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Let $ X $ be a smooth hypersurface of degree $ d > 1 $ in $ \mathbb{P}^{n+1} $. What can be said about the stability (Slope/Gieseker) of the cotangent bundle of $ X $?

The closest reference I could find for the first non-trivial case of $ n = 2 $ is some hard-to-parse theorems of Biswas, Chaput, Morougane saying that the restriction of the cotangent bundle of ambient projective space to $ X $ is stable.

Reference: https://hal.science/hal-01138190/document#:~:text=It%20is%20known%20that%20the,restriction%20ΩY%20%7CX%20is%20stable.

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    $\begingroup$ You should look at the papers of Olivier Benoist and Zhiyu Tian. Stability is known, and that is one way to prove separable rational connectedness in the Fano range (even in positive characteristic where the methods of Koll’ar — Miyaoka — Mori do not apply). $\endgroup$
    – Jason Starr
    Commented Jul 20, 2023 at 10:03

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