Seminario riservato al gruppo di ricerca, prima parte. Subvarieties of Grassmannian (and especially Fano varieties) obtained from section of homogeneous vector bundles are far from being classified. A case of particular interest is given by the Fano varieties of K3 type, for their deep links with hyperk\"ahler geometry. In this talk we will give a survey of some recent techniques we developed to study the Hodge theory of this particular class of varieties, and we will present some new examples.