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Everything is over the complex numbers. Let $ X = \text{Gr}(k,n) $ be a Grassmannian variety and with tangent sheaf $ T_X $.

(1) Is $ T_X $ simple, i.e. is $\text{Hom} ( T_X, T_X) = \mathbb{C} $?

(2) Is $ T_X $ (slope) stable?

Motivation: If (1) is true then $ T_X $ cannot split as a sum of two vector bundles of lower rank, akin to the famous example of $ T_{\mathbb{P}^2} $ which is proved rather easily using Chern classes. (2) is of course, stronger than (1).

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  • $\begingroup$ (1) is true, and follows for instance from Borel-Bott-Weil theorem. $\endgroup$
    – Sasha
    Commented Feb 4, 2022 at 9:49
  • $\begingroup$ @Sasha could you elaborate please? I thought Borel-Weil-Bott gives the cohomology of certain line bundles. How does (1) follow? $\endgroup$
    – Cranium Clamp
    Commented Feb 4, 2022 at 9:57
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    $\begingroup$ There is an extension of BBW to irreducible equivarant vector bundles on arbitrary $G/P$, see for instance Section 3 in [Kuznetsov, Alexander. Exceptional collections for Grassmannians of isotropic lines. Proc. Lond. Math. Soc. (3) 97 (2008), no. 1, 155--182]. $\endgroup$
    – Sasha
    Commented Feb 4, 2022 at 10:56
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    $\begingroup$ One possible starting point to the literature is arxiv.org/abs/0901.2350. $\endgroup$
    – pbelmans
    Commented Feb 4, 2022 at 11:22
  • $\begingroup$ Ok, if someone posts an answer with a short summary of keywords from which it follows, I'll accept it. $\endgroup$
    – Cranium Clamp
    Commented Feb 4, 2022 at 23:08

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