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!!!



References on complete intersections in Grassmanian
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).


