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!!!