Some properties of Fano manifolds that are zeros of sections in   homogenous vector bundles over Grassmannians
  Let $X$ be a Fano manifold which is the zero scheme of a general global section $s$ in an irreducible homogenous vector bundle over a Grassmannian. We prove that the restriction of the Pl\"ucker embedding embeds $X$ projectively normal, and that every small deformation of $X$ comes from a deformation of the section $s$. These results are strengthened in the case of Fano 4-folds. 
Oliver K\"uchle