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I am looking for a modern, maybe shorter or even easier, reference for Theorem II of Homogeneous vector bundles (R. Bott, Annals of mathematics, 1957). This is a theorem where the Dolbeault cohomology of a class of compact complex homogeneous manifolds is described in terms of the Lie algebras of the groups that define the homogeneous manifold.

I am trying to understand the proof of the mentioned theorem, but I am having a hard time understanding many aspects of Bott's approach. Now I am wondering if I can find a modern textbook with this result. Any recommendation will be welcome. Thanks!

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    $\begingroup$ I sent you my few pages of notes on Bott's proof by email. The proof for flag varieties is in Cap and Slovak, Parabolic Geometries I, and lots of other places. $\endgroup$
    – Ben McKay
    Commented Nov 11, 2022 at 13:50
  • $\begingroup$ Dear @BenMcKay, thank you very much for your notes. They will surely help me understand Bott's proof. $\endgroup$
    – Max Reinhold Jahnke
    Commented Nov 11, 2022 at 14:28

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