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Are there any (there should be) references on complete intersections in Grassmanian? Especially on calculating the cohomolgy of some sheaves naturally associated to the complete intersection. For example, if $X$ is the complete intersection, how to compute $\rm{Hom}^i(X,\mathcal{O}_X),\rm{Hom}^i(X,\mathcal{O}_X(n)),\rm{Hom}^i(X,\Omega^n_X)$?

Any suggestions are welcome!!!

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Maybe you should give a couple of details more on what you mean by " sheaves naturally associated to the complete intersection". The cohomology of Grassmannians is generated by schubert classes, so there's a lot that can help you (see Griffiths-Harris, for instance) but we need more info if you want real help. – IMeasy Sep 5 '13 at 9:55
@IMeasy Thank you for pointing out, I have made the edition. – Li Yutong Sep 5 '13 at 14:03

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You can look at the Koszul resolution of $\mathcal O_X$; all the terms will be direct sums of $\mathcal O_G(-n_i)$, where $\mathcal O_G(1)$ is the ample generator of $\mathrm{Pic}(G)$ and $n_i>0$ (by $G$ I denote the Grassmannian in question). The spectral sequence will be very easy to deal with thanks to Borel-Weil-Bott theorem (or Kodaira vanishing theorem).

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Thank you! I will try to look at it in more details. – Li Yutong Sep 5 '13 at 19:39

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