SOME PROPERTIES OF FANO MANIFOLDS THAT ARE ZEROS OF SECTIONS IN HOMOGENEOUS VECTOR-BUNDLES OVER GRASSMANNIANS

Let X be a Fano manifold which is the zero scheme of a general global section s in an irreducible homogeneous vector bundle over a Grassmann ian. We prove that the restriction of the Plucker embedding embeds X p rojectively normal, and that every small deformation of X comes from a deformation of the section s. These results are strengthened in the c ase of Fano 4-folds.