ (Fin 3)) (unitSphere 6),
      letI := atlas
      IsManifold 𝓘(ℂ, EuclideanSpace ℂ (Fin 3)) 1 (unitSphere 6) := by
  sorry

end Mathoverflow1973