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Let k be an arbitrary field. Let $G(2,4)_k$ be the Grassmannian of 2-planes in 4-space over that field. Let $\mathcal{E}$ be the tautological quotient bundle on the Grassmannian. I am trying to calculate $H^∗(G(2,4)_k,\mathcal{E}(d))$.

I think it might be related to Kempf vanishing theorem but I do not see how.

enter link description herehttps://en.wikipedia.org/wiki/Kempf_vanishing_theorem

I have asked this question in Math Stack Exchange but did not get a response.

Another post partially answered this (without twisting tautological bundle).

Cohomology for quotient bundle

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    $\begingroup$ I believe the method suggested by Allen Knutson in the linked question works also for this one. The flag manifold has multiple tautological line bundles. The pushforward of one of them is the tautological vector bundle, and the pullback of $\mathcal O(1)$ is another one. You can then use the projection formula to calculate sheaf cohomology. $\endgroup$
    – Will Sawin
    Commented Jan 19, 2021 at 22:21

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