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 since the statement in wikipedia is about vanishing of cohomology of line bundles and quotient bundle is of rank 2.



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