On the ampleness of the cotangent bundles of complete intersections

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Based on a geometric interpretation of Brotbek's symmetric differential forms, for the intersection family X of generalized Fermat-type hypersurfaces in Pₖⁿ defined over any field K, we reconstruct explicit symmetric differential forms by applying Cramer's rule, skipping cohomology arguments, and we further exhibit unveiled families of lower degree symmetric differential forms on all possible intersections of X with coordinate hyperplanes. Thereafter, we develop what we call the `moving coefficients method' to prove a conjecture made by Olivier Debarre: for generic c N/2 hypersurfaces H₁,,Hc⊂ PCⁿ of degrees d₁,,dc sufficiently large, the intersection X:=H₁ ∩ ∩ Hc has ample cotangent bundle Ωₓ, and concerning effectiveness, the lower bound d₁,,dc Nⁿ² works. Lastly, thanks to known results about the Fujita Conjecture, we establish the very-ampleness of SymκΩₓ for all κ 64 (∑ᵢ₌₁ᶜ dᵢ)².