Subvarieties of hypersurface sections of generalized Grassmannians

Consider a generalized Grassmannian G/P⊂P embedded in a projective space by a complete linear system of a positive generator of the Picard group. For a very general hypersurface H⊂P, we study subvarieties of G/P∩H that are not of general type (or not of positive geometric genus). When the degree of the hypersurface is not small, we show that, under a certain condition on the parabolic subgroup P, such subvarieties are union of lines. Our result is a generalization of Clemens-Ran's result concerning G/P=Pn.