# References on complete intersections in Grassmanian

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

## 1 Answer

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